Constrained multi-objective optimization problems: Methodologies, algorithms and applications

doi:10.1016/j.knosys.2024.111998

Knowledge-Based Systems 知识型系统

Volume 299, 5 September 2024, 111998
第 299 卷，2024 年 9 月 5 日，111998

, , , ,

^a: Faculty of Mathematics and Computer Science, Fernuniversität in Hagen, Hagen, 58097, Germany
德国哈根，58097，哈根 Fernuniversität 数学与计算机科学学院

^b: School of Data Science, Qingdao University of Science and Technology, No. 99, Songling Road, Qingdao, 266061, China
青岛市松岭路99号青岛科技大学数据科学学院邮编：266061

^c: School of Mathematics and Statistics, Minnan Normal University, China
中国闽南师范大学数学与统计学院

^d: School of Systems Science, Beijing Jiaotong University, No. 3, Shangyuan Village, Beijing, 100028, China
北京交通大学系统科学学院，中国北京上元村 3 号，邮编 100028

Received 3 April 2024, Revised 9 May 2024, Accepted 22 May 2024, Available online 29 May 2024, Version of Record 1 June 2024.
2024 年 4 月 3 日收到，2024 年 5 月 9 日修订，2024 年 5 月 22 日接受，2024 年 5 月 29 日在线提供，2024 年 6 月 1 日备案。

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Abstract 摘要

Constrained multi-objective optimization problems (CMOPs) are widespread in practical applications such as engineering design, resource allocation, and scheduling optimization. It is high challenging for CMOPs to balance the convergence and diversity due to conflicting objectives and complex constraints. Researchers have developed a variety of constrained multi-objective optimization algorithms (CMOAs) to find a set of optimal solutions, including evolutionary algorithms and machine learning-based methods. These algorithms exhibit distinct advantages in solving different categories of CMOPs. Recently, constrained multi-objective evolutionary algorithms (CMOEAs) have emerged as a popular approach, with several literature reviews available. However, there is a lack of comprehensive-view survey on the methods of CMOAs, limiting researchers to track the cutting-edge investigations in this research direction. Therefore, this paper reviews the latest algorithms for handling CMOPs. A new classification method is proposed to divide literature, containing classical mathematical methods, evolutionary algorithms and machine learning methods. Subsequently, it reviews the modeling and algorithms of CMOPs in the context of practical applications. Lastly, the paper gives potential research directions with respect to CMOPs. This paper is able to provide guidance and inspiration for scholars studying CMOPs.
约束多目标优化问题（CMOPs）在工程设计、资源分配和调度优化等实际应用中非常普遍。由于目标冲突和约束条件复杂，CMOPs 要在收敛性和多样性之间取得平衡极具挑战性。研究人员开发了多种约束多目标优化算法（CMOAs），包括进化算法和基于机器学习的方法，以找到一组最优解。这些算法在求解不同类别的 CMOP 时表现出明显的优势。最近，约束多目标进化算法（CMOEAs）成为一种流行的方法，已有多篇文献对此进行了综述。然而，目前还缺乏对 CMOEA 方法的全面综述，这限制了研究人员追踪该研究方向的前沿研究。因此，本文综述了处理 CMOP 的最新算法。本文提出了一种新的分类方法来划分文献，包括经典数学方法、进化算法和机器学习方法。随后，本文结合实际应用回顾了 CMOP 的建模和算法。最后，本文给出了 CMOPs 的潜在研究方向。本文能够为研究 CMOPs 的学者提供指导和启发。

Keywords 关键词

Constrained multi-objective optimization problems

Evolutionary algorithms

Machine learning

Applications

受约束多目标优化问题进化算法机器学习应用

1. Introduction 1.导言

Constrained multi-objective optimization problems are widely present in scientific research and practical engineering, including optimal wireless sensor location design [1], wind farm layout optimization [2], and portfolio optimization [3]. Without loss of generality, CMOPs are described as follows [4], (1)

minimize \vec{F} (\vec{x}) = {[f_{1} (\vec{x}), f_{2} (\vec{x}), \dots, f_{k} (\vec{x})]}^{T}

\begin{aligned} subject to & g_{i} (\vec{x}) \leq 0, for i = 1, 2, \dots, M \\ h_{j} (\vec{x}) = 0, for j = 1, 2, \dots, L \\ \vec{x} = {[x_{1}, x_{2}, \dots, x_{D}]}^{T} \in S, \end{aligned}

where

S

is the decision space and

\vec{x}

is the D-dimension decision vector,

\vec{x} \in S

\vec{F} (\vec{x})

is the objective vector that contains

k

objectives,

f_{i} (\vec{x})

is the

i

-th objective function,

g_{i} (\vec{x})

and

h_{j} (\vec{x})

are the inequality and equality constraint functions,

M

and

L

are the numbers of inequality constraint functions and equality constraint functions.
约束多目标优化问题广泛存在于科学研究和实际工程中，包括无线传感器位置优化设计[1]、风电场布局优化[2]和投资组合优化[3]等。在不失一般性的前提下，CMOPs 的描述如下[4]， (1)

minimize \vec{F} (\vec{x}) = {[f_{1} (\vec{x}), f_{2} (\vec{x}), \dots, f_{k} (\vec{x})]}^{T}

\begin{aligned} subject to & g_{i} (\vec{x}) \leq 0, for i = 1, 2, \dots, M \\ h_{j} (\vec{x}) = 0, for j = 1, 2, \dots, L \\ \vec{x} = {[x_{1}, x_{2}, \dots, x_{D}]}^{T} \in S, \end{aligned}

其中

S

为决策空间，

\vec{x}

为 D 维决策向量，

\vec{x} \in S

\vec{F} (\vec{x})

为目标向量，包含

k

个目标，

f_{i} (\vec{x})

为

i

-th 目标函数，

g_{i} (\vec{x})

和

h_{j} (\vec{x})

为不等式约束函数和等式约束函数，

M

和

L

为不等式约束函数和等式约束函数的个数。

The total constraint violation degree of the decision variables quantifies the extent to which the constraints are violated by the solution. This degree can be denoted as [5], [6], [7], (2)

C V (\vec{x}) = \sum_{i = 1}^{M} max {0, g_{i} (\vec{x})} + \sum_{j = 1}^{L} max {0, | h_{j} (\vec{x}) | - σ},

where

σ

is a positive tolerance value to relax the equality constraints. If

C V (\vec{x})

= 0, then

\vec{x}

is a feasible solution (FS); otherwise,

\vec{x}

is an infeasible solution (IFS). All the feasible solutions form the feasible region.
决策变量的总约束违反度量化了解决方案违反约束的程度。这个程度可以表示为 [5]、[6]、[7]， (2)

C V (\vec{x}) = \sum_{i = 1}^{M} max {0, g_{i} (\vec{x})} + \sum_{j = 1}^{L} max {0, | h_{j} (\vec{x}) | - σ},

其中

σ

是放宽相等约束的正公差值。如果

C V (\vec{x})

= 0，则

\vec{x}

为可行解（FS）；否则，

\vec{x}

为不可行解（IFS）。所有可行解构成可行区域。

For any two solutions

{\vec{x}}_{1}

and

{\vec{x}}_{2}

, if

f_{i} ({\vec{x}}_{1}) \leq f_{i} ({\vec{x}}_{2})

for each

i \in {1, \dots, k}

and

f_{j} ({\vec{x}}_{1}) < f_{j} ({\vec{x}}_{2})

for at least one

j \in {1, \dots, k}

, then

{\vec{x}}_{1}

dominates

{\vec{x}}_{2}

; otherwise,

{\vec{x}}_{1}

and

{\vec{x}}_{2}

are non-dominated solutions. A set of non-dominated feasible solutions is referred to as the Pareto optimal set (POS). Each solution within the POS is a Pareto optimal solution. The projection of POS in the objective space forms the Pareto front (PF). The PF of a constrained multi-objective optimization problem is typically denoted as CPF, while that of an unconstrained problem is denoted as UPF. Existing CMOPs can be broadly categorized into four types based on the relationship between CPF and UPF [8]:
对于任意两个解

{\vec{x}}_{1}

和

{\vec{x}}_{2}

，如果

f_{i} ({\vec{x}}_{1}) \leq f_{i} ({\vec{x}}_{2})

对每个

i \in {1, \dots, k}

和

f_{j} ({\vec{x}}_{1}) < f_{j} ({\vec{x}}_{2})

至少有一个

j \in {1, \dots, k}

，那么

{\vec{x}}_{1}

支配

{\vec{x}}_{2}

；否则，

{\vec{x}}_{1}

和

{\vec{x}}_{2}

是非支配解。非支配可行解的集合称为帕累托最优集（POS）。POS 中的每个解都是帕累托最优解。POS 在目标空间的投影形成帕累托前沿（PF）。有约束多目标优化问题的 PF 通常称为 CPF，而无约束问题的 PF 则称为 UPF。根据 CPF 和 UPF 的关系，现有的 CMOP 大致可分为四种类型 [8]：

•
Type I. CPF and UPF overlap completely, and UPF is fully feasible, as shown in Fig. 1(a);
如图 1（a）所示，CPF 和 UPF 完全重叠，UPF 完全可行；
•
Type II. CPF is a subset of UPF, and UPF is partially feasible, as shown in Fig. 1(b);
第二类。CPF 是 UPF 的子集，UPF 部分可行，如图 1（b）所示；
•
Type III. CPF partially overlaps with UPF, and UPF is partially feasible, as shown in Fig. 1(c);
类型 III。CPF 与 UPF 部分重叠，UPF 部分可行，如图 1（c）所示；
•
Type IV. CPF is completely separate from UPF, and UPF is completely unfeasible, as shown in Fig. 1(d).
第四类。如图 1（d）所示，CPF 完全独立于 UPF，而 UPF 则完全不可行。

The task of solving CMOPs is to identify a set of Pareto non-dominated solutions that exhibit convergence and uniform distribution. However, the difficulty of achieving this objective is significant [9]:
求解 CMOP 的任务是找出一组表现出收敛性和均匀分布的帕累托非支配解。然而，实现这一目标的难度很大 [9]：

1.
Feasibility: The constraints may render a large portion of the search space infeasible, thereby complicating the task of locating a small feasible region, as shown in Fig. 2(a).
可行性：如图 2(a)所示，约束条件可能会导致搜索空间的很大一部分不可行，从而使定位一个小的可行区域的任务变得复杂。
2.
Diversity: The discontinuity of the feasible region results in the true CPF exhibiting a fractured characteristic. This fracture in the CPF complicates the algorithm’s ability to cover its entire range, making it a challenge to obtain a complete CPF, as shown in Fig. 2(b).
多样性：可行区域的不连续性导致真正的 CPF 呈现断裂特征。如图 2(b) 所示，CPF 中的这种断裂使算法覆盖整个范围的能力变得复杂，因此要获得完整的 CPF 是一项挑战。
3.
Convergence: The solution is trapped in the external feasible region, making it difficult to towards the true CPF across the infeasible region, as shown in Fig. 2(c).
收敛：如图 2(c) 所示，解被困在外部可行区域，因此很难在不可行区域找到真正的 CPF。

Facing the aforementioned challenges, it is crucial to achieve a balance between convergence, diversity, and feasibility when solving CMOPs. In the past three decades, methods based on classical mathematical optimization were first introduced to constrained multi-objective optimization [11], [12]. However, constrained multi-objective evolutionary algorithms have gradually replaced them due to their greater efficiency in dealing with nonlinear or non-convex problems [13].
面对上述挑战，在求解 CMOP 时实现收敛性、多样性和可行性之间的平衡至关重要。在过去的三十年中，基于经典数学优化的方法首次被引入约束多目标优化[11]、[12]。然而，由于约束多目标进化算法在处理非线性或非凸问题时效率更高，因此逐渐取代了这些算法[13]。

CMOEAs are heuristic-based methods that combine constraint handling techniques (CHTs) and multi-objective evolutionary algorithms. CHTs are used to deal with constraints, while multi-objective evolutionary algorithms are the main driving algorithm used to approximate the PF. However, it is difficult for methods based on evolution to better extract the characteristics of the problem, and thus cannot adaptively adjust strategies or CHTs to flexibly solve complex and changing CMOPs via the characteristics of the problem. Methods based on machine learning have become a new trend expected to overcome the drawbacks of evolution-based methods with the emergence of the new wave of machine learning.
CMOEAs 是一种基于启发式的方法，它结合了约束处理技术（CHTs）和多目标进化算法。约束处理技术用于处理约束，而多目标进化算法则是用于近似 PF 的主要驱动算法。然而，基于进化的方法很难更好地提取问题的特征，因此无法根据问题的特征自适应地调整策略或 CHT，从而灵活地解决复杂多变的 CMOP。随着新一轮机器学习浪潮的兴起，基于机器学习的方法有望克服基于进化论方法的弊端，成为一种新趋势。

The current review on CMOPs focuses on CMOEAs, and the methods involved are not comprehensive [14]; therefore, this paper reviews mathematical methods, evolutionary algorithms, and machine learning perspectives to form a new systematic review. The contribution of this paper can be summarized as follows:
目前关于 CMOP 的综述主要集中于 CMOEA，所涉及的方法并不全面[14]；因此，本文从数学方法、进化算法和机器学习的角度进行综述，形成一篇新的系统综述。本文的贡献可概括如下：

•
This paper more scientifically and comprehensively categorizes advanced CMOAs into three categories: (1) methods based on classical mathematics [15], [16]; (2) methods based on evolution [17], [18]; (3) methods based on machine learning [19], [20]; as shown in Fig. 3. Each category is introduced and analyzed in detail, and the advantages and disadvantages of various CMOAs are also summarized. Some promising future research directions will be discussed.
本文更科学、全面地将先进的CMOA分为三类：（1）基于经典数学的方法[15]，[16]；（2）基于进化论的方法[17]，[18]；（3）基于机器学习的方法[19]，[20]；如图3所示。图 3 详细介绍和分析了每一类方法，并总结了各种 CMOA 的优缺点。此外，还将讨论一些有前景的未来研究方向。
•
This paper introduce the application of constrained multi-objective optimization in some typical fields more comprehensively than previous studies. These cases provide guidance to relevant practitioners in selecting suitable optimization algorithms.
与以往的研究相比，本文更全面地介绍了约束多目标优化在一些典型领域中的应用。这些案例为相关从业人员选择合适的优化算法提供了指导。
•
While the current CMOAs have demonstrated strong performance in solving many conventional CMOPs, there are still challenges arising from the increasing complexity of CMOPs. These challenges include how to better extract features from CMOPs, how to autonomously recommend CMOAs, how to handle dynamic interval CMOPs, and how to handle large-scale CMOPs.
虽然目前的 CMOAs 在解决许多传统 CMOP 时表现出了很强的性能，但随着 CMOP 的复杂性不断增加，仍然存在一些挑战。这些挑战包括如何更好地从 CMOP 中提取特征、如何自主推荐 CMOA、如何处理动态区间 CMOP 以及如何处理大规模 CMOP。

The rest of this paper is organized as follows: Section 2 classifies the existing CMOAs and discusses the advantages and disadvantages of each type of algorithm, Section 3 presents some real-world applications of CMOAs, Section 4 discusses the future trends and challenges for constrained multi-objective optimization.
本文接下来的内容安排如下：第 2 节对现有的 CMOA 进行了分类，并讨论了每种算法的优缺点；第 3 节介绍了 CMOA 在现实世界中的一些应用；第 4 节讨论了约束多目标优化的未来趋势和挑战。

2. State-of-the-art CMOAs
2.最先进的 CMOA

This section provides a comprehensive review of the current research landscape of CMOAs, delineating their respective strengths and weaknesses. Based on their fundamental design principles and solution strategies, CMOAs are categorized into three distinct categories. Collectively, these categories encompass the current state-of-the-art CMOAs.
本节全面回顾了 CMOA 目前的研究状况，划分了它们各自的优缺点。根据其基本设计原则和解决策略，CMOA 被分为三个不同的类别。总体而言，这些类别涵盖了当前最先进的 CMOA。

2.1. Methods based on classical mathematics
2.1.基于经典数学的方法

These methods are typically based on classical mathematical optimization theories, including linear programming [21],

ɛ

-constraint methods [22], and fuzzy optimization theory [23]. These algorithms apply classical mathematical optimization theories in innovative ways to tackle complex real-world problems. By combining different mathematical theories, these algorithms can provide more robust and efficient solutions.
这些方法通常基于经典数学优化理论，包括线性规划[21]、

ɛ

约束方法[22]和模糊优化理论[23]。这些算法以创新的方式应用经典数学优化理论来解决复杂的实际问题。通过结合不同的数学理论，这些算法可以提供更稳健、更高效的解决方案。

Mavrotas [15] introduced an augmented

ɛ

-constraint method (AUGMECON), which precludes the generation of weak Pareto optimal solutions and expedites the process. This method eliminates superfluous iterations and efficaciously applies the

ɛ

-constraint method to yield Pareto optimal solutions in multi-objective mathematical programming (MOMP). Liu et al. [24] combined the selection-based constraint handling with fuzzy membership functions, and constructed a new constraint handling method–multi-objective fuzzy selection, which is used to select two candidate schemes from a given group, and the selection rules are as follows: (1) Given two feasible solutions, select the one with better objective function value; (2) Given two infeasible solutions, select the one with smaller violation of constraint conditions; (3) If one solution is feasible and the other is not, select the FS. Saha et al. [25] proposed a novel fuzzy rule-based penalty function approach for single-objective nonlinear constrained optimization problems, a fuzzy inference system based on Mamdani-type IF-THEN rules, which does not require complex explicit mathematical formulas, and designed a parameter-free robust penalty function for constraint handling. Li et al. [26] proposed a fuzzy search bias (FSB) guided CHT, where the search bias is guided by a membership function, which is used to satisfy fuzzy knowledge, that is, solutions that violate larger constraints tend to search for solutions that violate smaller constraints, and solutions that violate smaller constraints tend to search for solutions with better objective function values.
Mavrotas [15] 提出了一种增强的

ɛ

-约束方法（AUGMECON），它能排除弱帕累托最优解的产生，并加快了过程。该方法消除了多余的迭代，并有效地应用

ɛ

- 约束方法在多目标数学规划（MOMP）中生成帕累托最优解。Liu等人[24]将基于选择的约束处理与模糊成员函数相结合，构建了一种新的约束处理方法--多目标模糊选择法，用于从给定组中选择两个候选方案，其选择规则如下：(1）给定两个可行方案，选择目标函数值较好的方案；（2）给定两个不可行方案，选择违反约束条件较小的方案；（3）如果一个方案可行，另一个方案不可行，则选择 FS。Saha 等人[25] 针对单目标非线性约束优化问题提出了一种新颖的基于模糊规则的惩罚函数方法，即一种基于 Mamdani 型 IF-THEN 规则的模糊推理系统，它不需要复杂的显式数学公式，并设计了一种用于约束处理的无参数鲁棒惩罚函数。Li 等人[26]提出了一种模糊搜索偏置（FSB）引导的 CHT，搜索偏置由成员函数引导，成员函数用于满足模糊知识，即违反较大约束条件的解倾向于搜索违反较小约束条件的解，违反较小约束条件的解倾向于搜索目标函数值更好的解。

Moreover, Yang et al. [16] devised an algorithm for fuzzy multi-objective linear fractional programming (FMOLFP) issues, employing the superiority and inferiority measure method (SIMM). This method handles fuzzy numbers within constraints to maximize the degree of membership and introduces linear objective programming for problems with fuzzy fractional objectives. Cocchi and Lapucci [27] proposed an enhancement to the classical generalized Lagrangian method, targeting equality and inequality constrained multi-objective optimization problems to improve computational efficiency. Ansary and Panda [28] developed a globally convergent sequential quadratic programming (SQP) approach for multi-objective optimization with inequality constraints, securing a feasible descent direction through linear approximation of all objectives and constraints, and addressing constraint violations with a non-differentiable penalty function. Yang et al. [29] transformed a nonlinear optimization model into a mixed-integer linear programming problem via linearization, applied the augmented generalized

ɛ

-constraint method for solution, and employed fuzzy clustering to select the final solution from the Pareto optimal set based on decision-maker preferences. Han et al. [30] proposed a new fuzzy constraint handling technique based on [24], and introduced a new concept called “fuzzy dominance”, which accurately describes and quantifies the comprehensive difference between solutions from the perspective of objectives and constraints, and embedded the proposed fuzzy constraint handling technique into a decomposition-based multi-objective evolutionary algorithm as an implementation.
此外，Yang 等人[16]采用优劣度量法（SIMM）设计了一种模糊多目标线性分数编程（FMOLFP）问题的算法。该方法在约束条件内处理模糊数，以最大化成员度，并为模糊分数目标问题引入了线性目标编程。Cocchi 和 Lapucci [27] 针对平等和不平等约束的多目标优化问题，提出了经典广义拉格朗日法的增强方法，以提高计算效率。Ansary 和 Panda [28] 针对不等式约束的多目标优化问题开发了一种全局收敛的连续二次编程（SQP）方法，通过对所有目标和约束进行线性逼近来确保可行的下降方向，并用无差别惩罚函数来解决违反约束的问题。Yang 等人[29] 通过线性化将非线性优化模型转化为混合整数线性规划问题，采用增强广义

ɛ

约束法求解，并根据决策者的偏好采用模糊聚类从帕累托最优集中选择最终解。Han等人[30]在[24]的基础上提出了一种新的模糊约束处理技术，并引入了 "模糊优势 "这一新概念，从目标和约束的角度准确地描述和量化了解之间的综合差异，并将所提出的模糊约束处理技术嵌入到基于分解的多目标进化算法中作为实现。

Besides, Lai et al. [31] established multi-objective approximate gradient projection (MAGP) and linear multi-objective approximate gradient projection (LMAGP) sequential optimality conditions without constraint qualifications for multi-objective constrained optimization problems. Pérez-Cañedo et al. [32] extended the concepts of dominance and Pareto optimality to the intuitionistic fuzzy domain, using a dictionary ordering criterion for triangular intuitionistic fuzzy numbers (TIFNs) and an intuitionistic fuzzy

ɛ

-constraint approach for multi-objective linear programming with TIFNs. Sharma et al. [33] proposed a method for a multi-objective two-layer chance-constrained optimization problem in an intuitionistic fuzzy environment, representing technological coefficients and objective function coefficients with Triangular Intuitionistic fuzzy numbers and resource coefficients with normal random variables, utilizing component-wise optimization to convert intuitionistic fuzzy objectives into their deterministic equivalents. Alouane and Boulif [34] proposed a fuzzy inference engine-based method that can switch between different forms of transformation functions (TFs) for selecting the best constraint ordering. TFs are a type of constraint-handling approach that operates in the phenotypic space. The TF adaptively adjusts its definition by incorporating the constraint order according to the priority of the constraints. The fuzzy inference engine then selects the most suitable form of the TF based on the problem characteristics and the constraint complexity, thus improving the solution efficiency and quality.
此外，Lai 等人[31] 为多目标约束优化问题建立了无约束条件的多目标近似梯度投影（MAGP）和线性多目标近似梯度投影（LMAGP）顺序最优条件。Pérez-Cañedo 等人[32]将支配性和帕累托最优性的概念扩展到直觉模糊领域，使用三角直觉模糊数（TIFN）的字典排序准则和直觉模糊

ɛ

- 约束方法来处理具有 TIFN 的多目标线性规划问题。Sharma 等人[33] 针对直觉模糊环境下的多目标双层机会约束优化问题提出了一种方法，用三角直觉模糊数表示技术系数和目标函数系数，用正态随机变量表示资源系数，利用分量优化将直觉模糊目标转换为其确定性等价物。Alouane 和 Boulif [34] 提出了一种基于模糊推理引擎的方法，可在不同形式的变换函数（TFs）之间切换，以选择最佳约束排序。变换函数是一种在表型空间中运行的约束处理方法。TF 可根据约束条件的优先级自适应地调整其定义，将约束顺序纳入其中。然后，模糊推理引擎会根据问题特征和约束复杂度选择最合适的 TF 形式，从而提高解决方案的效率和质量。

In general, methods based on classical mathematics rely on established optimization theories and require exact mathematical models and deterministic algorithms to solve multi-objective optimization problems. Although precise and appropriate for problems with certain properties like differentiability and convexity, they may struggle with complex, nonlinear, or non-convex challenges.
一般来说，基于经典数学的方法依赖于既定的优化理论，需要精确的数学模型和确定性算法来解决多目标优化问题。尽管这些方法精确且适用于具有某些特性（如可微性和凸性）的问题，但它们可能难以应对复杂、非线性或非凸性的挑战。

2.2. Methods based on evolution
2.2.基于进化的方法

These methods explore the solution space to identify feasible solutions by mimicking natural evolutionary processes. They are divided into three categories.
这些方法模仿自然进化过程，探索解决方案空间，以确定可行的解决方案。它们分为三类。

2.2.1. Methods based on multi-population coevolution
2.2.1.基于多种群协同进化的方法

These methods typically rely on two or three populations, each assigned a different task. The populations communicate with each other to achieve a better balance of convergence, feasibility, and diversity. One population acts as the main population and performs a constrained search to find feasible solutions and the CPF, while the other populations act as auxiliaries and perform an unconstrained search to find the UPF. Additionally, auxiliary populations can aid the main population in escaping from infeasible regions or local optima by transferring knowledge. This method can be classified as weak or strong coevolution, depending on the level of coevolution among the populations.
这些方法通常依靠两到三个种群，每个种群分配不同的任务。种群之间相互交流，以更好地平衡收敛性、可行性和多样性。其中一个种群作为主种群，执行约束搜索以找到可行的解决方案和 CPF，而其他种群作为辅助种群，执行无约束搜索以找到 UPF。此外，辅助种群还可以通过传授知识帮助主种群摆脱不可行区域或局部最优状态。根据种群间共同进化的程度，这种方法可分为弱共同进化和强共同进化。

(1)Strong coevolution: We use a two-population example, in which the two populations are typically merged before the mating selection stage and undergo their own mating selection processes. The populations are then combined with their own offspring and those of other populations, sharing all the offspring generated by all populations, as illustrated in Fig. 4(a). This method enhances the cooperation ability of the population, utilizes the information between the populations fully, but may also result in population homogenization and premature convergence.
(1)强共同进化：我们以两个种群为例，两个种群通常在交配选择阶段之前合并，并经历各自的交配选择过程。如图 4（a）所示，然后种群与自己的后代和其他种群的后代合并，共享所有种群产生的所有后代。这种方法提高了种群的合作能力，充分利用了种群间的信息，但也可能导致种群同质化和过早趋同。

Ming et al. [35] introduced a dual-population approach to handle infeasible solutions; the main population applies an adaptive penalty, while the secondary employs constrained non-dominated sorting genetic algorithm II(C-NSGA-II) to modify the objective vector of such solutions. During mating selection, both populations combine and utilize binary tournament selection for choosing parents. This algorithm efficiently leverages IFS data, yet overlooks Pareto optimal solutions at the constraint edge. Liu et al. [36] proposed a bidirectional coevolution (BiCo) algorithm to explore the feasible region boundary from both sides, helping to find Pareto optimal solutions at the boundary. Restrictive mating selection based on CV and angle density promotes strong coevolution. Jiao et al. [37] proposed a multiform optimization (MFO) algorithm that addresses the complexity of CMOPs by focusing on Pareto solutions at constraint boundaries and search space partitioning. The algorithm divides the problem into dual tasks: the main and auxiliary populations perform constraint and

ɛ

-constraint searches, with the

ɛ

boundary decreasing over iterations. Based on [37], Qiao et al. [38] proposed a double-balanced evolutionary multi-task optimization (DBEMTO) algorithm, which uses an adaptive mechanism to achieve the balance between intra-task and inter-task, enabling the population to utilize suitable knowledge, and also uses an effective environmental selection method to exploit the information of parents and offspring, enhancing the effect of strong coevolution. Li et al. [39] proposed a two-archive based constrained multi-objective competitive swarm optimizer (TACSO). The main population aims at feasible solutions, while the auxiliary population aims at convergence. The adaptive

δ

influences mate selection from the convergence archive. Using strong coevolution, it exploits elite insights to avoid local optima.
Ming 等人[35]引入了一种双种群方法来处理不可行解；主种群采用自适应惩罚，而副种群则采用约束非支配排序遗传算法 II（C-NSGA-II）来修改此类解的目标向量。在交配选择过程中，两个种群结合起来，利用二元锦标赛选择来选择亲本。该算法有效利用了 IFS 数据，但忽略了约束边缘的帕累托最优解。Liu 等人[36]提出了一种双向协同进化（BiCo）算法，从两侧探索可行区域边界，帮助找到边界处的帕累托最优解。基于 CV 和角度密度的限制性交配选择促进了强协同进化。Jiao 等人[37]提出了一种多形式优化（MFO）算法，通过关注约束边界的帕累托最优解和搜索空间划分来解决 CMOP 的复杂性问题。该算法将问题分为双重任务：主种群和辅助种群分别执行约束和

ɛ

- 约束搜索，其中

ɛ

边界随迭代递减。Qiao等[38]在[37]的基础上提出了双平衡进化多任务优化算法（DBEMTO），该算法利用自适应机制实现任务内和任务间的平衡，使种群能够利用合适的知识，同时利用有效的环境选择方法利用亲代和子代的信息，增强了强协同进化的效果。Li 等人[39]提出了一种基于双档案的约束多目标竞争性蜂群优化器（TACSO）。主种群以可行解为目标，辅助种群以收敛为目标。自适应

δ

会从收敛档案中影响配偶选择。通过强协同进化，它可以利用精英的洞察力来避免局部最优。

(2)Weak coevolution: Each population evolves independently, and does not share population information during the mating selection stage. After generating offspring, the populations are combined with their own offspring and the offspring of other populations, and share all the offspring produced by the populations, as shown in Fig. 4(b). This method maintains population diversity, avoids premature convergence, reduces computational overhead, but it may also underutilize the information between populations, leading to traps in local optima.
(2) 弱共同进化：每个种群独立进化，在交配选择阶段不共享种群信息。产生后代后，种群与自己的后代和其他种群的后代结合，共享种群产生的所有后代，如图 4（b）所示。这种方法保持了种群的多样性，避免了过早收敛，减少了计算开销，但也可能没有充分利用种群之间的信息，导致陷入局部最优。

Tian et al. [17] first proposed a coevolutionary constrained multi-objective optimization (CCMO) framework, which is flexible and lightweight due to the weak cooperation between populations, and does not require new selection strategies and parameters, thus reducing the computational complexity. Yang et al. [40] proposed a cooperative evolutionary algorithm for dealing with steady-state CMOPs, which differs from the above algorithms [17], [35], [36], [37], [38], [39], [41] in that it finally outputs the auxiliary population. Wang et al. [42] proposed a cooperative multi-objective evolutionary algorithm with propulsive population (CMOEA-PP), where the main population searches for feasible PF and maintains diversity. The auxiliary population initially overlooks constraints, considers them, and searches only for corner and central solutions to speed up convergence in the later stage. Qiao et al. [43] proposed an evolutionary multitasking (EMT)-based constrained multi-objective optimization (EMCMO) algorithm, which determines which populations are useful knowledge based on the four relationships of CPF and UPF in Fig. 1.
Tian等[17]首次提出了一种协同进化约束多目标优化（CCMO）框架，该框架由于种群间的弱合作而灵活轻便，不需要新的选择策略和参数，从而降低了计算复杂度。Yang等人[40]提出了一种处理稳态CMOP的合作进化算法，它与上述算法[17]、[35]、[36]、[37]、[38]、[39]、[41]的不同之处在于它最终输出辅助种群。Wang 等人[42]提出了一种带推进种群的合作多目标进化算法（CMOEA-PP），其中主种群搜索可行的 PF 并保持多样性。辅助种群最初忽略约束，考虑约束，只搜索角解和中心解，以加快后期的收敛速度。Qiao 等人[43]提出了一种基于进化多任务（EMT）的约束多目标优化（EMCMO）算法，该算法根据图 1 中 CPF 和 UPF 的四种关系确定哪些种群是有用的知识。

Moreover, Ming et al. [44] proposed a tri-population (TriP) based coevolutionary framework, in which the first and second populations target the original problem and the unconstrained problem respectively, and use weak coevolution to handle simple CMOPs; the third population targets the constraint relaxed problem and uses constraint relaxation techniques to evolve. The cooperation of the three populations preserves the advantages of both weak coevolution and constraint relaxation. Li et al. [45] proposed a CMOEA with the two-archive weak cooperation (CMOEA-TWC), which is based on CCMO [17], uses the density-based SDE to enhance the diversity of the auxiliary population, and uses the strict constraint domination principle to improve the feasibility of the main population. Kong et al. [46] proposed a dynamic dual-population coevolution multi-objective evolutionary algorithm (DDCMEA), rooted in CCMO [17]. This method dynamically modulates offspring counts for both populations to balance convergence with feasibility. To boost offspring diversity, genetic and differential evolution algorithms serve as search mechanisms for the auxiliary and main populations, respectively. Cao et al. [47] proposed a coevolutionary constrained multi-objective algorithm with learning constraint boundary, which is based on CCMO [17] and transforms the CMOP into an unconstrained one by taking the constraint violation as an additional objective. The auxiliary population selects solutions according to the learning constraint boundary (LCB). Song et al. [48] proposed a Self-adaptive epsilon constrained multi-objective optimization (SaE-CMO), which is based on CCMO [17], both populations adopt the newly proposed self-adaptive

ɛ

method, and use different comparison criteria to select the next population from the mating pool.
此外，Ming 等人[44] 提出了一种基于三种群（TriP）的协同演化框架，其中第一和第二种群分别以原始问题和无约束问题为目标，使用弱协同演化来处理简单的 CMOP；第三种群以约束松弛问题为目标，使用约束松弛技术进行演化。三个种群的合作保留了弱协同进化和约束松弛的优点。Li 等人[45]在 CCMO[17] 的基础上提出了一种双存档弱合作的 CMOEA（CMOEA-TWC），利用基于密度的 SDE 增强辅助种群的多样性，利用严格约束支配原理提高主种群的可行性。Kong 等人[46]在 CCMO[17] 的基础上提出了一种动态双种群协同进化多目标进化算法（DDCMEA）。该方法动态调节两个种群的后代数量，以平衡收敛性和可行性。为了提高后代的多样性，遗传算法和差分进化算法分别作为辅助种群和主要种群的搜索机制。Cao 等人[47]提出了一种具有学习约束边界的协同进化约束多目标算法，该算法基于 CCMO [17]，通过将违反约束作为附加目标，将 CMOP 转化为无约束算法。辅助种群根据学习约束边界（LCB）选择解。Song 等人 [48]在 CCMO [17] 的基础上提出了自适应ε约束多目标优化（SaE-CMO），两个种群都采用了新提出的自适应

ɛ

方法，并使用不同的比较标准从交配池中选择下一个种群。

2.2.2. Methods based on multi-stage
2.2.2.基于多阶段的方法

These methods operate through the sequential application of different strategies or mechanisms at each stage, aiming to progressively refine the solution set towards the PF. Typically, the early stages utilize more information from the objective function, which helps to balance the feasible and infeasible regions and avoid falling into local optima. During the later stages, the focus shifts to information from the constraint conditions, thereby improving the quality of feasible solutions.
这些方法通过在每个阶段依次应用不同的策略或机制来运行，目的是逐步完善解集，以达到目标函数。通常情况下，早期阶段会利用更多来自目标函数的信息，这有助于平衡可行和不可行区域，避免陷入局部最优。在后期阶段，重点将转移到来自约束条件的信息上，从而提高可行解决方案的质量。

Fan et al. [18] first proposed a push and pull search (PPS) framework, which divides the search process into two distinct stages: push and pull search stages. The push stage is the early stage, which can estimate the constraints in CMOPs and thus set the parameters for the constraint handling method in the pull stage; the pull stage is the later stage, which uses an improved

ɛ

-constraint handling method to pull the population into the CPF. Jiao et al. [49] proposed a feasible proportion control technique, which can control the proportion of feasible solutions in the population. Different from [18], in the early stage, the feasible ratio control technique is used to handle the constraints, and in the later stage, the commonly used differential evolution algorithm (DE) is used to speed up the convergence. Fan et al. [50] based on [18], combined the multi-objective to multi-objective (M2M) decomposition method with the push and pull search framework (PPS-M2M). Yu et al. [51] proposed a purpose-directed two-phase multi-objective differential evolution (PDTP-MDE) algorithm. Tian et al. [52] proposed a two-phase evolutionary algorithm that can switch between two phases according to the current population state. Yu et al. [53] proposed a dynamic selection preference-assisted constrained multi-objective differential evolutionary algorithm. Ming et al. [54] proposed a two-stage framework to improve the efficiency and generality of CMOPs. Xu et al. [55] proposed a CMOEA (C-TPEA) with two-stage constraint handling. In the early stage, the grid constraint decomposition method framework can well maintain the convergence and diversity; in the later stage, the

ɛ

-constraint strategy is introduced to further improve the feasibility and quality of the solution. Ma et al. [56] proposed a strategy to rank the constraint handling priorities based on the impact on the unconstrained PF, and added the constraint conditions one by one according to the constraint handling priority, and processed them at different stages of evolution.
Fan 等人[18]首次提出了推拉搜索（PPS）框架，将搜索过程分为两个不同的阶段：推和拉搜索阶段。推阶段是早期阶段，可以估计 CMOP 中的约束，从而为拉阶段的约束处理方法设置参数；拉阶段是后期阶段，使用改进的

ɛ

- 约束处理方法将种群拉入 CPF。Jiao 等人[49]提出了一种可行比例控制技术，可以控制可行解在群体中的比例。与[18]不同的是，前期采用可行比例控制技术处理约束，后期采用常用的微分进化算法（DE）加速收敛。Fan 等人[50] 在[18]的基础上，将多目标对多目标（M2M）分解法与推拉搜索框架（PPS-M2M）相结合。Yu 等人[51]提出了一种目的导向的两阶段多目标差分进化（PDTP-MDE）算法。Tian 等人[52]提出了一种两阶段进化算法，可根据当前种群状态在两个阶段之间切换。Yu 等[53] 提出了一种动态选择偏好辅助约束多目标差分进化算法。Ming 等人[54] 提出了一种两阶段框架，以提高 CMOP 的效率和通用性。Xu 等人 [55] 提出了一种两阶段约束处理的 CMOEA（C-TPEA）。在前期，网格约束分解法框架能很好地保持收敛性和多样性；在后期，引入

ɛ

约束策略，进一步提高解的可行性和质量。Ma 等人 [56] 提出了一种基于对无约束 PF 影响的约束处理优先级排序策略，并根据约束处理优先级逐一添加约束条件，在不同的演化阶段进行处理。

Moreover, Jiao et al. [57] proposed two-types weight adjustments based on decomposition-based multi-objective evolutionary algorithm (MOEA/D). Feng et al. [58] proposed a three-phase hybrid driving strategy for selecting mating parents. In the first phase, tournament selection for feasibility is used to select mating parents. In the second phase, tournament selection for diversity is used to select mating parents. During the third phase, the population is distributed in the proximity of the PF. The main objective of this phase is to further optimize the population towards the PF while maintaining stability. Tournament selection for feasibility is utilized to select mating parents. Zhang et al. [59] proposed a non-parametric technique for handling constraints in the later stage. This approach makes full use of the hidden information in the feasible non-dominated set. Sun et al. [60] addressed the complex multi-constraint situation. In the early stage, they obtained the priority among the constraints based on the relationship between the PF of each constraint and their common PF. In the middle stage, they evaluated each constraint according to the priority and further explored the objective space. In the late stage, they fully considered the feasibility, improved the quality of the solutions obtained in the previous two stages, and finally obtained solutions with good convergence, feasibility, and diversity. Song et al. [61] proposed a two-stage differential evolution algorithm (TSCMODE). Liang et al. [62] explored and exploited the relationship between CPF and UPF to solve CMOPs, taking the early stage as the learning stage, measuring the relationship between CPF and UPF; the later stage as the evolutionary stage, calculating specific evolutionary strategies based on the learned relationship, to improve the utilization efficiency of objective information.
此外，Jiao 等人[57] 基于基于分解的多目标进化算法（MOEA/D）提出了两种权重调整方法。Feng 等人[58]提出了一种用于选择交配亲本的三阶段混合驱动策略。在第一阶段，使用可行性锦标赛选择来选择交配亲本。在第二阶段，使用多样性锦标赛选择来选择交配亲本。在第三阶段，种群分布在 PF 附近。该阶段的主要目标是在保持稳定的同时，进一步优化种群，使其趋向 PF。在选择交配亲本时，会利用锦标赛选择可行性。Zhang 等人[59]提出了一种非参数技术，用于处理后期阶段的约束条件。这种方法充分利用了可行非支配集中的隐藏信息。Sun 等人 [60] 解决了复杂的多约束条件问题。在早期阶段，他们根据各约束条件的 PF 与它们的共同 PF 之间的关系来获得各约束条件的优先级。在中期阶段，他们根据优先级对各约束条件进行评估，并进一步探索目标空间。在后期阶段，他们充分考虑了可行性，改进了前两个阶段得到的解的质量，最终得到了具有良好收敛性、可行性和多样性的解。Song 等人[61]提出了一种两阶段差分进化算法（TSCMODE）。梁等人[61]提出了一种两阶段差分进化算法（TSCMODE）。 [62]探索并利用 CPF 和 UPF 之间的关系求解 CMOP，将前期作为学习阶段，测量 CPF 和 UPF 之间的关系；后期作为进化阶段，根据学习到的关系计算具体的进化策略，提高客观信息的利用效率。

Besides, Liu et al. [63] aimed at expensive multi-objective constrained optimization problems, proposed a surrogate-assisted two-stage differential evolution (SA-TSDE). Zou et al. [64] aimed at complex feasible regions, proposed a more flexible two-stage evolutionary algorithm based on automatic regulation (ARCMO), which performs fast global search in the early stage, and passes the population information to the later stage. The later stage consists of two dynamically complementary subprocesses: exploration subprocess and convergence subprocess. Xia et al. [65] proposed a novel CMOEA with two-stage resources allocation, which dynamically allocates resources to two stages. Feng et al. [66] proposed an adaptive tradeoff evolutionary algorithm (ATEA), which divides the search process into three stages. In the early stage, it guides the constraint relaxation technique to perform global search, making the population quickly cross the infeasible region and approach the feasible region. In the middle stage, it further detects and estimates the constraint conditions, to retain more feasible individuals and competitive infeasible individuals, thus making the population accurately identify all possible feasible regions, and gradually extend to the feasible boundary. In the later stage, it explores the areas that are insufficiently explored in the previous stages, accelerates the population convergence, and escapes from the local optimum state. Chen et al. [67] aimed at dynamic constrained multi-objective optimization problems, proposed a two-stage diversity compensation strategy. Huang et al. [68] proposed a framework for a CMOEA that utilizes global and local FS searches. The algorithm uses a global search operator in the early stage and a local search operator in the middle stage. The first two stages are not subject to constraint restrictions, and the feasible solutions are stored in the FeasiblePool for environmental selection. In the later stage, the information regarding these feasible solutions is reused to guide the population evolution while considering various constraint conditions.
此外，Liu 等人[63] 针对昂贵的多目标约束优化问题，提出了一种代理辅助两阶段差分进化算法（SA-TSDE）。Zou 等人[64]针对复杂的可行区域，提出了一种更灵活的基于自动调节的两阶段进化算法（ARCMO），该算法在早期阶段执行快速全局搜索，并将种群信息传递给后期阶段。后期阶段由两个动态互补的子过程组成：探索子过程和收敛子过程。Xia 等人[65]提出了一种具有两阶段资源分配的新型 CMOEA，可动态地将资源分配给两个阶段。Feng 等人[66]提出了一种自适应权衡进化算法（ATEA），它将搜索过程分为三个阶段。在早期阶段，它引导约束松弛技术进行全局搜索，使种群快速穿过不可行区域并接近可行区域。中期，进一步检测和估计约束条件，保留更多的可行个体和竞争性的不可行个体，从而使种群准确识别所有可能的可行区域，并逐渐扩展到可行边界。在后一阶段，它探索前一阶段探索不足的区域，加速种群收敛，摆脱局部最优状态。Chen 等人[67] 针对动态约束多目标优化问题，提出了两阶段多样性补偿策略。Huang 等人[68] 提出了一种利用全局和局部 FS 搜索的 CMOEA 框架。该算法在早期阶段使用全局搜索算子，在中期阶段使用局部搜索算子。前两个阶段不受约束条件限制，可行解存储在可行池中供环境选择。在后期阶段，考虑到各种约束条件，这些可行解的相关信息将被重新用于指导种群进化。

2.2.3. Methods based on chaotic optimization
2.2.3.基于混沌优化的方法

These methods typically use chaotic maps to generate pseudorandom numbers that are mapped as decision variables for global optimization problems [69]. They are usually combined with a genetic algorithm to increase the diversity of the population through chaotic initialization, or adjust the algorithm parameters through chaotic mapping, so as to improve the search efficiency and solution quality of the algorithm.
这些方法通常使用混沌映射生成伪随机数，并将其映射为全局优化问题的决策变量[69]。它们通常与遗传算法相结合，通过混沌初始化增加种群的多样性，或通过混沌映射调整算法参数，从而提高算法的搜索效率和求解质量。

Aslimani et al. [70] proposed a new method based on chaotic search. The method is characterized by fast and accurate convergence, as well as parallel and independent decomposition of the target space. Li et al. [71] proposed the parallel chaotic optimization algorithm (PCOA) as an improvement to the chaotic optimization algorithm (COA). PCOA is a population-based optimization algorithm that reduces sensitivity to initial values. Zhang et al. [72] proposed a multi-objective evolutionary algorithm (LCMO) based on a weakly cooperative evolutionary framework and multiple chaos operators. The hybrid chaos operator is used to improve the distribution uniformity of candidate solution populations, and the quadratic selection criterion refines the Pareto level. Takubo and Kanazaki [73] proposed an integrated optimization method based on CMOEA and non-intrusive polynomial chaos expansion (PCE) for solving robust constrained multi-objective optimization problems under time-series dynamics. Yacoubi et al. [74] proposed a multi-objective chaos game optimization algorithm (MOCGO/DR), by embedding theoretical concepts such as multi-objective, decomposition, and stochastic learning in the traditional CGO algorithm. Rauf et al. [75] proposed the multiple population-based chaotic differential evolution (MPC-DE) algorithm. The algorithm divides the population into two sub-populations using an enhanced chaos-based population initialization method. The first sub-population uses a sinusoidal chaotic initialization method, while the second sub-population uses a tent-mapped chaotic initialization method. Each sub-population adopts an improved mutation strategy based on a two-dimensional chaotic mapping. The proposed mutation strategy generates mutation vectors, and a corresponding selection criterion is used to select the optimal progeny generated by each sub-population.
Aslimani 等人[70] 提出了一种基于混沌搜索的新方法。该方法的特点是收敛速度快、精度高，以及目标空间的并行和独立分解。Li 等人[71]提出了并行混沌优化算法（PCOA），作为对混沌优化算法（COA）的改进。PCOA 是一种基于群体的优化算法，可降低对初始值的敏感性。Zhang 等人[72]提出了一种基于弱合作进化框架和多混沌算子的多目标进化算法（LCMO）。混合混沌算子用于改善候选解群的分布均匀性，二次选择准则则用于完善帕累托水平。Takubo 和 Kanazaki [73] 提出了一种基于 CMOEA 和非侵入式多项式混沌扩展（PCE）的集成优化方法，用于解决时间序列动态下的鲁棒约束多目标优化问题。Yacoubi 等人[74]通过在传统 CGO 算法中嵌入多目标、分解和随机学习等理论概念，提出了一种多目标混沌博弈优化算法（MOCGO/DR）。Rauf 等人[75]提出了基于多种群的混沌微分进化算法（MPC-DE）。该算法使用增强的基于混沌的种群初始化方法将种群分为两个子种群。第一个子种群使用正弦混沌初始化方法，而第二个子种群使用帐篷映射混沌初始化方法。每个子种群都采用基于二维混沌映射的改进突变策略。所提出的突变策略会产生突变向量，并使用相应的选择标准来选择每个子群体产生的最优后代。

In general, methods based on evolution are known for their simplicity, flexibility, and minimal parameter requirements, effectively balancing constraints and goals while preserving population diversity. However, they can incur high computational costs, especially with archived populations during co-evolution, and may struggle with constraint-objective balance, causing poor convergence. Additionally, reliance on prior knowledge when addressing constraints may hinder performance in complex scenarios.
一般来说，基于进化的方法以其简单、灵活和参数要求最低而著称，能在保持种群多样性的同时有效平衡约束和目标。然而，这些方法可能会产生较高的计算成本，尤其是在协同进化过程中使用存档种群时，而且可能难以实现约束条件与目标之间的平衡，导致收敛性较差。此外，在处理约束条件时，对先验知识的依赖可能会妨碍复杂情况下的性能。

2.3. Methods based on machine learning
2.3.基于机器学习的方法

These methods address optimization challenges through a data-driven approach, capable of learning from historical data to predict outcomes. They are aptly suited for dynamic settings and online optimization issues. The methods are categorized into three distinct types: algorithm recommendation, intelligent parameter tuning, and predictive modeling.
这些方法通过数据驱动方法解决优化难题，能够从历史数据中学习并预测结果。它们非常适合动态设置和在线优化问题。这些方法分为三种不同类型：算法推荐、智能参数调整和预测建模。

2.3.1. Algorithm recommendation
2.3.1.算法建议

Tian et al. [76] proposed an automated algorithm selection method for choosing the most suitable evolutionary algorithm for a given MOP, where the input is the explicit and implicit features of multi-objective optimization problems sampled by Latin hypercube, and the output is the evolutionary algorithm with the best performance on multi-objective optimization problems. Qiao et al. [77] proposed the first evolution-based constrained multi-objective feature extraction method (ECMOFE), where two populations target the constraints and objectives respectively, and adopt two complementary evolutionary strategies to co-evolve. In the evolutionary process, the feature matrix is obtained by recording the success rate of the offspring population. Then, a dimensionality reduction method is designed to reduce the size of the matrix without losing important information, and then the matrix is transformed into a vector, and then trained by a classifier. Han et al. [78] proposed a multi-heuristic algorithm based on deep reinforcement learning, which dynamically selects the most suitable meta-heuristic algorithm (discrete cuckoo search and particle swarm optimization) based on the diversity and quality of the population during the population evolution process, and designs rewards based on the evolution effect, thereby guiding the algorithm to update the solutions more effectively and achieve global optimization.
Tian等人[76]提出了一种自动算法选择方法，用于为给定澳门威尼斯人官网作选择最合适的进化算法，其中输入是由拉丁超立方采样的多目标优化问题的显性和隐性特征，输出是在多目标优化问题上性能最佳的进化算法。Qiao 等人[77]首次提出了基于进化的约束多目标特征提取方法（ECMOFE），两个种群分别针对约束和目标，采用两种互补的进化策略共同进化。在进化过程中，通过记录子种群的成功率获得特征矩阵。然后，设计一种降维方法，在不丢失重要信息的情况下减小矩阵的大小，再将矩阵转化为向量，然后由分类器进行训练。Han 等人[78]提出了一种基于深度强化学习的多启发式算法，在种群进化过程中根据种群的多样性和质量动态选择最合适的元启发式算法（离散布谷鸟搜索和粒子群优化），并根据进化效果设计奖励，从而引导算法更有效地更新解，实现全局优化。

Moreover, Zuo et al. [19] proposed the process knowledge-guided constrained multi-objective autonomous evolutionary optimization method, which is different from [76], [77] in that it does not recommend CMOEAs, but intelligently recommends the subsequent guidance strategies according to the established mapping model and the current population state, and adopts appropriate strategies at different stages of population evolution, without changing the structure of the existing evolutionary algorithm, but using process knowledge to guide the population evolution. A multilayer perceptron is used to train the mapping model. Wang et al. [79] proposed a transfer-based method to enrich the algorithm library for constrained multi-objective optimization problems. This method calculates the similarity between problems based on the landscape features of the problems, and then calculates the transfer probability of intelligent optimization algorithms that solve similar problems based on the performance of each algorithm and the similarity between problems. Algorithms with high transfer probability will enrich the algorithm library for solving the current problem. This method uses a Softmax regression model to recommend algorithms for solving the current problem. Ming et al. [80] proposed an adaptive selection of auxiliary tasks for multi-task auxiliary constrained multi-objective optimization. The method transforms CMOPs into a multi-task optimization problem. The original CMOPs serve as the main task, while several auxiliary tasks use different algorithmic strategies. The method uses reinforcement learning and deep learning techniques to dynamically select the most suitable auxiliary tasks based on the current state of the population, thereby improving the optimization results.
此外，Zuo 等人[19]提出了过程知识引导的约束多目标自主进化优化方法，与[76]、[77]不同的是，该方法不推荐 CMOEA，而是根据建立的映射模型和当前种群状态智能推荐后续引导策略，在种群进化的不同阶段采用合适的策略，不改变现有进化算法的结构，而是利用过程知识引导种群进化。多层感知器用于训练映射模型。Wang 等人[79]提出了一种基于转移的方法，以丰富约束多目标优化问题的算法库。该方法根据问题的景观特征计算问题间的相似性，然后根据各算法的性能和问题间的相似性计算解决相似问题的智能优化算法的转移概率。转移概率高的算法将丰富解决当前问题的算法库。该方法使用 Softmax 回归模型来推荐解决当前问题的算法。Ming 等人[80]提出了一种多任务辅助约束多目标优化的辅助任务自适应选择方法。该方法将 CMOPs 转化为多任务优化问题。原始的 CMOPs 作为主要任务，而多个辅助任务则使用不同的算法策略。该方法利用强化学习和深度学习技术，根据群体的当前状态动态选择最合适的辅助任务，从而改善优化结果。

2.3.2. Intelligent parameter adjustment
2.3.2.智能参数调整

Fallahi et al. [81] proposed using a reinforcement learning algorithm, Q-learning, to intelligently tune the parameter values of DE and particle swarm optimization (PSO) algorithms. They also developed two new variants of DE and PSO, namely DE with Q-learning (DEQL) and PSO with Q-learning (PSOQL). Yu et al. [82] proposed the reinforcement learning-based multi-objective differential evolution (RLMODE) algorithm. During the evolution process, the offspring generated is evaluated and compared with its corresponding parents, the relationship between the offspring and parent can adjust the parameters of RLMODE by the reinforcement learning (RL) technique. The feedback mechanism can generate the most suitable parameters for RLMODE, thus pushing the population towards the feasible region.
Fallahi 等人[81]提出使用强化学习算法 Q-learning 来智能调整 DE 和粒子群优化（PSO）算法的参数值。他们还开发了 DE 和 PSO 的两个新变体，即带有 Q-learning 的 DE（DEQL）和带有 Q-learning 的 PSO（PSOQL）。Yu 等人[82]提出了基于强化学习的多目标差分进化（RLMODE）算法。在进化过程中，生成的子代与其对应的父代进行评估和比较，子代与父代之间的关系可通过强化学习（RL）技术调整 RLMODE 的参数。反馈机制可以为 RLMODE 生成最合适的参数，从而推动种群向可行区域进化。

2.3.3. Prediction generation
2.3.3.预测生成

Ma et al. [83] proposed an adaptive reference vector reinforcement learning method based on decomposition algorithms, which aims to solve the problem of decomposition-based algorithms being sensitive to the shape of the PF. In this method, the adaptation process of the reference vectors is regarded as a reinforcement learning task, where each reference vector can learn from the environmental feedback and choose the actions that best fit the problem characteristics. Moreover, the sampling operation of the reference points uses a distribution estimation learning model to generate new reference points. Wang et al. [84] proposed a model-based evolutionary constrained multi-objective optimization offspring generation strategy. This strategy uses a growing neural gas network to collect feasibility information during the evolutionary search process, and gradually builds a distribution model of the feasible region in the decision space. In order to make full use of infeasible solutions to assist the modeling of the feasible region, this strategy treats feasible solutions and infeasible solutions as two classes of samples, and uses supervised learning methods to train the network. It can use the learned network to generate promising offspring, thereby accelerating the convergence of the population to the optimal feasible region.
Ma 等人[83]提出了一种基于分解算法的自适应参考向量强化学习方法，旨在解决基于分解算法对 PF 的形状敏感的问题。在该方法中，参考向量的适应过程被视为强化学习任务，每个参考向量都能从环境反馈中学习，并选择最适合问题特征的行动。此外，参考点的采样操作使用分布估计学习模型来生成新的参考点。Wang 等人[84]提出了一种基于模型的进化约束多目标优化子代生成策略。该策略在进化搜索过程中利用生长神经气体网络收集可行性信息，并逐步建立决策空间中可行区域的分布模型。为了充分利用不可行解来辅助可行区域的建模，该策略将可行解和不可行解视为两类样本，并使用监督学习方法来训练网络。它可以利用学习到的网络生成有希望的后代，从而加速种群向最优可行区域的收敛。

Moreover, Ming et al. [20] proposed a new machine learning-based method, which adopts the determinantal point process to select the population and archive, instead of the traditional distance- or density-based selection strategies. The determinantal point process is a probabilistic model that can quantify the diversity of a subset, and it can effectively extract representative and balanced solutions from a candidate set [85] . To apply the determinantal point process in multi-objective optimization, the method designed two innovative kernel matrices, which reflect the convergence, diversity, and feasibility of the solutions in the population and archive, respectively, thus achieving efficient exploration and exploitation of the solution space. Based on [20], Hu et al. [86] proposed a reinforcement learning-based population state evaluation method, which uses the overall improvement degree as the reward signal to guide the neural network to select the appropriate parent population and archive for mutation operations. In addition, this method also introduced the idea of coevolution into the generation of training data, by sharing information among multiple populations, enhancing the diversity of the samples, and thus improving the accuracy of the neural network model. Liu et al. [87] proposed a CMOEA with Pareto estimation by neural network, using a dual population mechanism, and training the network with self-organizing map (SOM). The algorithm preserves neighborhood information, maps the population structure of the decision space to the objective space, and then uses neuron weights to estimate the PF.
此外，Ming 等人[20] 提出了一种基于机器学习的新方法，该方法采用行列式点过程来选择种群和存档，而不是传统的基于距离或密度的选择策略。行列式点过程是一种概率模型，可以量化子集的多样性，并能有效地从候选集中提取具有代表性的均衡解 [85] 。为了将行列式点过程应用于多目标优化，该方法设计了两个创新的核矩阵，分别反映了种群和归档中解的收敛性、多样性和可行性，从而实现了对解空间的有效探索和利用。在[20]的基础上，Hu 等人[86]提出了一种基于强化学习的种群状态评价方法，该方法以总体改进度作为奖励信号，引导神经网络选择合适的父种群和存档进行突变操作。此外，该方法还将协同进化的思想引入到训练数据的生成中，通过在多个种群间共享信息，增强样本的多样性，从而提高神经网络模型的准确性。Liu 等人[87]提出了一种通过神经网络进行帕累托估计的 CMOEA，采用双种群机制，并用自组织图（SOM）训练网络。该算法保留了邻域信息，将决策空间的种群结构映射到目标空间，然后利用神经元权重估计 PF。

In general, machine learning-based methods excel in adapting to complex environments and enhancing algorithm robustness, offering great promise. Yet, their performance hinges on feature extraction, demanding extensive data and computational power, which could escalate algorithm complexity and diminish interpretability.
一般来说，基于机器学习的方法在适应复杂环境和增强算法鲁棒性方面表现出色，前景广阔。然而，它们的性能取决于特征提取，需要大量的数据和计算能力，这可能会增加算法的复杂性，降低可解释性。

2.4. Discussion 2.4.讨论情况

The strengths and weaknesses of the above categories of existing CMOAs will be discussed here mainly and summarized in Table 1. Additionally, Fig. 5 displays a timeline of the development of current CMOAs.
本文将主要讨论上述几类现有中医药组织协定的优缺点，并在表 1 中进行总结。此外，图 5 显示了当前 CMOA 的发展时间表。

(1) Methods based on classical mathematics: In methods based on classical mathematics, solving constrained multi-objective optimization problems usually involves transforming multiple objectives into a single objective to leverage established optimization algorithms. These methods have a solid theoretical foundation, a clear computational procedure, and often converge quickly to a solution. Linear programming is a widely used technique that finds the optimal solution by linearizing objectives and constraints. The

ɛ

-constraint method selects one main objective from multiple objectives to optimize, while the other objectives are used as constraints. Fuzzy set theory, on the other hand, uses fuzzy logic to deal with uncertainty, allowing solutions to be elastic in the degree of satisfaction. However, it is important to note that these methods have limitations. First, multi-objective problems are often reduced to single-objective problems, which can result in misunderstandings or oversimplifications of the original problem. Additionally, determining the weight or

ɛ

value may require subjective judgment from the decision maker, which can make it difficult to ensure the quality of the solution. In practical applications, these methods may require more tuning and adaptation to obtain an effective solution, as they may not effectively handle nonlinear or non-convex problems due to their reliance on the linear and convex nature of the problem to ensure solution quality and algorithm convergence.
(1) 基于经典数学的方法：在基于经典数学的方法中，解决受约束的多目标优化问题通常涉及将多个目标转化为单一目标，以利用已有的优化算法。这些方法具有坚实的理论基础和清晰的计算程序，通常能快速收敛到一个解决方案。线性规划是一种广泛使用的技术，它通过将目标和约束线性化来找到最优解。0# 约束法是从多个目标中选择一个主要目标进行优化，其他目标则作为约束条件。而模糊集理论则利用模糊逻辑来处理不确定性，使解决方案在满足程度上具有弹性。不过，必须指出的是，这些方法都有局限性。首先，多目标问题往往被简化为单目标问题，这可能导致对原始问题的误解或过度简化。此外，确定权重或

ɛ

值可能需要决策者的主观判断，这就很难确保解决方案的质量。在实际应用中，这些方法可能需要更多的调整和适应才能获得有效的解决方案，因为它们依赖问题的线性和凸性来确保解决方案的质量和算法的收敛性，因此可能无法有效处理非线性或非凸性问题。

(2) Methods based on evolution: Methods based on evolution are more flexible than classical mathematics optimization methods and can handle nonlinear, non-convex, or non-differentiable problems. This is because evolution methods do not rely on the mathematical properties of the problem, but approach the optimal solution through iterative search. However, this approach also has its limitations. As trial-and-error based methods, they may require significant computational resources to find satisfactory solutions. This is particularly noticeable in problems with high dimensions or complex constraints, where the search space is vast, and the algorithm requires more time to explore. Additionally, the performance of the evolutionary methods is highly dependent on the proper setting of its parameters. If the parameters are not selected appropriately, the algorithm may fail to converge to the optimal solution, or the convergence rate may be very slow. The algorithm designer must possess a thorough understanding of the problem and extensive experience in parameter tuning.
(2) 基于演化的方法：与经典数学优化方法相比，基于演化的方法更加灵活，可以处理非线性、非凸或无差异问题。这是因为进化法并不依赖于问题的数学特性，而是通过迭代搜索来接近最优解。然而，这种方法也有其局限性。作为基于试错的方法，它们可能需要大量的计算资源才能找到令人满意的解决方案。这一点在高维度或复杂约束的问题中尤为明显，因为在这些问题中，搜索空间非常大，算法需要更多的时间来探索。此外，进化方法的性能在很大程度上取决于参数的正确设置。如果参数选择不当，算法可能无法收敛到最优解，或者收敛速度非常缓慢。算法设计者必须对问题有透彻的理解，并在参数调整方面拥有丰富的经验。

(3) Methods based on machine learning: According to the no free lunch theorem [88], any algorithm’s performance advantage on one class of problems becomes a disadvantage on another class of problems. Based on machine learning methods, this approach can use data and models to automatically learn and adapt to complex problem environments, improving the algorithm’s generalization and robustness. This approach can construct nonlinear mappings and strategies through deep learning or reinforcement learning methods, breaking through the limitations of traditional algorithms and improving the algorithm’s flexibility and innovation. Feature extraction, which is essential for training models, directly affects the algorithm’s performance. In addition, model training requires a lot of data and computing resources, which may increase the algorithm’s time and space complexity and make it difficult to achieve real-time and online optimization. Moreover, model design and parameter setting need to be reasonable, otherwise they may reduce the algorithm’s stability and interpretability and make it difficult to guarantee the algorithm’s reliability and controllability.
(3) 基于机器学习的方法：根据 "没有免费的午餐 "定理[88]，任何算法在一类问题上的性能优势在另一类问题上都会变成劣势。基于机器学习方法，这种方法可以利用数据和模型自动学习并适应复杂的问题环境，从而提高算法的泛化能力和鲁棒性。这种方法可以通过深度学习或强化学习方法构建非线性映射和策略，突破传统算法的局限性，提高算法的灵活性和创新性。特征提取是训练模型的关键，直接影响算法的性能。此外，模型训练需要大量的数据和计算资源，可能会增加算法的时间和空间复杂度，难以实现实时和在线优化。此外，模型设计和参数设置也需要合理，否则会降低算法的稳定性和可解释性，难以保证算法的可靠性和可控性。

Table 1. Advantages and limitations of various methods.
表 1.各种方法的优势和局限性。

Categories 类别	Advantages 优势	Limitations 局限性
Methods based on classical mathematics 基于经典数学的方法	The methods in solving CMOPs include a solid theoretical foundation, clear computational procedures, and the ability to converge quickly to a solution. 解决 CMOP 的方法包括扎实的理论基础、清晰的计算程序以及快速收敛求解的能力。	These methods may lead to misinterpretation or oversimplification of the original problem, and may also require subjective judgment to determine weights or values, which can affect the quality of the solution, and may not be able to deal effectively with nonlinear or nonconvex problems. 这些方法可能会导致对原始问题的误解或过度简化，还可能需要主观判断来确定权重或取值，从而影响求解质量，而且可能无法有效处理非线性或非凸问题。
Methods based on evolution 基于进化的方法	The methods are more flexible, can handle nonlinear, non-convex, or non-differentiable problems, and do not rely on the mathematical nature of the problem. 这些方法更加灵活，可以处理非线性、非凸性或无差异问题，而且不依赖于问题的数学性质。	They may demand extensive computational resources and sensitive parameter tuning to effectively converge on an optimal solution. 它们可能需要大量的计算资源和敏感的参数调整，才能有效地收敛到最优解。
Methods based on machine learning 基于机器学习的方法	The methods uses data and models to automatically learn and adapt to complex problem environments, improving the generalization and robustness of the algorithms. 这些方法利用数据和模型来自动学习和适应复杂的问题环境，从而提高算法的通用性和鲁棒性。	Feature extraction directly affects the performance of the algorithm, and also requires a large amount of data and computational resources, which may make the algorithm more complex in time and space and less interpretable. 特征提取直接影响算法的性能，同时也需要大量的数据和计算资源，这可能会使算法在时间和空间上更加复杂，可解释性降低。

Despite the excellent performance of existing algorithms in solving CMOPs, some research gaps still need to be considered:
尽管现有算法在求解 CMOP 方面表现出色，但仍需考虑一些研究空白：

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With the development of deep learning, large language models have achieved remarkable results in the field of natural language processing. Large language models can be pre-trained on large-scale text data, learning rich language knowledge, and then complete different downstream tasks by fine-tuning or zero-shot learning. Liu et al. [89] combined large language models with multi-objective optimization and used suitable prompt engineering to make the general large language models act as black-box search operators for MOEA/D in a zero-shot learning manner. They showed powerful performance. Their method can effectively handle the balance between constraints and objectives in multi-objective optimization problems and improve the search efficiency and solution quality by using the powerful expression of large language models. However, research on the combination of constrained multi-objective optimization and large language models is still lacking. We think this direction is challenging and promising, and also a new development trend of constrained multi-objective optimization in the future. Large language models can adapt to different problem types by using different prompt engineering as a general search operator, achieving an effective balance between constraints and objectives. This direction of research cannot only expand the application scope of large language models, but also provide a new idea and method for solving constrained multi-objective optimization problems.
随着深度学习的发展，大型语言模型在自然语言处理领域取得了令人瞩目的成果。大型语言模型可以在大规模文本数据上进行预训练，学习丰富的语言知识，然后通过微调或零点学习完成不同的下游任务。Liu 等人[89]将大型语言模型与多目标优化相结合，并使用适当的提示工程使通用大型语言模型以零点学习的方式充当 MOEA/D 的黑盒搜索算子。他们展示了强大的性能。他们的方法能有效处理多目标优化问题中约束与目标之间的平衡，并利用大语言模型的强大表达能力提高搜索效率和求解质量。然而，关于约束多目标优化与大型语言模型相结合的研究仍然缺乏。我们认为这一方向具有挑战性和发展前景，也是未来约束多目标优化的一个新的发展趋势。大型语言模型可以通过使用不同的提示工程作为一般搜索算子来适应不同的问题类型，实现约束和目标之间的有效平衡。这一研究方向不仅拓展了大语言模型的应用范围，也为约束多目标优化问题的求解提供了新的思路和方法。
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In this section, we mentioned the machine learning-based method that uses a data-driven approach to solve CMOPs. However, the performance of the machine learning-based method depends largely on the effect of feature extraction, which is the process of transforming raw data into meaningful information. The machine learning-based method will fail or be inefficient if feature extraction cannot fully reflect the data’s characteristics and the complexity of the CMOP. Therefore, feature engineering research is an urgent topic for CMOPs. The purpose of feature engineering research is to find an effective feature extraction method that can extract features from the original data, which are helpful for optimizing CMOPs, and reduce the dimensionality and complexity of the data, improving the computational efficiency and interpretability.
在本节中，我们提到了基于机器学习的方法，该方法采用数据驱动的方式来解决 CMOP 问题。然而，基于机器学习的方法的性能很大程度上取决于特征提取的效果，而特征提取是将原始数据转化为有意义信息的过程。如果特征提取不能充分反映数据的特征和 CMOP 的复杂性，基于机器学习的方法就会失败或效率低下。因此，特征工程研究是 CMOP 迫切需要解决的课题。特征工程研究的目的是找到一种有效的特征提取方法，既能从原始数据中提取有助于优化 CMOP 的特征，又能降低数据的维度和复杂度，提高计算效率和可解释性。
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Some methods based on machine learning use supervised learning to train mapping models, but a general dataset for researchers to conduct experimental exploration is still lacking. Due to their diversity and complexity, CMOPs make it difficult to find a general dataset that can cover different problem characteristics and scenarios. Researchers usually use some artificially constructed test functions or some datasets from practical applications to conduct experiments, but these datasets often fail to reflect the reality and universality of CMOPs. Therefore, a meaningful research direction is to design a general dataset that can simulate different aspects of CMOPs, such as the type, number, difficulty, and dynamics of constraints. Such a dataset can provide researchers with a fair and comprehensive evaluation platform and promote the development and improvement of machine learning-based methods.
一些基于机器学习的方法使用监督学习来训练绘图模型，但仍缺乏供研究人员进行实验探索的通用数据集。由于 CMOP 的多样性和复杂性，很难找到一个能涵盖不同问题特征和场景的通用数据集。研究人员通常使用一些人为构建的测试函数或一些实际应用中的数据集来进行实验，但这些数据集往往无法反映 CMOP 的现实性和普遍性。因此，一个有意义的研究方向是设计一个能模拟 CMOP 不同方面的通用数据集，如约束的类型、数量、难度和动态。这样的数据集可以为研究人员提供一个公平、全面的评估平台，促进基于机器学习方法的发展和完善。

Table 2. Summary of practical applications of CMOAs.
表 2.CMOA 的实际应用摘要。

Scenarios 场景	Problems 问题	Algorithms 算法	The number of objectives 目标数量	The number of constraints 制约因素的数量
Engineering design problems 工程设计问题	vibration platform design problem [90] 振动平台设计问题 [90]	AT-CMOEA [91], CMOEA-MS [52] 联合国气候变化框架公约秘书处[91]，联合国气候变化框架公约管理部门[52]	2	5
	the process synthesis and design problem [92] 工艺综合和设计问题 [92]	AT-CMOEA [91]	2	2
	software engineering [93] 软件工程 [93］	ToM [94]	2,3,4	20
	welded beam design [95] 焊接梁设计 [95］	PPTA [96], GLS-CMOEA [68], [97] PPTA [96]、GLS-CMOEA [68]、[97］	2	3
	antenna design optimization [98] 天线设计优化 [98]	MOEA/D-2WA [57] 海洋环境部/D-2WA [57]	3	2
	robot gripper optimization [99] 机器人抓手优化 [99]	PPS [18]	2	8
	optimal wireless sensor location design [1] 最佳无线传感器位置设计 [1]	DCCMO [1]	2	2,9
	dual source compressed air piping optimization problem [100] 双源压缩空气管道优化问题 [100]	MODE-CHS [101], DDCMEA [46], MODE-PS [102] mode-chs [101], ddcmea [46], mode-ps [102].	2	4
	servo turntable system [103] 伺服转盘系统 [103]	MOPSO [103]	4	3
Resource optimization problems 资源优化问题	water distribution network optimization [104] 配水管网优化 [104]	C-TAEA [39]	4	2
	groundwater remediation design [105] 地下水修复设计 [105]	AMALGAM [105] AMALGAM [105］	2	4
	thermocline filling layer thermal energy storage [106] 温跃层填充层热能储存[106]	[106]	2	3
	test resource allocation optimization [107] 测试资源配置优化 [107]	ECHTs [107] 非物质文化遗产 [107]	3	3
	combined cooling heating and power (CCHP) microgrids optimization [29] 冷热电联供（CCHP）微电网优化 [29]	MILP [29]	2	17
	economic load dispatch problem (EDP) [75] 经济负荷调度问题 (EDP) [75]	MPC-DE [75]	2	8
Scheduling optimization problems 调度优化问题	multi-product batch plant problem [108] 多产品配料站问题 [108]	CMME [109], ARCMO [64], AT-CMOEA [91]	3	3
	dual-resource constrained flexible job shop scheduling problem [110] 双资源约束灵活作业车间调度问题 [110]	TS-MOPSO [110]	3	19
	short-term crude oil scheduling problem [111] 短期原油调度问题 [111]	NSGAII-APE [111]	5	10
	urban bus scheduling problem [112] 城市公交调度问题 [112]	ShiP [113]	2	5
	combined economic emission dispatch (CEED) problem [114] 联合经济排放调度（CEED）问题 [114]	CMOPEO-EED [115], [116], [117] cmopeo-eed [115]、[116]、[117]	2	9
	optimal reactive power dispatch (ORPD) problem [118] 最佳无功功率调度（ORPD）问题 [118]	[118]	2	5
Path planning problems 路径规划问题	unmanned aerial vehicle (UAV) planning [119] 无人驾驶飞行器（UAV）规划 [119]	M2M-DW [119]	2	4
	path planning for emergency material scheduling [120] 应急物资调度的路径规划 [120]	[120]	3	8
	fuzzy business restricted multi-objective multi-index transportation problem [121] 模糊商业受限多目标多指标运输问题 [121]	[121]	2	3
	robot trajectory planning [122] 机器人轨迹规划 [122]	MOGA [122]	2	5
	constrained multi-objective solid traveling salesman problems [123] 受约束的多目标固体旅行推销员问题 [123]	R-MOGA [123] 俄文-MOGA [123]	2	1
Other problems 其他问题	process synthesis problems [124] 过程合成问题 [124]	CAEAD [125], MCCMO [126] 《儿童权利公约》[125]，《气象公约》[126]	2	9
	polyester polymerization process problems [127] 聚酯聚合工艺问题 [127]	IARVEA [127]	4	3
	gasoline blending process problem [30] 汽油混合工艺问题[30］	MOEA/D-FCHT [30] 海洋和海洋法司 [30]	2	10

3. Applications 3.应用

This section presents the applications of CMOAs in some representative domains, such as engineering design, resource optimization, scheduling optimization, and path planning. These domains involve problems that are of practical significance and value. Table 2 provides detailed information on the practical application of some CMOAs.
本节将介绍 CMOA 在一些代表性领域的应用，如工程设计、资源优化、调度优化和路径规划。这些领域涉及的问题都具有实际意义和价值。表 2 提供了一些 CMOAs 实际应用的详细信息。

3.1. Engineering design problems
3.1.工程设计问题

Engineering design problems are those that can be optimized or improved in engineering practice. The objective is to identify the optimal or near-optimal design solutions that meet certain performance, safety, economic, and relevant requirements. It mainly includes gearbox design [128], heat exchanger network design [129], disc brake design [130], spring design problems [131], twin-rod truss design [132], car side impact problem [133], reducer design problem [134], the simply supported I-beam design problem [135], bulk carrier design [136], pressure vessel design [137], reactor network design [138], vibration platform design problem [90], the process synthesis and design problem [92], software engineering [93], welded beam design [95], antenna design optimization [98], robot gripper optimization [99], optimal wireless sensor location design [1], dual source compressed air piping optimization problem [100], servo turntable system [103], turning process parameters optimization [139], Al-Li alloy thin-wall workpieces design [140].
工程设计问题是指在工程实践中可以优化或改进的问题。其目的是找出最优或接近最优的设计方案，以满足一定的性能、安全、经济和相关要求。它主要包括齿轮箱设计问题[128]、热交换器网络设计问题[129]、盘式制动器设计问题[130]、弹簧设计问题[131]、双杆桁架设计问题[132]、汽车侧面撞击问题[133]、减速机设计问题[134]、简支撑工字钢设计问题[135]、散货船设计问题[136]、压力容器设计问题[137]、反应堆网络设计问题[138]、振动平台设计问题[90]、工艺综合与设计问题[92]、软件工程[93]、焊接梁设计[95]、天线设计优化[98]、机器人抓手优化[99]、无线传感器位置优化设计[1]、双源压缩空气管道优化问题[100]、伺服转盘系统[103]、车削工艺参数优化[139]、铝锂合金薄壁工件设计[140]。

Kohli and Arora [97] introduced chaos theory into the GWO algorithm with the aim of accelerating its global convergence speed to solve the spring design problem, gear train design problem, welded beam design problem, pressure vessel design problem. Ming et al. [109] proposed a novel constrained multi-objective optimization evolutionary algorithm with enhanced mating and environmental selections (CMME) for gearbox design and heat exchanger network design problems. Zou et al. [125] proposed a dual-population constrained optimization algorithm based on surrogate evolution and degeneration (CAEAD) to solve the heat exchanger network design problem. Kong et al. [46] proposed a dynamic dual-population cooperative evolutionary multi-objective evolutionary algorithm to solve the disc brake design problem and the dual-source compressed air pipeline optimization problem. Yang et al. [141] proposed a novel CMOEA assisted by an additional objective function (CMAOO) to solve the disc brake problem. Gu et al. [142] proposed an improved constrained dominance principle (ICDP) embedded in MOEA/D (MOEA/D-ICDP) to solve the spring design problem and the disc brake problem. Yuan et al. [143] proposed a criterion-based constrained evolutionary algorithm (CCEA) for evaluating the value of infeasible solutions to solve the twin-rod truss design problem. Xia et al. [65] proposed a novel two-stage resource allocation CMOEA (CMOEA-TSRA) to solve the car side impact problem, the reducer design problem and the disc brake problem. Zou et al. [64] proposed a more flexible two-stage evolutionary algorithm based on automatic adjustment to solve the disc brake design problem, the simply supported I-beam design problem, the gearbox design problem, and the car side impact problem.
Kohli 和 Arora [97] 将混沌理论引入 GWO 算法，旨在加快其全局收敛速度，以解决弹簧设计问题、齿轮系设计问题、焊接梁设计问题和压力容器设计问题。Ming 等人[109]针对齿轮箱设计和热交换器网络设计问题提出了一种具有增强配对和环境选择（CMME）的新型约束多目标优化进化算法。Zou 等人[125]提出了一种基于代理进化和退化的双种群约束优化算法（CAEAD）来解决热交换器网络设计问题。Kong 等[46]提出了一种动态双群合作进化多目标进化算法，用于解决盘式制动器设计问题和双源压缩空气管道优化问题。Yang 等人[141]提出了一种由附加目标函数（CMAOO）辅助的新型 CMOEA，用于解决盘式制动器问题。Gu 等人[142] 提出了一种嵌入 MOEA/D 的改进约束支配原理（ICDP），用于解决弹簧设计问题和盘式制动器问题。Yuan 等人[143] 提出了一种基于准则的约束进化算法 (CCEA)，用于评估不可行解的值，以解决双杆桁架设计问题。Xia 等人[65] 提出了一种新型的两阶段资源分配 CMOEA（CMOEA-TSRA），用于解决汽车侧面碰撞问题、减速器设计问题和盘式制动器问题。Zou 等人[64]提出了一种基于自动调整的更灵活的两阶段进化算法，用于解决盘式制动器设计问题、简支撑工字钢设计问题、变速箱设计问题和汽车侧面碰撞问题。

Moreover, Yang et al. [101] proposed a multi-objective differential evolution algorithm based on dominance and constraint handling switching mechanism (MODE-CHS) to solve the disc brake design problem and the dual-source compressed air pipeline optimization problem. Sun et al. [60] proposed a multi-stage CMOEA with constraint condition priority (C3M) to solve the bulk carrier design, pressure vessel design, and reactor network design problems. Ma et al. [56] proposed a multi-stage evolutionary algorithm with stepwise addition of constraints (MSCMO) to solve the reducer design problem, the disc brake design problem, and the car side impact problem. Bao et al. [91] proposed an archive-based two-stage evolutionary algorithm (AT-CMOEA) to solve the vibration platform design problem and the process synthesis and design problem. Xiang et al. [94] proposed a tradeoff model (ToM) to solve the constrained optimal software product selection problem in search-based software engineering. Tian et al. [52] proposed CMOEA with a two-stage framework (CMOEA-MS) to solve the car side impact problem and the vibration platform design problem. To effectively handle CMOPs with complex infeasible regions, Qin et al. [96] proposed a dual-archive assisted push-pull evolutionary algorithm (PPTA) to solve the welded beam design problem. Huang et al. [68] proposed a CMOEA framework based on global and local FS search (GLS-CMOEA) to solve the reducer design problem, the welded beam design problem, and the disc brake design problem. To solve high-constrained multi-objective optimization problems, Jiao et al. [57] proposed two types of weight adjustment methods based on MOEA/D (MOEA/D-2WA) to solve the antenna design optimization problem.
此外，Yang 等人[101] 提出了一种基于支配和约束处理切换机制的多目标差分进化算法（MODE-CHS），用于解决盘式制动器设计问题和双源压缩空气管道优化问题。Sun 等人[60] 提出了一种具有约束条件优先权（C3M）的多阶段 CMOEA，用于解决散货船设计、压力容器设计和反应器网络设计问题。Ma 等人[56]提出了一种逐步添加约束的多阶段进化算法（MSCMO），用于解决减速器设计问题、盘式制动器设计问题和汽车侧面碰撞问题。Bao等人[91]提出了一种基于归档的两阶段进化算法（AT-CMOEA）来解决振动平台设计问题和工艺综合设计问题。Xiang 等人[94]提出了一种权衡模型（ToM），用于解决基于搜索的软件工程中的约束最优软件产品选择问题。Tian 等人[52]提出了两阶段框架（CMOEA-MS）的 CMOEA，用于解决汽车侧面碰撞问题和振动平台设计问题。为了有效处理具有复杂不可行区域的 CMOP，Qin 等人[96]提出了一种双存档辅助推拉进化算法（PPTA）来解决焊接梁设计问题。Huang 等人[68]提出了一种基于全局和局部 FS 搜索（GLS-CMOEA）的 CMOEA 框架，用于解决减速器设计问题、焊接梁设计问题和盘式制动器设计问题。为了解决高约束多目标优化问题，Jiao 等人[57] 提出了两种基于 MOEA/D 的权重调整方法（MOEA/D-2WA）来解决天线设计优化问题。

Besides, Song et al. [48] proposed an adaptive

ɛ

-constraint multi-objective optimization and adopted weak cooperative evolution, to solve the bulk carrier design problem and the reactor network design problem. Jiao et al. [37] proposed a multi-task optimization framework to solve the antenna array synthesis problem. Fan et al. [18] proposed an improved

ɛ

-constraint handling push-pull search framework for robot gripper optimization problem. Yu et al. [1] proposed a dual-population constrained multi-objective optimization algorithm (DCCMO) to find the optimal wireless sensor location. Yang et al. [102] proposed a multi-objective differential evolution algorithm based on partition selection (MODE-PS), to solve the dual-source compressed air pipeline optimization problem. Zhang et al. [103] proposed a switching nonlinear autoregressive moving average model (MOPSO) for continuous electromechanical servo turntable system, and developed a multi-objective particle swarm optimization algorithm, which implements the global best position selection of each particle and the maintenance of the external archive in the same program, to optimize the performance of the servo turntable system. Gadagi and Adake [139] considered the process parameters such as spindle speed, cutting depth and feed rate, and performed turning on glass fiber reinforced composite materials, and optimized the turning process parameters by genetic algorithm and particle swarm optimization techniques, to improve the turning quality and efficiency. Yue et al. [140] studied the impact of milling parameters on surface roughness and specific cutting energy. They developed a multi-objective optimization model and employed a multi-objective particle swarm optimization algorithm with a chaotic local search and adaptive inertia weights to find the best milling parameter combination.
此外，Song 等人[48] 提出了一种自适应

ɛ

约束多目标优化方法，并采用弱合作进化，解决了散装载波设计问题和电抗器网络设计问题。Jiao 等人[37] 提出了一种多任务优化框架来解决天线阵列合成问题。Fan 等人[18] 针对机器人抓手优化问题提出了一种改进的

ɛ

约束处理推拉搜索框架。Yu 等人[1]提出了一种双人口约束多目标优化算法（DCCMO）来寻找最佳无线传感器位置。Yang等[102]提出了一种基于分区选择的多目标差分进化算法（MODE-PS），用于解决双源压缩空气管道优化问题。Zhang 等人[103]提出了连续机电伺服转台系统的切换非线性自回归移动平均模型（MOPSO），并开发了多目标粒子群优化算法，在同一程序中实现了每个粒子的全局最佳位置选择和外部档案的维护，优化了伺服转台系统的性能。Gadagi 和 Adake[139]考虑了主轴转速、切削深度和进给速率等工艺参数，对玻璃纤维增强复合材料进行了车削加工，并通过遗传算法和粒子群优化技术对车削工艺参数进行了优化，提高了车削质量和效率。Yue 等人[140]研究了铣削参数对表面粗糙度和比切削能的影响。他们建立了一个多目标优化模型，并采用了一种带有混沌局部搜索和自适应惯性权重的多目标粒子群优化算法来寻找最佳铣削参数组合。

3.2. Resource optimization problems
3.2.资源优化问题

As the concept of environmental protection becomes more popular, resource optimization problems have attracted widespread attention. Effectively utilizing resources, reducing resource waste, and minimizing environmental pollution are the objectives of a constrained multi-objective optimization problem. It mainly includes water resources optimization [144], green mix design of rubberized concrete [145], integrated coal mine energy system [146], wastewater control problem [147], hybrid renewable energy systems [148], wind farm layout optimization [2], water resource management problem [149], water distribution network optimization [104], groundwater remediation design [105], thermocline filling layer thermal energy storage [106], test resource allocation optimization [107], CCHP microgrids optimization [29], EDP [75].
随着环保理念的深入人心，资源优化问题引起了广泛关注。有效利用资源、减少资源浪费、降低环境污染是约束多目标优化问题的目标。它主要包括水资源优化[144]、橡胶混凝土绿色混合设计[145]、煤矿综合能源系统[146]、废水控制问题[147]、混合可再生能源系统[148]、风电场布局优化[2]、水资源管理问题[149]、配水管网优化[104]、地下水修复设计[105]、温跃层填充层热能存储[106]、试验资源配置优化[107]、CCHP 微电网优化[29]、EDP[75]等。

Tian et al. [52] proposed a two-stage evolutionary algorithm that adapts the fitness assessment strategy during the evolutionary process to balance objective optimization and constraint satisfaction for solving water resource optimization problems. Golafshani et al. [145] used machine learning ensemble models and multi-objective gray wolf optimizer to estimate the composition of rubber concrete and achieve its green design. Wang et al. [79] proposed a migration-based method of enriching the algorithm library. This method selects and adds suitable algorithms to the library based on problem similarity and algorithm performance, and then uses Softmax regression to create an optimal algorithm combination that solves the coal mine comprehensive energy system optimization problem. Hou et al. [150] proposed a multi-state constrained multi-objective differential evolution with variable neighborhood strategy (MSCMODE-VNS) for wastewater control optimization. Ming et al. [148] designed a CMOEA based on two-stage and two-population (DD-CMOEA) for hybrid renewable energy system (HRES) optimization. Yin et al. [151] developed a code design optimization framework, jointly optimizing the wind farm layout and pumped storage power station configuration, and constructed an adaptive neuro-fuzzy inference system (ANFIS) trained by evolution to predict the power output of offshore wind farms. Qin et al. [96] proposed a dual-archive assisted push-pull evolutionary algorithm to solve the water resources management problem.
Tian等人[52]提出了一种两阶段进化算法，在进化过程中调整适合度评估策略，以平衡目标优化和约束满足，用于解决水资源优化问题。Golafshani 等人[145]利用机器学习集合模型和多目标灰狼优化器来估计橡胶混凝土的成分并实现其绿色设计。Wang 等人[79]提出了一种基于迁移的丰富算法库的方法。该方法根据问题相似度和算法性能，选择合适的算法并添加到算法库中，然后使用 Softmax 回归创建最优算法组合，解决煤矿综合能源系统优化问题。Hou等[150]提出了一种用于污水控制优化的多状态约束多目标微分进化与可变邻域策略（MSCMODE-VNS）。Ming 等人[148] 设计了一种基于两阶段和两群体的 CMOEA（DD-CMOEA），用于混合可再生能源系统（HRES）优化。Yin 等人[151]开发了一个代码设计优化框架，联合优化了风电场布局和抽水蓄能电站配置，并构建了一个经过进化训练的自适应神经模糊推理系统（ANFIS）来预测海上风电场的功率输出。Qin 等人[96]提出了一种双档案辅助推拉进化算法来解决水资源管理问题。

Moreover, Li et al. [39] proposed a parameter-free dual-archive evolutionary algorithm (C-TAEA), and developed a constrained mating selection mechanism, to solve the water distribution network optimization problem. Ouyang et al. [105] proposed a multi-algorithm gene adaptive multi-objective method (AMALGAM), as a multi-objective optimization solver, by incorporating the uncertainty of surrogate modeling into the optimization model through chance-constrained programming (CCP), to solve the groundwater remediation design problem of high-density non-aqueous phase liquid contaminated sites. Marti et al. [106] used constrained multi-objective optimization method to optimize the energy efficiency and material cost of a packed bed thermal energy storage system with air as the heat transfer fluid. Su et al. [107] proposed a multi-objective test resource allocation problem model with preset reliability, and developed several enhanced constraint handling techniques (ECHTs) derived from the new lower bound, to correct and reduce the violation of constraints, to solve the test resource allocation optimization problem. Yang et al. [29] developed a model for multi-objective optimal dispatch in CHP microgrids. The model integrates renewable energy sources, energy storage systems, and incentive demand response to enable economic optimization and peak load reduction. The problem was solved using an mixed-integer linear programming (MILP) based on the extended

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-constraints approach. Rauf et al. [75] proposed a multiple swarm-based chaotic differential evolutionary algorithm (MPC-DE) for the EDP.
此外，Li 等人[39] 提出了一种无参数双存档进化算法（C-TAEA），并开发了一种受约束的配对选择机制，用于解决配水管网优化问题。欧阳等[105]提出了一种多算法基因自适应多目标方法（AMALGAM），作为一种多目标优化求解器，通过机会约束编程（CCP）将代用模型的不确定性纳入优化模型，解决了高密度非水相液体污染场地的地下水修复设计问题。Marti 等人[106]采用约束多目标优化方法优化了以空气为传热流体的填料床蓄热系统的能效和材料成本。Su 等人[107]提出了一种具有预设可靠性的多目标试验资源分配问题模型，并开发了几种由新下限衍生的增强约束处理技术（ECHTs）来纠正和减少约束的违反，从而解决试验资源分配优化问题。Yang 等人[29] 建立了热电联产微电网多目标优化调度模型。该模型整合了可再生能源、储能系统和激励需求响应，以实现经济优化和峰值负荷降低。该问题采用基于扩展

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约束方法的混合整数线性规划（MILP）来解决。Rauf 等人[75] 针对 EDP 提出了一种基于多蜂群的混沌差分进化算法（MPC-DE）。

3.3. Scheduling optimization problems
3.3.调度优化问题

Scheduling optimization involves efficiently organizing tasks or activities within constrained time and resources to meet specific objectives. It mainly includes multi-product batch plant problem [108], dual-resource constrained flexible job shop scheduling problem [110], short-term crude oil scheduling problem [111], urban bus scheduling problem [112], CEED problem [114], ORPD problem [118].
调度优化涉及在有限的时间和资源范围内有效组织任务或活动，以实现特定目标。它主要包括多产品批量工厂问题[108]、双资源约束灵活作业车间调度问题[110]、短期原油调度问题[111]、城市公交调度问题[112]、CEED问题[114]、ORPD问题[118]。

Ming et al. [109] proposed a novel CMOEA, which improved the mechanisms of crossover and environment selection, adopted two innovative ranking strategies and a novel individual density estimation method, thus effectively solving the multi-product mixing plant problem. Zou et al. [64] proposed a more flexible two-phase evolutionary algorithm based on adaptive regulation, thus effectively solving the multi-product batch plant problem. Bao et al. [91] effectively solved the multi-product batch plant problem by proposing an archive-based two-phase evolutionary algorithm. Shi et al. [110] considered the problem of increased worker boredom caused by repetitive work assignments, and constructed an efficiency function to characterize the impact of worker boredom, established a two-layer dictionary model, with effectively completing all production tasks as the primary optimization objective, and improving worker job satisfaction as the secondary optimization objective, proposed a two-phase multi-objective particle swarm optimization algorithm that adopts a three-dimensional representation scheme (TS-MOPSO), to solve the dual-resource constrained flexible job shop scheduling problem. Hou et al. [111] proposed an adaptive enhanced selection pressure algorithm based on NSGA-II-CDP (NSGAII-APE), which could effectively enhance the selection pressure in the later iterations, thus effectively solving the problem of simultaneous processing of low-melting-point oil and high-melting-point oil.
Ming等人[109]提出了一种新型的CMOEA，改进了交叉和环境选择机制，采用了两种创新的排序策略和一种新型的个体密度估计方法，从而有效地解决了多产品混合工厂问题。Zou 等[64]提出了一种基于自适应调节的更灵活的两阶段进化算法，从而有效地解决了多产品批量工厂问题。Bao 等人[91]提出了一种基于档案的两阶段进化算法，有效地解决了多产品批量工厂问题。Shi 等人[110]考虑了重复性工作分配导致工人厌烦情绪增加的问题，构建了表征工人厌烦情绪影响的效率函数，建立了以有效完成所有生产任务为首要优化目标、以提高工人工作满意度为次要优化目标的双层字典模型，提出了采用三维表示方案的两阶段多目标粒子群优化算法（TS-MOPSO），解决了双资源约束柔性作业车间调度问题。Hou等[111]提出了一种基于NSGA-II-CDP的自适应增强选择压力算法（NSGAII-APE），该算法能有效增强后期迭代的选择压力，从而有效解决了低熔点油和高熔点油同时加工的问题。

Moreover, Ma and Wang [113] proposed a constraint handling technique called shift-based penalty (ShiP) to solve the vehicle scheduling problem of urban bus routes. Chen et al. [115] proposed a constrained multi-objective population extremal optimization algorithm named CMOPEO-EED, which found the Pareto optimal solutions of the renewable energy multi-objective economic emission dispatch problem by minimizing the total cost and environmental emission, two conflicting objectives. El-Shorbagy and Mousa [116] proposed a new optimization system named constrained multi-objective equilibrium optimizer, which combined the dominance criterion to handle the multi-objective functions, and reduced the size of the PF to a reasonable one by clustering analysis, applied repair methods to handle the particles’ constraint conditions and feasibility, thus better solving the CEED problem. Yu et al. [117] designed a new DE mutation strategy and Gaussian mutation to maintain the convergence and diversity of feasible solutions, thus solving the CEED problem. Mohseni-Bonab et al. [118] studied the stochastic multi-objective ORPD (SMO-ORPD) problem in wind-integrated power systems, considering the uncertainty of load and wind power generation, and used the

ɛ

-constraint method and fuzzy satisfaction method to select the best trade-off solution.
此外，Ma 和 Wang[113]提出了一种约束处理技术，称为基于移位的惩罚（ShiP），用于解决城市公交线路的车辆调度问题。Chen 等人[115]提出了一种名为 CMOPEO-EED 的约束多目标群体极值优化算法，通过最小化总成本和环境排放这两个相互冲突的目标，找到了可再生能源多目标经济排放调度问题的帕累托最优解。El-Shorbagy 和 Mousa[116]提出了一种名为约束多目标均衡优化器的新优化系统，该优化器结合支配准则处理多目标函数，并通过聚类分析将 PF 的大小减小到合理范围，应用修复方法处理粒子的约束条件和可行性，从而更好地解决了 CEED 问题。Yu等[117]设计了一种新的DE突变策略和高斯突变，以保持可行解的收敛性和多样性，从而解决了CEED问题。Mohseni-Bonab 等[118]研究了风电一体化电力系统中的随机多目标 ORPD（SMO-ORPD）问题，考虑了负荷和风力发电的不确定性，采用

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约束法和模糊满足法选择最佳折衷解。

3.4. Path planning problems
3.4.路径规划问题

Path planning involves identifying the most appropriate route from start to finish, considering various constraints and context-specific criteria such as distance, speed, safety, and energy efficiency. It mainly includes UAV planning [119], vehicle routing problems with time windows [152], path planning for emergency material scheduling [120], fuzzy business restricted multi-objective multi-index transportation problem [121], UAV path planning problem for disaster relief [82], robot trajectory planning [122], constrained multi-objective solid traveling salesman problems [123].
路径规划涉及确定从起点到终点的最合适路线，同时考虑各种约束条件和特定环境标准，如距离、速度、安全性和能效。它主要包括无人机规划问题[119]、带时间窗的车辆路由问题[152]、紧急物资调度的路径规划问题[120]、模糊业务受限多目标多指标运输问题[121]、用于救灾的无人机路径规划问题[82]、机器人轨迹规划问题[122]、受约束多目标固体旅行推销员问题[123]。

Peng and Qiu [119] proposed a decomposition-based constrained multi-objective evolutionary algorithm (M2M-DW) with a local infeasibility utilization mechanism for UAV path planning. Hou et al. [150] proposed a multi-state constrained multi-objective differential evolution algorithm with a variable neighborhood strategy to improve the search efficiency of complex feasible regions, which was used to solve the multi-objective vehicle routing problem with time windows. Tian et al. [17] proposed a weakly cooperative coevolutionary framework to improve the solving ability for the multi-objective vehicle routing problem with time windows through bi-directional information exchange. Wang et al. [120] established a drone-based emergency medical service queue location model, which used fuzzy theory to deal with the fuzziness of drone endurance time and demand arrival rate under the priority queuing strategy, and used multi-objective optimization methods to balance the total cost, system efficiency and fair response time, effectively solving the emergency medical service path planning problem. Latpate and Kurade [121] proposed a two-stage fuzzy business constrained multi-objective multi-index transportation problem, where the supply, demand and requirement were triangular fuzzy numbers, and adopted a fuzzy non-dominated sorting genetic algorithm, which used the alpha-cut technique to deal with the uncertainty of the problem parameters, and performed the frontier non-dominance sorting of the parent population according to the uncertainty level, effectively solving the fuzzy business constrained multi-objective multi-index transportation problem. Yu et al. [82] proposed a reinforcement learning-based multi-objective differential evolution algorithm, which can dynamically adjust the main parameters of the algorithm through reinforcement learning techniques, effectively solving the disaster relief drone path planning problem. Chen and Pham [122] proposed a technique based on constrained multi-objective genetic algorithm (MOGA) for solving the general electric motor-driven parallel manipulator problem. Maity et al. [123] proposed a rough multi-objective genetic algorithm (R-MOGA) for solving the constrained multi-objective solid traveling salesman problem in rough, fuzzy rough and stochastic rough environments.
Peng 和 Qiu [119] 针对无人机路径规划提出了一种基于分解的约束多目标进化算法（M2M-DW），该算法具有局部不可行性利用机制。Hou等[150]提出了一种多状态约束多目标差分进化算法，采用可变邻域策略提高复杂可行区域的搜索效率，用于解决带时间窗的多目标车辆路由问题。Tian等[17]提出了一种弱合作协同进化框架，通过双向信息交换提高了带时间窗的多目标车辆路由问题的求解能力。Wang 等[120]建立了基于无人机的急救医疗服务队列定位模型，利用模糊理论处理了优先排队策略下无人机续航时间和需求到达率的模糊性，并采用多目标优化方法平衡了总成本、系统效率和公平响应时间，有效解决了急救医疗服务路径规划问题。Latpate 和 Kurade[121]提出了一个两阶段模糊业务约束多目标多指标运输问题，其中供应量、需求量和要求量均为三角形模糊数，并采用模糊非支配排序遗传算法，利用阿尔法切技术处理问题参数的不确定性，根据不确定性程度对父种群进行前沿非支配排序，有效地解决了模糊业务约束多目标多指标运输问题。Yu 等人 [82]提出了一种基于强化学习的多目标差分进化算法，该算法可通过强化学习技术动态调整算法的主要参数，有效解决了救灾无人机路径规划问题。Chen 和 Pham[122]提出了一种基于约束多目标遗传算法（MOGA）的技术，用于解决普通电机驱动并联机械手问题。Maity 等人[123]提出了一种粗糙多目标遗传算法（R-MOGA），用于求解粗糙、模糊粗糙和随机粗糙环境下的约束多目标固体旅行推销员问题。

3.5. Other problems 3.5.其他问题

Here, we discuss some problems that do not belong to any of the previous categories. These include process synthesis problems [124], synchronous optimized pulse-width modulation problem for three-stage inverters [153], [154], [155], synchronous optimized pulse-width modulation problem for five-stage inverters [153], [154], [155], synchronous optimized pulse-width modulation problem for thirteen-stage inverters [153], [154], [155], polyester polymerization process problems [127], gasoline blending process problem [30], portfolio optimization [3], agricultural planting structure optimization [156], capacitated multi-facility location problem [157], the landing trajectory design of supersonic transport (SST) [73].
在此，我们将讨论一些不属于上述任何类别的问题。这些问题包括工艺合成问题 [124]、三级逆变器同步优化脉宽调制问题 [153]、[154]、[155]、五级逆变器同步优化脉宽调制问题 [153]、[154]、[155]、十三级逆变器同步优化脉宽调制问题 [153]、[154]、[155]、聚酯聚合工艺问题[127]、汽油混合工艺问题[30]、投资组合优化[3]、农业种植结构优化[156]、容纳性多设施选址问题[157]、超音速运输（SST）着陆轨迹设计[73]。

Zou et al. [125] proposed a dual-population constrained optimization algorithm based on the idea of surrogate evolution and degeneration, and can effectively solve the process synthesis problem and the synchronous optimization pulse width modulation problem of five-level and three-level inverters. Zou et al. [126] proposed a new multi-population CMOEA (MCCMO) that uses two new techniques: activation dormancy detection and combine occasion detection (COD). These techniques accelerate the optimization process and determine the best time to find SubCPF. As a result, they effectively solve the process synthesis problem and the synchronous optimization pulse width modulation problem of five-level and thirteen-level inverters. Zhu et al. [127] aimed at the four-objective optimization problem in the polyesterification process, proposed an interactive adaptive reference vector guided multi-objective optimization evolutionary algorithm (IARVEA). Han et al. [30] proposed a fuzzy constraint handling technique, which uses fuzzy set theory to accurately characterize the difference of solutions in objective function values and constraint violation degrees, and embeds it into MOEA/D (MOEA/D-FCHT), thus effectively solving the gasoline blending process problem.
Zou 等人[125]提出了一种基于代演化和退化思想的双种群约束优化算法，能有效解决五电平和三电平逆变器的过程合成问题和同步优化脉宽调制问题。Zou 等人[126]提出了一种新的多群体 CMOEA（MCCMO），它使用了两种新技术：激活休眠检测和结合场合检测（COD）。这些技术加速了优化过程，并确定了寻找 SubCPF 的最佳时间。因此，它们有效地解决了五级和十三级逆变器的过程合成问题和同步优化脉宽调制问题。Zhu 等人[127]针对聚酯化过程中的四目标优化问题，提出了一种交互式自适应参考向量引导的多目标优化进化算法（IARVEA）。Han等人[30]提出了一种模糊约束处理技术，利用模糊集理论准确表征解在目标函数值和约束违反程度上的差异，并将其嵌入到MOEA/D（MOEA/D-FCHT）中，从而有效地解决了汽油混合工艺问题。

Moreover, Lwin et al. [3] extended the Markowitz mean–variance portfolio optimization model, and proposed an efficient learning-guided hybrid multi-objective evolutionary algorithm. This algorithm effectively solves the constrained portfolio optimization problem under the extended mean–variance framework. Peykani et al. [158] proposed a novel Fuzzy MultiPeriod Multi-Objective Portfolio Optimization (FMPMOPO) model that is capable to be used under data ambiguity and practical constraints including budget constraint, cardinality constraint, and bound constraint, and thus is applicable to the presence of uncertain data in portfolio optimization. Yang et al. [156] utilized the SIMM method to handle fuzzy numbers in the constraints. They then introduced a linear objective planning method to solve the problem of fractional objectives as fuzzy objectives, in order to optimize the agricultural planting structure. Ozgen and Gulsun [157] addressed the multi-objective, capacitated, multi-facility location problem by integrating multiple perspectives to evaluate quantitative and qualitative factors and by combining a two-stage probabilistic linear programming approach with fuzzy hierarchical analysis. Takubo and Kanazaki [73] proposed an integrated optimization method based on CMOEA and non-intrusive PCE for the landing trajectory design of supersonic transport with wind uncertainty.
此外，Lwin 等人[3] 扩展了马科维茨均值-方差投资组合优化模型，并提出了一种高效的学习引导混合多目标进化算法。该算法有效地解决了扩展均值方差框架下的约束组合优化问题。Peykani 等人[158]提出了一种新颖的模糊多期多目标组合优化（Fuzzy MultiPeriod Multi-Objective Portfolio Optimization，FMPMOPO）模型，该模型能够在数据模糊性和实际约束条件（包括预算约束、卡数约束和约束条件）下使用，因此适用于存在不确定数据的组合优化。Yang 等人[156]利用 SIMM 方法处理了约束条件中的模糊数。然后，他们引入了线性目标规划方法来解决作为模糊目标的分数目标问题，以优化农业种植结构。Ozgen 和 Gulsun [157] 通过综合多角度评估定量和定性因素，并将两阶段概率线性规划方法与模糊层次分析法相结合，解决了多目标、有能力、多设施选址问题。Takubo 和 Kanazaki [73] 提出了一种基于 CMOEA 和非侵入式 PCE 的综合优化方法，用于具有风不确定性的超音速运输着陆轨迹设计。

4. Future research directions
4.未来研究方向

CMOAs have achieved success on multiple CMOPs, but there are still several research directions to explore in the field of constrained multi-objective optimization. This section introduces the important topics in this area.
CMOA 在多个 CMOP 上取得了成功，但在约束多目标优化领域仍有多个研究方向有待探索。本节将介绍这一领域的重要课题。

4.1. Feature extraction based on constrained multi-objective optimization
4.1.基于约束多目标优化的特征提取

Section 3 discussed how feature extraction affects the performance of the machine learning-based methods for CMOPs. Feature extraction is the process of extracting information that reflects the essence and characteristics of a problem from the original data. [19], [76], [77] and some other studies have proposed methods for feature extraction, but feature extraction for CMOPs still faces these difficulties:
第 3 节讨论了特征提取如何影响基于机器学习的 CMOP 方法的性能。特征提取是从原始数据中提取反映问题本质和特征的信息的过程。虽然[19]、[76]、[77]和其他一些研究已经提出了特征提取的方法，但用于 CMOP 的特征提取仍然面临着这些困难：

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The feature space of CMOPs is usually high-dimensional, containing various information such as objective functions, constraints, and decision variables. Such a feature space makes the computational complexity of feature extraction high and increases the difficulty of feature selection.
CMOP 的特征空间通常是高维的，包含目标函数、约束条件和决策变量等各种信息。这样的特征空间使得特征提取的计算复杂度很高，也增加了特征选择的难度。
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The features of CMOPs often have correlation or redundancy, affecting the effectiveness and stability of feature extraction. The correlation between features indicates a certain statistical relationship between features, for example, linear correlation, nonlinear correlation, or monotonic correlation.
CMOP 的特征往往具有相关性或冗余性，影响特征提取的有效性和稳定性。特征之间的相关性表示特征之间存在某种统计关系，例如线性相关、非线性相关或单调相关。
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The features of CMOPs often have nonlinear or non-convex properties, resulting in the non-uniqueness or incomparability of the feature extraction results. The nonlinearity or non-convexity of features implies that the shape or structure of the feature space does not conform to the linear or convex assumptions, for example, curves, surfaces, or multi-peaks.
CMOP 的特征通常具有非线性或非凸特性，从而导致特征提取结果的非唯一性或不可比性。特征的非线性或非凸性意味着特征空间的形状或结构不符合线性或凸性假设，例如曲线、曲面或多峰。
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The features of CMOPs often change with the problem, requiring the feature extraction method to have adaptability and generalization. The change of features indicates that the feature space changes with the input, output, parameters, objectives, and constraints of the problem, for example, dynamic, uncertain, or multimodal.
CMOP 的特征往往会随着问题的变化而变化，这就要求特征提取方法具有适应性和通用性。特征的变化表明特征空间会随着问题的输入、输出、参数、目标和约束条件的变化而变化，例如动态、不确定或多模态。

No universal feature extraction method exists that extracts the features of all constrained multi-objective optimization problems. To address this problem, we propose borrowing the word vector generation methods from natural language processing, for example, Word2Vec [159], FastText [160], ELMo [161], and BERT [162]. We believe that BERT is a promising direction, as it constructs a similar CMOPs feature extraction pre-training model that meets various downstream CMOPs tasks. We believe that graph neural network (GNN) [163] better represents the relationships between individuals in the population, and GNN is a promising direction. GNN is a kind of neural network that handles graph-structured data, using the topology and node attributes of the graph to learn the feature representation of the nodes. We view the population of CMOPs as a graph, in which each individual is a node, the similarity or distance between each individual is an edge, the objective value or constraint violation of each individual is a node attribute, and use GNN to learn the feature representation of the population, extracting features that are helpful for CMOPs solutions.
目前还没有一种通用的特征提取方法可以提取所有约束多目标优化问题的特征。为解决这一问题，我们建议借鉴自然语言处理中的词向量生成方法，例如 Word2Vec [159]、FastText [160]、ELMo [161] 和 BERT [162]。我们认为 BERT 是一个很有前景的方向，因为它构建了一个类似于 CMOPs 特征提取的预训练模型，可以满足 CMOPs 的各种下游任务。我们认为图神经网络（GNN）[163] 更好地代表了种群中个体之间的关系，因此 GNN 是一个很有前景的方向。GNN 是一种处理图结构数据的神经网络，它利用图的拓扑结构和节点属性来学习节点的特征表示。我们将 CMOPs 群体视为一个图，其中每个个体是一个节点，每个个体之间的相似度或距离是一条边，每个个体的目标值或违反约束的情况是一个节点属性，并利用 GNN 学习群体的特征表示，提取有助于 CMOPs 解决方案的特征。

4.2. Autonomous recommendation of constrained multi-objective optimization algorithms
4.2.约束多目标优化算法的自主推荐

According to the no free lunch theorem [88], no algorithm has a performance advantage on one class of problems, that does not turn into a disadvantage on another class of problems. Therefore, to solve different constrained multi-objective optimization problems, it is necessary to select appropriate algorithms according to the characteristics of the problems. Currently, there are two types of methods for autonomous recommendation of algorithms, one is to automatically match the corresponding algorithms for different problems [76], [77], [79], the other is to automatically match the corresponding constraint handling techniques or meta-heuristic algorithms according to the state of the population during the evolution [19], [78], [80]. These two methods have their own advantages and disadvantages: the former can adapt to diverse problems, but requires a lot of prior knowledge and human intervention; the latter can adaptively adjust the algorithm parameters and strategies, but requires longer running time and more computing resources. However, there is no algorithm that combines these two methods, which may improve the efficiency and effectiveness of autonomous recommendation. With the development of deep learning, large language models have achieved remarkable results in the field of natural language processing. Liu et al. [89] combined large language models with multi-objective optimization, using suitable prompt engineering, to make the general large language models act as black-box search operators for MOEA/D with zero-shot learning, showing powerful performance. Their method can effectively handle the balance between constraints and objectives in multi-objective optimization problems, while using the strong expressive power of large language models to improve the search efficiency and quality of solutions. However, there is still a lack of research on the combination of constrained multi-objective optimization and large language models. We believe that this is a new direction, by combining large language models for algorithm autonomous recommendation. Specifically, we can consider the following aspects:
根据 "天下没有免费的午餐 "定理[88]，任何算法在一类问题上的性能优势，在另一类问题上都不会变成劣势。因此，要解决不同的约束多目标优化问题，必须根据问题的特点选择合适的算法。目前，自主推荐算法的方法有两种，一种是针对不同问题自动匹配相应的算法[76]、[77]、[79]，另一种是根据种群在演化过程中的状态自动匹配相应的约束处理技术或元启发式算法[19]、[78]、[80]。这两种方法各有利弊：前者能适应各种问题，但需要大量的先验知识和人工干预；后者能自适应地调整算法参数和策略，但需要更长的运行时间和更多的计算资源。然而，目前还没有一种算法能将这两种方法结合起来，从而提高自主推荐的效率和效果。随着深度学习的发展，大型语言模型在自然语言处理领域取得了显著的成果。Liu 等人[89]将大语言模型与多目标优化相结合，采用合适的提示工程，使一般大语言模型充当零点学习的 MOEA/D 的黑盒搜索算子，显示出强大的性能。他们的方法可以有效处理多目标优化问题中约束和目标之间的平衡，同时利用大语言模型的强大表达能力提高搜索效率和求解质量。然而，关于约束多目标优化与大型语言模型相结合的研究还很缺乏。我们认为，将大语言模型与算法自主推荐相结合是一个新的方向。具体来说，我们可以考虑以下几个方面：

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How to use the pre-training and fine-tuning capabilities of large language models to extract the features and semantic information of constrained multi-objective optimization problems, to facilitate the decision-making of autonomous algorithm recommendation.
如何利用大语言模型的预训练和微调能力，提取约束多目标优化问题的特征和语义信息，促进自主算法推荐的决策。
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How to design suitable prompts, so that the large language models can generate appropriate algorithms or parameters, as well as corresponding constraint handling techniques or meta-heuristic algorithms, according to the different problems and population states.
如何设计合适的提示，使大语言模型能够根据不同的问题和群体状态，生成合适的算法或参数，以及相应的约束处理技术或元启发式算法。
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How to evaluate and optimize the performance of the large language models on constrained multi-objective optimization problems.
如何在受约束的多目标优化问题上评估和优化大型语言模型的性能。

4.3. Dynamic constrained multi-objective problems
4.3.动态约束多目标问题

Dynamic constrained multi-objective problems (DCMOPs) refer to multi-objective optimization problems in which the objective functions, constraints, or parameters change over time in dynamic environments. These problems have multiple conflicting objectives and constraints, and require the finding of optimal solutions under different environmental states. DCMOPs have a wide range of application scenarios in engineering, transportation, intelligent manufacturing, and so on, such as fluid catalytic cracking operation optimization [164], ore dressing process operation index optimization [165], and optimal control problems [166]. The main difficulty in solving DCMOPs is how to ensure the convergence and diversity of the population in dynamic environments. This means being able to track the dynamic Pareto optimal solution set in a timely manner, while maintaining the uniform and wide coverage of the solution set. At present, some dynamic CMOEAs have been proposed [167], [168], [169], [170], that are mainly divided into two categories: diversity introduction methods and prediction methods [171]. Diversity introduction methods [172], [173] improve the adaptability of the algorithm by increasing the diversity of the population, but may reduce the convergence speed. Prediction methods [174], [175] use prediction models to predict the advantageous population in the changing environment, which can significantly improve the convergence of the population, but the performance of the prediction model is affected by the training cycle. Therefore, how to design more efficient algorithms is still a worthwhile research problem.
动态约束多目标问题（DCMOPs）是指目标函数、约束条件或参数在动态环境中随时间变化的多目标优化问题。这些问题具有多个相互冲突的目标和约束条件，需要在不同的环境状态下找到最优解。DCMOP 在工程、交通、智能制造等领域有着广泛的应用场景，如流体催化裂化操作优化 [164]、选矿工艺操作指标优化 [165] 和最优控制问题 [166]。求解 DCMOPs 的主要困难在于如何在动态环境中确保群体的收敛性和多样性。这意味着要能及时跟踪动态帕累托最优解集，同时保持解集的均匀性和广泛覆盖性。目前，一些动态 CMOEAs 已被提出 [167]、[168]、[169]、[170]，主要分为两类：多样性引入方法和预测方法 [171]。多样性引入法 [172]、[173] 通过增加种群的多样性来提高算法的适应性，但可能会降低收敛速度。预测方法[174]、[175]利用预测模型预测变化环境中的优势种群，可以显著提高种群的收敛速度，但预测模型的性能会受到训练周期的影响。因此，如何设计更高效的算法仍是一个值得研究的问题。

4.4. Dynamic interval constrained multi-objective problems
4.4.动态区间约束多目标问题

DCMOPs and interval multi-objective optimization problems (IMOPs) can be combined to form dynamic interval constrained multi-objective problems (DI-CMOPs). DI-CMOPs have applications in many practical scenarios, such as steelmaking continuous casting, hot rolling [176], robot path planning [177], multi-period portfolio selection problems [178], [179]. The objective functions of these problems are usually interval values, which means that traditional methods, such as selection based on non-inferior solutions, detection and response of optimization problem changes, and so on, are not applicable to these problems. In addition, as the range of interval parameters changes over time, the non-inferior solutions of IMOPs may become inferior solutions in DI-CMOPs. Therefore, solving DI-CMOPs requires addressing the following challenges: accurately detecting the changes of optimization problems, quickly tracking the PF after the changes, and providing candidate solutions with good diversity and approximation in a timely manner [180].
DCMOPs 和区间多目标优化问题 (IMOPs) 可以组合成动态区间约束多目标问题 (DI-CMOPs)。DI-CMOPs 在许多实际场景中都有应用，如炼钢连铸、热轧 [176]、机器人路径规划 [177]、多周期组合选择问题 [178]、[179]。这些问题的目标函数通常都是区间值，这意味着基于非劣解的选择、优化问题变化的检测和响应等传统方法不适用于这些问题。此外，随着时间的推移，区间参数的范围会发生变化，IMOP 的非劣解可能会变成 DI-CMOP 的劣解。因此，求解 DI-CMOPs 需要解决以下难题：准确检测优化问题的变化、快速跟踪变化后的 PF 以及及时提供具有良好多样性和近似性的候选解 [180]。

4.5. Large-scale constrained multi-objective problems
4.5.大规模约束多目标问题

Large-scale constrained multi-objective optimization problems (Large-Scale CMOPs) are optimization problems with a large number of decision variables and multiple conflicting objectives. These problems are very common in engineering, science, and management fields, such as multi-objective vehicle routing problem [181] and configuration software optimization [182], but traditional optimization methods or single evolutionary algorithms are often difficult to effectively solve them. Large-Scale CMOPs face the following main challenges:
大规模约束多目标优化问题（Large-Scale CMOPs）是指具有大量决策变量和多个冲突目标的优化问题。这类问题在工程、科学和管理领域非常常见，如多目标车辆路由问题 [181] 和配置软件优化 [182]，但传统优化方法或单一进化算法往往难以有效解决。大规模 CMOP 面临以下主要挑战：

•
Exponential growth of search space: As the number of decision variables increases, the size of the search space grows exponentially, resulting in the sparsity, diversity reduction of feasible solutions, and the complexity, uncertainty increase of the PF.
搜索空间的指数增长：随着决策变量数量的增加，搜索空间的大小呈指数增长，导致可行方案的稀疏性和多样性减少，以及 PF 的复杂性和不确定性增加。
•
Difficulty of constraint handling: In Large-Scale CMOPs, the number and complexity of constraints may also be high, that leads to the uneven and discontinuous distribution of feasible solutions, and the fuzzy and irregular constraint boundaries. These make it difficult for evolutionary algorithms to satisfy and optimize the constraints. Moreover, the evaluation of constraints may also be very expensive, which limits the computational cost and efficiency of evolutionary algorithms.
约束处理困难：在大型 CMOP 中，约束条件的数量和复杂程度也可能很高，从而导致可行解的分布不均匀、不连续，约束边界模糊且不规则。这使得进化算法难以满足和优化约束条件。此外，约束条件的评估也可能非常昂贵，从而限制了进化算法的计算成本和效率。
•
Interactivity among variables: In Large-Scale CMOPs, there may be strong correlation or dependency among decision variables, that affects the effectiveness of mutation and crossover operations of evolutionary algorithms, and the applicability of decomposition and dimensionality reduction techniques.
变量之间的交互性：在大型 CMOP 中，决策变量之间可能存在很强的相关性或依赖性，这会影响进化算法中突变和交叉操作的效果，以及分解和降维技术的适用性。
•
Redundancy and irrelevance of high-dimensional features: In Large-Scale CMOPs, the dimension of decision variables is very high, but not all variables have significant impact on the objective functions and constraints, i.e., some variables are redundant or irrelevant. These variables increase the search difficulty and computational overhead of evolutionary algorithms, and reduce the performance and stability of evolutionary algorithms.
高维特征的冗余性和无关性：在大规模CMOP中，决策变量的维度非常高，但并非所有变量都对目标函数和约束条件有重大影响，即有些变量是冗余的或无关的。这些变量增加了进化算法的搜索难度和计算开销，降低了进化算法的性能和稳定性。

4.6. Robust constrained multi-objective problems
4.6.稳健约束多目标问题

Robust constrained multi-objective problems (RC-MOPs) [183], [184] are a type of optimization problem that aim to find a set of solutions that can satisfy multiple objective functions and constraints in uncertain environments. The goal of this kind of problem is to find solutions that are robust, i.e., that they can maintain the feasibility of the constraints and the quality of the objective functions even when the input data is noisy or perturbed. Solving this kind of problem faces the following challenges:
鲁棒约束多目标问题（RC-MOPs）[183], [184]是一种优化问题，其目的是在不确定的环境中找到一组能满足多个目标函数和约束条件的解决方案。这类问题的目标是找到稳健的解决方案，即即使输入数据存在噪声或扰动，也能保持约束条件的可行性和目标函数的质量。解决这类问题面临以下挑战：

•
How to define and measure robustness, i.e., how to quantify the degree of sensitivity of the solution to uncertainty, and how to assess the capability of the solution to withstand interference. Different definitions and measurement methods of robustness may result in different robust PFs.
如何定义和测量稳健性，即如何量化解决方案对不确定性的敏感程度，以及如何评估解决方案抵御干扰的能力。不同的稳健性定义和测量方法可能会产生不同的稳健性 PF。
•
How to handle constraints, i.e., how to ensure the robustness and diversity of the solution while satisfying the constraints. The presence of constraints increases the complexity and discontinuity of the search space, thus increasing the difficulty of solving.
如何处理约束条件，即如何在满足约束条件的同时确保解的稳健性和多样性。约束条件的存在增加了搜索空间的复杂性和不连续性，从而增加了求解的难度。

Therefore, it is necessary to design reasonable robustness mechanisms and robustness measurement methods, and combine them with constraint handling techniques, to balance convergence and diversity, and thus effectively solve robust CMOPs.
因此，有必要设计合理的鲁棒性机制和鲁棒性测量方法，并将其与约束处理技术相结合，平衡收敛性和多样性，从而有效求解鲁棒 CMOP。

4.7. Computationally intensive constrained multi-objective problems
4.7.计算密集型约束多目标问题

Computationally intensive constrained multi-objective problems (CICMOPs) are optimization problems with multiple possibly conflicting objective functions [185], [186]. They require a large amount of CPU time or memory space to solve, and they must satisfy some constraints. The calculation process of the objective function or the constraint function is very complex, which makes this kind of problem difficult. It may involve high-dimensional numerical integration, nonlinear differential equation solving, accurate physical simulation, and other techniques. These calculation processes usually require a lot of iteration or recursion, resulting in exponential growth of computation time and space. A typical example is the computational fluid dynamics problem. It requires solving the Navier–Stokes equation, a nonlinear partial differential equation system that describes the motion of fluids [187]. To solve this kind of problem effectively, researchers have proposed specialized methods, such as the surrogate model-based method [63], [188], [189], [190], [191]. The idea of this method is to use simple functions to approximate the complex objective function or constraint function, thereby reducing the computation cost and improving the solution efficiency. However, since the surrogate model is only approximate, it may have errors, and it needs to be constantly updated and improved. The computation cost of CICMOPs is very high. Therefore, parallel execution is an effective way to speed up [134]. It can use multi-core processors or distributed systems to execute multiple computation tasks simultaneously, and it can improve its scalability. However, when studying the parallelization mechanism of surrogate models, we need to consider the knowledge transfer: how to share and transfer useful information between different computation nodes, such as samples or model parameters, to improve the accuracy and stability of surrogate models.
计算密集型约束多目标问题（CICMOPs）是具有多个可能相互冲突的目标函数的优化问题 [185], [186]。解决这些问题需要大量的 CPU 时间或内存空间，而且必须满足某些约束条件。目标函数或约束函数的计算过程非常复杂，因此这类问题很难解决。它可能涉及高维数值积分、非线性微分方程求解、精确物理模拟等技术。这些计算过程通常需要大量的迭代或递归，导致计算时间和空间呈指数增长。计算流体动力学问题就是一个典型的例子。它需要求解纳维-斯托克斯方程，这是一个描述流体运动的非线性偏微分方程系统[187]。为了有效解决这类问题，研究人员提出了专门的方法，如基于代理模型的方法 [63]、[188]、[189]、[190]、[191]。这种方法的思路是用简单的函数来逼近复杂的目标函数或约束函数，从而降低计算成本，提高求解效率。但是，由于代用模型只是近似的，因此可能存在误差，需要不断更新和改进。CICMOP 的计算成本非常高。因此，并行执行是一种有效的加速方法 [134]。它可以利用多核处理器或分布式系统同时执行多个计算任务，并能提高其可扩展性。然而，在研究代用模型的并行化机制时，我们需要考虑知识传递：如何在不同计算节点之间共享和传递有用信息，如样本或模型参数，以提高代用模型的准确性和稳定性。

5. Conclusion 5.结论

This paper provides a comprehensive review of constrained multi-objective optimization problems, involving the state-of-the-art algorithms, applications, and future research directions. Specifically, it proposes a classification method that can scientifically divide algorithms based on their characteristics, comprising three categories. This classification focuses on mathematical methods, evolutionary algorithms, and machine learning methods, analyzing the advantages and disadvantages of each CMOA. Then, it introduces and summarizes some practical applications of constrained multi-objective optimization, providing an application paradigm for extending CMOAs to real-world scenarios. Furthermore, potential future research directions are discussed. Overall, this work provides a systematic and scientific investigation of CMOPs, and helps researchers quickly preview the latest research status.
本文全面回顾了约束多目标优化问题，涉及最先进的算法、应用和未来研究方向。具体而言，本文提出了一种分类方法，可根据算法的特点对其进行科学划分，包括三类算法。这种分类方法主要针对数学方法、进化算法和机器学习方法，分析了每种 CMOA 的优缺点。然后，介绍并总结了约束多目标优化的一些实际应用，为将 CMOA 扩展到现实世界的应用场景提供了一个应用范例。此外，还讨论了潜在的未来研究方向。总之，本研究对 CMOP 进行了系统而科学的探讨，有助于研究人员快速预览最新的研究状况。

CRediT authorship contribution statement
CRediT 作者贡献声明

Yuanyuan Hao: Supervision, Resources, Methodology, Writing – original draft, Writing – review & editing. Chunliang Zhao: Project administration, Methodology, Investigation. Yiqin Zhang: Data curation, Writing – review & editing. Yuanze Cao: Methodology, Investigation, Formal analysis, Data curation. Zhong Li: Supervision, Funding acquisition.
郝媛媛指导、资源、方法、写作--原稿、写作--审阅和编辑。赵春亮项目管理、方法论、调查。张益勤数据整理、写作--审阅和编辑。曹元泽方法学、调查、形式分析、数据整理。李忠：监督、获取资金。

Declaration of competing interest
利益冲突声明

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
作者声明，他们没有任何可能会影响本文所报告工作的已知经济利益或个人关系。

Acknowledgments 致谢

This work was supported in part by the Shandong Provincial Natural Science Foundation under grant ZR2023QF065, the Natural Science Foundation of Fujian Province (No. 2020J01800), and the PPP Program of DAAD , Germany under the grant number 57653779.
本文部分研究工作得到了山东省自然科学基金（ZR2023QF065）、福建省自然科学基金（2020J01800）和德国德意志学术交流中心（DAAD）PPP项目（57653779）的资助。

Data availability 数据可用性

No data was used for the research described in the article.
文章所述研究未使用任何数据。

References

[1]
Yu Q., Yang C., Dai G., Peng L., Chen X.
Synchronous wireless sensor and sink placement method using dual-population coevolutionary constrained multi-objective optimization algorithm
IEEE Trans. Ind. Inform. (2022)
Google Scholar
[2]
Sorkhabi S.Y.D., Romero D.A., Beck J.C., Amon C.H.
Constrained multi-objective wind farm layout optimization: Novel constraint handling approach based on constraint programming
Renew. Energy, 126 (2018), pp. 341-353
Google Scholar
[3]
Lwin K., Qu R., Kendall G.
A learning-guided multi-objective evolutionary algorithm for constrained portfolio optimization
Appl. Soft Comput., 24 (2014), pp. 757-772
View PDF View article View in Scopus Google Scholar
[4]
Wang Y., Zixing C., Yuren Z., Chixin X.
Constrained optimization evolutionary algorithms
J. Softw., 20 (1) (2009), pp. 11-29
Google Scholar
[5]
Liu Z.-Z., Wang Y., Wang B.-C.
Indicator-based constrained multiobjective evolutionary algorithms
IEEE Trans. Syst. Man. Cybern. Syst., 51 (9) (2021), pp. 5414-5426, 10.1109/TSMC.2019.2954491
View in Scopus Google Scholar
[6]
Li W., Gong W., Ming F., Wang L.
Constrained multi-objective evolutionary algorithm with an improved two-archive strategy
Knowl.-Based Syst., 246 (2022), Article 108732
View PDF View article View in Scopus Google Scholar
[7]
Xia M., Dong M.
A novel two-archive evolutionary algorithm for constrained multi-objective optimization with small feasible regions
Knowl.-Based Syst., 237 (2022), Article 107693
View PDF View article View in Scopus Google Scholar
[8]
Ma Z., Wang Y.
Evolutionary constrained multiobjective optimization: Test suite construction and performance comparisons
IEEE Trans. Evol. Comput., 23 (6) (2019), pp. 972-986
Crossref View in Scopus Google Scholar
[9]
Fan Z., Li W., Cai X., Li H., Wei C., Zhang Q., Deb K., Goodman E.
Difficulty adjustable and scalable constrained multiobjective test problem toolkit
Evol. Comput., 28 (3) (2020), pp. 339-378
Crossref Google Scholar
[10]
Fan Z., Li W., Cai X., Huang H., Fang Y., You Y., Mo J., Wei C., Goodman E.
An improved epsilon constraint-handling method in MOEA/D for CMOPs with large infeasible regions
Soft Comput., 23 (2019), pp. 12491-12510
Crossref View in Scopus Google Scholar
[11]
Hsiao Y.-T., Chiang H.-D., Liu C.-C., Chen Y.-L.
A computer package for optimal multi-objective VAR planning in large scale power systems
IEEE Trans. Power Syst., 9 (2) (1994), pp. 668-676
View in Scopus Google Scholar
[12]
Fu L., Fan J., Yuanchun L., Yantao T., Yisong D.
Study and application of a constrained multi-objective optimization algorithm
Proceedings of the IEEE International Vehicle Electronics Conference (IVEC’99)(Cat. No. 99EX257), IEEE (1999), pp. 305-307
View in Scopus Google Scholar
[13]
Coello C.A.C., Brambila S.G., Gamboa J.F., Tapia M.G.C.
Multi-objective evolutionary algorithms: Past, present, and future
Pardalos P.M., Rasskazova V., Vrahatis M.N. (Eds.), Black Box Optimization, Machine Learning, and No-Free Lunch Theorems, Springer International Publishing, Cham (2021), pp. 137-162
Crossref View in Scopus Google Scholar
[14]
Liang J., Ban X., Yu K., Qu B., Qiao K., Yue C., Chen K., Tan K.C.
A survey on evolutionary constrained multiobjective optimization
IEEE Trans. Evol. Comput., 27 (2) (2022), pp. 201-221
Crossref Google Scholar
[15]
Mavrotas G.
Effective implementation of the $ɛ$ -constraint method in multi-objective mathematical programming problems
Appl. Math. Comput., 213 (2) (2009), pp. 455-465, 10.1016/j.amc.2009.03.037
URL: https://www.sciencedirect.com/science/article/pii/S0096300309002574
View PDF View article View in Scopus Google Scholar
[16]
Yang G., Li X., Huo L., Liu Q.
A solving approach for fuzzy multi-objective linear fractional programming and application to an agricultural planting structure optimization problem
Chaos Solitons Fractals, 141 (2020), Article 110352
View PDF View article View in Scopus Google Scholar
[17]
Tian Y., Zhang T., Xiao J., Zhang X., Jin Y.
A coevolutionary framework for constrained multiobjective optimization problems
IEEE Trans. Evol. Comput., 25 (1) (2020), pp. 102-116
Google Scholar
[18]
Fan Z., Li W., Cai X., Li H., Wei C., Zhang Q., Deb K., Goodman E.
Push and pull search for solving constrained multi-objective optimization problems
Swarm Evol. Comput., 44 (2019), pp. 665-679
View PDF View article View in Scopus Google Scholar
[19]
Zuo M., Gong D., Wang Y., Ye X., Zeng B., Meng F.
Process knowledge-guided autonomous evolutionary optimization for constrained multiobjective problems
IEEE Trans. Evol. Comput. (2023)
Google Scholar
[20]
Ming F., Gong W., Li S., Wang L., Liao Z.
Handling constrained many-objective optimization problems via determinantal point processes
Inform. Sci. (2023), Article 119260
View PDF View article View in Scopus Google Scholar
[21]
Dantzig G.B.
Linear programming
Oper. Res., 50 (1) (2002), pp. 42-47
View in Scopus Google Scholar
[22]
Ehrgott M., Ruzika S.
Improved $ɛ$ -constraint method for multiobjective programming
J. Optim. Theory Appl., 138 (3) (2008), pp. 375-396
Crossref View in Scopus Google Scholar
[23]
Lodwick W.A., Kacprzyk J.
Fuzzy Optimization: Recent Advances and Applications, Springer (2010)
Google Scholar
[24]
Liu B., Fernandez F.V., Gao P., Gielen G.
A fuzzy selection based constraint handling method for multi-objective optimization of analog cells
2009 European Conference on Circuit Theory and Design, IEEE (2009), pp. 611-614
View in Scopus Google Scholar
[25]
Saha C., Das S., Pal K., Mukherjee S.
A fuzzy rule-based penalty function approach for constrained evolutionary optimization
IEEE Trans. Cybern., 46 (12) (2014), pp. 2953-2965
Google Scholar
[26]
Li Z., Chen S., Zhang S., Jiang S., Gu Y., Nouioua M.
FSB-EA: Fuzzy search bias guided constraint handling technique for evolutionary algorithm
Expert Syst. Appl., 119 (2019), pp. 20-35
View PDF View article Crossref Google Scholar
[27]
Cocchi G., Lapucci M.
An augmented Lagrangian algorithm for multi-objective optimization
Comput. Optim. Appl., 77 (1) (2020), pp. 29-56
Crossref View in Scopus Google Scholar
[28]
Ansary M.A.T., Panda G.
A sequential quadratic programming method for constrained multi-objective optimization problems
J. Appl. Math. Comput., 64 (1) (2020), pp. 379-397
Crossref View in Scopus Google Scholar
[29]
Yang X., Leng Z., Xu S., Yang C., Yang L., Liu K., Song Y., Zhang L.
Multi-objective optimal scheduling for CCHP microgrids considering peak-load reduction by augmented $ɛ$ -constraint method
Renew. Energy, 172 (2021), pp. 408-423
View PDF View article View in Scopus Google Scholar
[30]
Han D., Du W., Jin Y., Du W., Yu G.
A fuzzy constraint handling technique for decomposition-based constrained multi-and many-objective optimization
Inf. Sci., 597 (2022), pp. 318-340
View PDF View article View in Scopus Google Scholar
[31]
Lai K.K., Maurya J.K., Mishra S.K.
Multiobjective approximate gradient projection method for constrained vector optimization: Sequential optimality conditions without constraint qualifications
J. Comput. Appl. Math., 410 (2022), Article 114122
View PDF View article View in Scopus Google Scholar
[32]
Pérez-Cañedo B., Verdegay J.L., Concepción-Morales E.R., et al.
An $ɛ$ -constraint method for multiobjective linear programming in intuitionistic fuzzy environment
Int. J. Intell. Syst., 2023 (2023)
Google Scholar
[33]
Sharma K., Singh V.P., Ebrahimnejad A., Chakraborty D.
Solving a multi-objective chance constrained hierarchical optimization problem under intuitionistic fuzzy environment with its application
Expert Syst. Appl., 217 (2023), Article 119595
View PDF View article View in Scopus Google Scholar
[34]
Alouane B., Boulif M.
Fuzzy constraint prioritization to solve heavily constrained problems with the genetic algorithm
Eng. Appl. Artif. Intell., 119 (2023), Article 105768
View PDF View article View in Scopus Google Scholar
[35]
Ming M., Trivedi A., Wang R., Srinivasan D., Zhang T.
A dual-population-based evolutionary algorithm for constrained multiobjective optimization
IEEE Trans. Evol. Comput., 25 (4) (2021), pp. 739-753
Crossref View in Scopus Google Scholar
[36]
Liu Z.-Z., Wang B.-C., Tang K.
Handling constrained multiobjective optimization problems via bidirectional coevolution
IEEE Trans. Cybern., 52 (10) (2021), pp. 10163-10176
Google Scholar
[37]
Jiao R., Xue B., Zhang M.
A multiform optimization framework for constrained multiobjective optimization
IEEE Trans. Cybern. (2022)
Google Scholar
[38]
Qiao K., Liang J., Yu K., Wang M., Qu B., Yue C., Guo Y.
A self-adaptive evolutionary multi-task based constrained multi-objective evolutionary algorithm
IEEE Trans. Emerg. Top. Computat. Intell. (2023)
Google Scholar
[39]
Li L., He C., Lin J., Pan L.
Two-archive-based competitive swarm optimizer for constrained multiobjective optimization
2023 5th International Conference on Data-Driven Optimization of Complex Systems, DOCS (2023), pp. 1-6, 10.1109/DOCS60977.2023.10294727
View in Scopus Google Scholar
[40]
Yang Y., Liu J., Tan S.
A multi-objective evolutionary algorithm for steady-state constrained multi-objective optimization problems
Appl. Soft Comput., 101 (2021), Article 107042
View PDF View article View in Scopus Google Scholar
[41]
Li M., Yang S., Liu X.
Pareto or non-Pareto: Bi-criterion evolution in multiobjective optimization
IEEE Trans. Evol. Comput., 20 (5) (2015), pp. 645-665
View in Scopus Google Scholar
[42]
Wang J., Li Y., Zhang Q., Zhang Z., Gao S.
Cooperative multiobjective evolutionary algorithm with propulsive population for constrained multiobjective optimization
IEEE Trans. Syst. Man. Cybern. Syst., 52 (6) (2021), pp. 3476-3491
Crossref View in Scopus Google Scholar
[43]
Qiao K., Yu K., Qu B., Liang J., Song H., Yue C.
An evolutionary multitasking optimization framework for constrained multiobjective optimization problems
IEEE Trans. Evol. Comput., 26 (2) (2022), pp. 263-277
Crossref View in Scopus Google Scholar
[44]
Ming F., Gong W., Wang L., Lu C.
A tri-population based co-evolutionary framework for constrained multi-objective optimization problems
Swarm Evol. Comput., 70 (2022), Article 101055
View PDF View article View in Scopus Google Scholar
[45]
Li Y., Feng X., Yu H.
A constrained multiobjective evolutionary algorithm with the two-archive weak cooperation
Inform. Sci., 615 (2022), pp. 415-430
View PDF View article Google Scholar
[46]
Kong X., Yang Y., Lv Z., Zhao J., Fu R.
A dynamic dual-population co-evolution multi-objective evolutionary algorithm for constrained multi-objective optimization problems
Appl. Soft Comput., 141 (2023), Article 110311
View PDF View article View in Scopus Google Scholar
[47]
Cao J., Yan Z., Chen Z., Zhang J.
A coevolutionary constrained multi-objective algorithm with a learning constraint boundary
Appl. Soft Comput., 148 (2023), Article 110845, 10.1016/j.asoc.2023.110845
URL: https://www.sciencedirect.com/science/article/pii/S1568494623008633
View PDF View article View in Scopus Google Scholar
[48]
Song S., Zhang K., Zhang L., Wu N.
A dual-population algorithm based on self-adaptive epsilon method for constrained multi-objective optimization
Inform. Sci., 655 (2024), Article 119906
View PDF View article View in Scopus Google Scholar
[49]
Jiao R., Zeng S., Li C.
A feasible-ratio control technique for constrained optimization
Inform. Sci., 502 (2019), pp. 201-217
View PDF View article View in Scopus Google Scholar
[50]
Fan Z., Wang Z., Li W., Yuan Y., You Y., Yang Z., Sun F., Ruan J.
Push and pull search embedded in an M2M framework for solving constrained multi-objective optimization problems
Swarm Evol. Comput., 54 (2020), Article 100651
View PDF View article View in Scopus Google Scholar
[51]
Yu K., Liang J., Qu B., Yue C.
Purpose-directed two-phase multiobjective differential evolution for constrained multiobjective optimization
Swarm Evol. Comput., 60 (2021), Article 100799
View PDF View article View in Scopus Google Scholar
[52]
Tian Y., Zhang Y., Su Y., Zhang X., Tan K.C., Jin Y.
Balancing objective optimization and constraint satisfaction in constrained evolutionary multiobjective optimization
IEEE Trans. Cybern., 52 (9) (2021), pp. 9559-9572
Google Scholar
[53]
Yu K., Liang J., Qu B., Luo Y., Yue C.
Dynamic selection preference-assisted constrained multiobjective differential evolution
IEEE Trans. Syst. Man. Cybern. Syst., 52 (5) (2021), pp. 2954-2965
Crossref View in Scopus Google Scholar
[54]
Ming F., Gong W., Zhen H., Li S., Wang L., Liao Z.
A simple two-stage evolutionary algorithm for constrained multi-objective optimization
Knowl.-Based Syst., 228 (2021), Article 107263
View PDF View article View in Scopus Google Scholar
[55]
Xu H., Cai X., Li Z., Fan Z.
A two-phase constrained multi-objective evolutionary algorithm based on the constrained decomposition approach
2021 IEEE 7th International Conference on Cloud Computing and Intelligent Systems, CCIS, IEEE (2021), pp. 154-158
Crossref View in Scopus Google Scholar
[56]
Ma H., Wei H., Tian Y., Cheng R., Zhang X.
A multi-stage evolutionary algorithm for multi-objective optimization with complex constraints
Inform. Sci., 560 (2021), pp. 68-91
View PDF View article Crossref View in Scopus Google Scholar
[57]
Jiao R., Zeng S., Li C., Ong Y.-S.
Two-type weight adjustments in MOEA/D for highly constrained many-objective optimization
Inform. Sci., 578 (2021), pp. 592-614
View PDF View article View in Scopus Google Scholar
[58]
Feng X., Pan A., Ren Z., Fan Z.
Hybrid driven strategy for constrained evolutionary multi-objective optimization
Inform. Sci., 585 (2022), pp. 344-365
View PDF View article View in Scopus Google Scholar
[59]
Zhang K., Xu Z., Yen G.G., Zhang L.
Two-stage multi-objective evolution strategy for constrained multi-objective optimization
IEEE Trans. Evol. Comput. (2022)
Google Scholar
[60]
Sun R., Zou J., Liu Y., Yang S., Zheng J.
A multi-stage algorithm for solving multi-objective optimization problems with multi-constraints
IEEE Trans. Evol. Comput. (2022)
Google Scholar
[61]
Song E., Han G., Zhang G.
A two-stage differential evolution for constrained multi-objective optimization problems
2022 18th International Conference on Computational Intelligence and Security, CIS, IEEE (2022), pp. 12-17
Crossref View in Scopus Google Scholar
[62]
Liang J., Qiao K., Yu K., Qu B., Yue C., Guo W., Wang L.
Utilizing the relationship between unconstrained and constrained pareto fronts for constrained multiobjective optimization
IEEE Trans. Cybern. (2022)
Google Scholar
[63]
Liu Y., Liu J., Jin Y., Li F., Zheng T.
A surrogate-assisted two-stage differential evolution for expensive constrained optimization
IEEE Trans. Emerg. Top. Computat. Intell. (2023)
Google Scholar
[64]
Zou J., Luo J., Liu Y., Yang S., Zheng J.
A flexible two-stage constrained multi-objective evolutionary algorithm based on automatic regulation
Inform. Sci., 634 (2023), pp. 227-243
View PDF View article View in Scopus Google Scholar
[65]
Xia M., Chong Q., Dong M.
A constrained multi-objective evolutionary algorithm with two-stage resources allocation
Swarm Evol. Comput., 79 (2023), Article 101313
View PDF View article View in Scopus Google Scholar
[66]
Feng J., Liu S., Yang S., Zheng J., Liu J.
An adaptive tradeoff evolutionary algorithm with composite differential evolution for constrained multi-objective optimization
Swarm Evol. Comput., 83 (2023), Article 101386
View PDF View article View in Scopus Google Scholar
[67]
Chen G., Guo Y., Wang Y., Liang J., Gong D., Yang S.
Evolutionary dynamic constrained multiobjective optimization: Test suite and algorithm
IEEE Trans. Evol. Comput. (2023)
Google Scholar
[68]
Huang W., Zou J., Liu Y., Yang S., Zheng J.
Global and local feasible solution search for solving constrained multi-objective optimization
Inform. Sci., 649 (2023), Article 119467
View PDF View article View in Scopus Google Scholar
[69]
Yang D., Li G., Cheng G.
On the efficiency of chaos optimization algorithms for global optimization
Chaos Solitons Fractals, 34 (4) (2007), pp. 1366-1375
View PDF View article View in Scopus Google Scholar
[70]
Aslimani N., El-Ghazali T., Ellaia R.
A new chaotic-based approach for multi-objective optimization
Algorithms, 13 (9) (2020), p. 204
Crossref View in Scopus Google Scholar
[71]
Li Q., Liu L., Yuan X., et al.
Multiobjective parallel chaos optimization algorithm with crossover and merging operation
Math. Probl. Eng., 2020 (2020)
Google Scholar
[72]
Zhang L., Yu R., Liu J., Xu Y., Jiang H., Xu P., Lin P., Luo Y.
A constrained multi-objective evolutionary algorithm based on weak cooperation framework and multi-chaotic operators
2022 Global Conference on Robotics, Artificial Intelligence and Information Technology, GCRAIT (2022), pp. 102-106, 10.1109/GCRAIT55928.2022.00030
View in Scopus Google Scholar
[73]
Takubo Y., Kanazaki M.
Robust constrained multi-objective evolutionary algorithm based on polynomial chaos expansion for trajectory optimization
2022 IEEE Congress on Evol. Comput., CEC, IEEE (2022), pp. 1-9
Crossref Google Scholar
[74]
Yacoubi S., Manita G., Chhabra A., Korbaa O., Mirjalili S.
A multi-objective chaos game optimization algorithm based on decomposition and random learning mechanisms for numerical optimization
Appl. Soft Comput., 144 (2023), Article 110525
View PDF View article View in Scopus Google Scholar
[75]
Rauf H.T., Gao J., Almadhor A., Haider A., Zhang Y.-D., Al-Turjman F.
Multi population-based chaotic differential evolution for multi-modal and multi-objective optimization problems
Appl. Soft Comput., 132 (2023), Article 109909
View PDF View article View in Scopus Google Scholar
[76]
Tian Y., Peng S., Rodemann T., Zhang X., Jin Y.
Automated selection of evolutionary multi-objective optimization algorithms
2019 IEEE Symposium Series on Computational Intelligence, SSCI, IEEE (2019), pp. 3225-3232
Crossref View in Scopus Google Scholar
[77]
Qiao K., Yu K., Qu B., Liang J., Yue C., Ban X.
Feature extraction for recommendation of constrained multi-objective evolutionary algorithms
IEEE Trans. Evol. Comput. (2022)
Google Scholar
[78]
Han Z., Yang Y., Ye H.
A deep reinforcement learning based multiple meta-heuristic methods approach for resource constrained multi-project scheduling problem
2022 7th International Conference on Intelligent Computing and Signal Processing, ICSP, IEEE (2022), pp. 26-29
Crossref View in Scopus Google Scholar
[79]
Wang Y., Zuo M., Gong D.
Migration-based algorithm library enrichment for constrained multi-objective optimization and applications in algorithm selection
Inform. Sci., 649 (2023), Article 119593
View PDF View article View in Scopus Google Scholar
[80]
Ming F., Gong W., Gao L.
Adaptive auxiliary task selection for multitasking-assisted constrained multi-objective optimization [feature]
IEEE Comput. Intell. Mag., 18 (2) (2023), pp. 18-30
Crossref View in Scopus Google Scholar
[81]
Fallahi A., Bani E.A., Niaki S.T.A.
A constrained multi-item EOQ inventory model for reusable items: Reinforcement learning-based differential evolution and particle swarm optimization
Expert Syst. Appl., 207 (2022), Article 118018
View PDF View article View in Scopus Google Scholar
[82]
Yu X., Xu P., Wang F., Wang X.
Reinforcement learning-based differential evolution algorithm for constrained multi-objective optimization problems
Eng. Appl. Artif. Intell., 131 (2024), Article 107817
View PDF View article View in Scopus Google Scholar
[83]
Ma L., Li N., Guo Y., Wang X., Yang S., Huang M., Zhang H.
Learning to optimize: reference vector reinforcement learning adaption to constrained many-objective optimization of industrial copper burdening system
IEEE Trans. Cybern. (2021)
Google Scholar
[84]
Wang C., Huang H., Zhang X.
Growing neural gas network for offspring generation in evolutionary constrained multi-objective optimization
IEEE Trans. Emerg. Top. Computat. Intell. (2023)
Google Scholar
[85]
Kulesza A., Taskar B., et al.
Determinantal point processes for machine learning
Found. Trends Mach. Learn., 5 (2–3) (2012), pp. 123-286
Crossref View in Scopus Google Scholar
[86]
Hu Z., Gong W., Pedrycz W., Li Y.
Deep reinforcement learning assisted co-evolutionary differential evolution for constrained optimization
Swarm Evol. Comput., 83 (2023), Article 101387
View PDF View article View in Scopus Google Scholar
[87]
Liu Z., Zhao P., Cao J., Zhang J., Chen Z.
A constrained multi-objective evolutionary algorithm with Pareto estimation via neural network
Expert Syst. Appl., 237 (2024), Article 121718
View PDF View article View in Scopus Google Scholar
[88]
Wolpert D.H., Macready W.G.
No free lunch theorems for optimization
IEEE Trans. Evol. Comput., 1 (1) (1997), pp. 67-82
View in Scopus Google Scholar
[89]
Liu F., Lin X., Wang Z., Yao S., Tong X., Yuan M., Zhang Q.
Large language model for multi-objective evolutionary optimization
(2023)
arXiv preprint arXiv:2310.12541
Google Scholar
[90]
Ray T., Tai K., Seow K.C.
Multiobjective design optimization by an evolutionary algorithm
Eng. Optim., 33 (4) (2001), pp. 399-424
Crossref View in Scopus Google Scholar
[91]
Bao Q., Wang M., Dai G., Chen X., Song Z., Li S.
An archive-based two-stage evolutionary algorithm for constrained multi-objective optimization problems
Swarm Evol. Comput., 75 (2022), Article 101161
View PDF View article View in Scopus Google Scholar
[92]
Kocis G.R., Grossmann I.E.
Global optimization of nonconvex mixed-integer nonlinear programming (MINLP) problems in process synthesis
Ind. Eng. Chem. Res., 27 (8) (1988), pp. 1407-1421
Crossref View in Scopus Google Scholar
[93]
Xiang Y., Yang X., Zhou Y., Huang H.
Enhancing decomposition-based algorithms by estimation of distribution for constrained optimal software product selection
IEEE Trans. Evol. Comput., 24 (2) (2019), pp. 245-259
Crossref View in Scopus Google Scholar
[94]
Xiang Y., Yang X., Huang H., Wang J.
Balancing constraints and objectives by considering problem types in constrained multiobjective optimization
IEEE Trans. Cybern. (2021)
Google Scholar
[95]
Santana-Quintero L.V., Hernández-Díaz A.G., Molina J., Coello C.A.C., Caballero R.
DEMORS: A hybrid multi-objective optimization algorithm using differential evolution and rough set theory for constrained problems
Comput. Oper. Res., 37 (3) (2010), pp. 470-480
View PDF View article View in Scopus Google Scholar
[96]
Qin C., Ming F., Gong W., Gu Q.
Constrained multi-objective optimization via two archives assisted push–pull evolutionary algorithm
Swarm Evol. Comput., 75 (2022), Article 101178
View PDF View article View in Scopus Google Scholar
[97]
Kohli M., Arora S.
Chaotic grey wolf optimization algorithm for constrained optimization problems
J. Comput. Des. Eng., 5 (4) (2018), pp. 458-472
View PDF View article Crossref View in Scopus Google Scholar
[98]
Vallozzi L., Vandendriessche W., Rogier H., Hertleer C., Scarpello M.
Design of a protective garment GPS antenna
Microw. Opt. Technol. Lett., 51 (6) (2009), pp. 1504-1508
Crossref View in Scopus Google Scholar
[99]
Saravanan R., Ramabalan S., Ebenezer N.G.R., Dharmaraja C.
Evolutionary multi criteria design optimization of robot grippers
Appl. Soft Comput., 9 (1) (2009), pp. 159-172
View PDF View article View in Scopus Google Scholar
[100]
Yang Y., Liu J., Tan S., Wang H.
Application of constrained multi-objective evolutionary algorithm in multi-source compressed-air pipeline optimization problems
IFAC-PapersOnLine, 51 (18) (2018), pp. 168-173
View PDF View article Google Scholar
[101]
Yang Y., Liu J., Tan S., Liu Y.
A multi-objective differential evolution algorithm based on domination and constraint-handling switching
Inform. Sci., 579 (2021), pp. 796-813
View PDF View article View in Scopus Google Scholar
[102]
Yang Y., Liu J., Tan S.
A partition-based constrained multi-objective evolutionary algorithm
Swarm Evol. Comput., 66 (2021), Article 100940
View PDF View article View in Scopus Google Scholar
[103]
Zhang Q., Wang Q., Li G.
Switched system identification based on the constrained multi-objective optimization problem with application to the servo turntable
Int. J. Control Autom. Syst., 14 (5) (2016), pp. 1153-1159
Crossref View in Scopus Google Scholar
[104]
Mala-Jetmarova H., Sultanova N., Savic D.
Lost in optimisation of water distribution systems? A literature review of system operation
Environ. Model. Softw., 93 (2017), pp. 209-254
View PDF View article View in Scopus Google Scholar
[105]
Ouyang Q., Lu W., Hou Z., Zhang Y., Li S., Luo J.
Chance-constrained multi-objective optimization of groundwater remediation design at DNAPLs-contaminated sites using a multi-algorithm genetically adaptive method
J. Contam. Hydrol., 200 (2017), pp. 15-23
View PDF View article View in Scopus Google Scholar
[106]
Marti J., Geissbühler L., Becattini V., Haselbacher A., Steinfeld A.
Constrained multi-objective optimization of thermocline packed-bed thermal-energy storage
Appl. Energy, 216 (2018), pp. 694-708
View PDF View article View in Scopus Google Scholar
[107]
Su Z., Zhang G., Yue F., Zhan D., Li M., Li B., Yao X.
Enhanced constraint handling for reliability-constrained multiobjective testing resource allocation
IEEE Trans. Evol. Comput., 25 (3) (2021), pp. 537-551
Crossref View in Scopus Google Scholar
[108]
Dhiman G., Kumar V.
Multi-objective spotted hyena optimizer: a multi-objective optimization algorithm for engineering problems
Knowl.-Based Syst., 150 (2018), pp. 175-197
View PDF View article View in Scopus Google Scholar
[109]
Ming F., Gong W., Wang L., Gao L.
A constrained many-objective optimization evolutionary algorithm with enhanced mating and environmental selections
IEEE Trans. Cybern. (2022)
Google Scholar
[110]
Shi J., Chen M., Ma Y., Qiao F.
A new boredom-aware dual-resource constrained flexible job shop scheduling problem using a two-stage multi-objective particle swarm optimization algorithm
Inform. Sci., 643 (2023), Article 119141
View PDF View article View in Scopus Google Scholar
[111]
Hou Y., Zhang Y., Wu N., Zhu Q.
Constrained multi-objective optimization of short-term crude oil scheduling with dual pipelines and charging tank maintenance requirement
Inform. Sci., 588 (2022), pp. 381-404
View PDF View article Google Scholar
[112]
Zuo X., Chen C., Tan W., Zhou M.
Vehicle scheduling of an urban bus line via an improved multiobjective genetic algorithm
IEEE Trans. Intell. Transp. Syst., 16 (2) (2014), pp. 1030-1041
Google Scholar
[113]
Ma Z., Wang Y.
Shift-based penalty for evolutionary constrained multiobjective optimization and its application
IEEE Trans. Cybern. (2021)
Google Scholar
[114]
Güvenc U., Sönmez Y., Duman S., Yörükeren N.
Combined economic and emission dispatch solution using gravitational search algorithm
Sci. Iran., 19 (6) (2012), pp. 1754-1762
View PDF View article View in Scopus Google Scholar
[115]
Chen M.-R., Zeng G.-Q., Lu K.-D.
Constrained multi-objective population extremal optimization based economic-emission dispatch incorporating renewable energy resources
Renew. Energy, 143 (2019), pp. 277-294
View PDF View article View in Scopus Google Scholar
[116]
El-Shorbagy M.A., Mousa A.
Constrained multiobjective equilibrium optimizer algorithm for solving combined economic emission dispatch problem
Complexity, 2021 (2021), pp. 1-14
Google Scholar
[117]
Yu X., Yu X., Lu Y., Yen G.G., Cai M.
Differential evolution mutation operators for constrained multi-objective optimization
Appl. Soft Comput., 67 (2018), pp. 452-466
View PDF View article View in Scopus Google Scholar
[118]
Mohseni-Bonab S.M., Rabiee A., Mohammadi-Ivatloo B.
Voltage stability constrained multi-objective optimal reactive power dispatch under load and wind power uncertainties: A stochastic approach
Renew. Energy, 85 (2016), pp. 598-609
View PDF View article View in Scopus Google Scholar
[119]
Peng C., Qiu S.
A decomposition-based constrained multi-objective evolutionary algorithm with a local infeasibility utilization mechanism for UAV path planning
Appl. Soft Comput., 118 (2022), Article 108495
View PDF View article View in Scopus Google Scholar
[120]
Wang X., Zhao J., Cheng C., Qi M.
A multi-objective fuzzy facility location problem with congestion and priority for drone-based emergency deliveries
Comput. Ind. Eng., 179 (2023), Article 109167
View PDF View article Google Scholar
[121]
Latpate R., Kurade S.S.
Multi-objective multi-index transportation model for crude oil using fuzzy NSGA-II
IEEE Trans. Intell. Transp. Syst., 23 (2) (2020), pp. 1347-1356
Google Scholar
[122]
Chen C.-T., Pham H.-V.
Trajectory planning in parallel kinematic manipulators using a constrained multi-objective evolutionary algorithm
Nonlinear Dynam., 67 (2012), pp. 1669-1681
Crossref View in Scopus Google Scholar
[123]
Maity S., Roy A., Maiti M.
A rough multi-objective genetic algorithm for uncertain constrained multi-objective solid travelling salesman problem
Granul. Comput., 4 (1) (2019), pp. 125-142
Crossref View in Scopus Google Scholar
[124]
Kocis G., Grossmann I.
A modelling and decomposition strategy for the minlp optimization of process flowsheets
Comput. Chem. Eng., 13 (7) (1989), pp. 797-819, 10.1016/0098-1354(89)85053-7
URL: https://www.sciencedirect.com/science/article/pii/0098135489850537
View PDF View article View in Scopus Google Scholar
[125]
Zou J., Sun R., Yang S., Zheng J.
A dual-population algorithm based on alternative evolution and degeneration for solving constrained multi-objective optimization problems
Inform. Sci., 579 (2021), pp. 89-102
View PDF View article View in Scopus Google Scholar
[126]
Zou J., Sun R., Liu Y., Hu Y., Yang S., Zheng J., Li K.
A multi-population evolutionary algorithm using new cooperative mechanism for solving multi-objective problems with multi-constraint
IEEE Trans. Evol. Comput. (2023)
Google Scholar
[127]
Zhu X., Jiang C., Hao K., Wang R.
High-dimensional interactive adaptive RVEA for multi-objective optimization of polyester polymerization process
Inform. Sci., 650 (2023), Article 119707
View PDF View article View in Scopus Google Scholar
[128]
Kumar A., Wu G., Ali M.Z., Luo Q., Mallipeddi R., Suganthan P.N., Das S.
A benchmark-suite of real-world constrained multi-objective optimization problems and some baseline results
Swarm Evol. Comput., 67 (2021), Article 100961
View PDF View article View in Scopus Google Scholar
[129]
Kumar A., Wu G., Ali M.Z., Mallipeddi R., Suganthan P.N., Das S.
A test-suite of non-convex constrained optimization problems from the real-world and some baseline results
Swarm Evol. Comput., 56 (2020), Article 100693
View PDF View article View in Scopus Google Scholar
[130]
Tawhid M.A., Savsani V.
A novel multi-objective optimization algorithm based on artificial algae for multi-objective engineering design problems
Appl. Intell., 48 (2018), pp. 3762-3781
Crossref View in Scopus Google Scholar
[131]
Kotinis M.
A particle swarm optimizer for constrained multi-objective engineering design problems
Eng. Optim., 42 (10) (2010), pp. 907-926
Crossref View in Scopus Google Scholar
[132]
Chiandussi G., Codegone M., Ferrero S., Varesio F.E.
Comparison of multi-objective optimization methodologies for engineering applications
Comput. Math. Appl., 63 (5) (2012), pp. 912-942
View PDF View article View in Scopus Google Scholar
[133]
Jain H., Deb K.
An evolutionary many-objective optimization algorithm using reference-point based nondominated sorting approach, part II: Handling constraints and extending to an adaptive approach
IEEE Trans. Evol. Comput., 18 (4) (2013), pp. 602-622
Google Scholar
[134]
He C., Zhang Y., Gong D., Ji X.
A review of surrogate-assisted evolutionary algorithms for expensive optimization problems
Expert Syst. Appl. (2023), Article 119495
View PDF View article View in Scopus Google Scholar
[135]
Huang H.-Z., Gu Y.-K., Du X.
An interactive fuzzy multi-objective optimization method for engineering design
Eng. Appl. Artif. Intell., 19 (5) (2006), pp. 451-460
View PDF View article View in Scopus Google Scholar
[136]
Parsons M.G., Scott R.L.
Formulation of multicriterion design optimization problems for solution with scalar numerical optimization methods
J. Ship Res., 48 (01) (2004), pp. 61-76
Crossref View in Scopus Google Scholar
[137]
Kannan B., Kramer S.N.
An augmented Lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design
(1994)
Google Scholar
[138]
Gujarathi A.M., Purohit S., Srikanth B.
Optimization of reactor network design problem using jumping gene adaptation of differential evolution
Journal of Physics: Conference Series, vol. 622, no. 1, IOP Publishing (2015), Article 012044
Crossref View in Scopus Google Scholar
[139]
Gadagi A., Adake C.
A constrained multi-objective optimization of turning process parameters by genetic algorithm and particle swarm optimization techniques
Mater. Today: Proc., 42 (2021), pp. 1207-1212
View PDF View article Google Scholar
[140]
Yue H., Guo C., Li Q., Zhao L., Hao G.
Milling parameters optimization of Al-Li alloy thin-wall workpieces using response surface methodology and particle swarm optimization
CMES Comput. Model. Eng. Sci., 124 (3) (2020), pp. 937-952
Crossref View in Scopus Google Scholar
[141]
Yang Y., Huang P.-Q., Kong X., Zhao J.
A constrained multi-objective evolutionary algorithm assisted by an additional objective function
Appl. Soft Comput., 132 (2023), Article 109904
View PDF View article View in Scopus Google Scholar
[142]
Gu Q., Bai J., Li X., Xiong N., Lu C.
A constrained multi-objective evolutionary algorithm based on decomposition with improved constrained dominance principle
Swarm Evol. Comput., 75 (2022), Article 101162
View PDF View article View in Scopus Google Scholar
[143]
Yuan J., Liu H.-L., He Z.
A constrained multi-objective evolutionary algorithm using valuable infeasible solutions
Swarm Evol. Comput., 68 (2022), Article 101020
View PDF View article View in Scopus Google Scholar
[144]
Musselman K., Talavage J.
A tradeoff cut approach to multiple objective optimization
Oper. Res., 28 (6) (1980), pp. 1424-1435
Crossref View in Scopus Google Scholar
[145]
Golafshani E.M., Arashpour M., Kashani A.
Green mix design of rubbercrete using machine learning-based ensemble model and constrained multi-objective optimization
J. Clean. Prod., 327 (2021), Article 129518
Google Scholar
[146]
Wang Y., Hu H., Sun X., Zhang Y., Gong D.
Unified operation optimization model of integrated coal mine energy systems and its solutions based on autonomous intelligence
Appl. Energy, 328 (2022), Article 120106
View PDF View article View in Scopus Google Scholar
[147]
Qiao J.-F., Hou Y., Zhang L., Han H.-G.
Adaptive fuzzy neural network control of wastewater treatment process with multiobjective operation
Neurocomputing, 275 (2018), pp. 383-393
View PDF View article View in Scopus Google Scholar
[148]
Ming M., Wang R., Ishibuchi H., Zhang T.
A novel dual-stage dual-population evolutionary algorithm for constrained multiobjective optimization
IEEE Trans. Evol. Comput., 26 (5) (2021), pp. 1129-1143
Google Scholar
[149]
Ray T., Tai K., Seow C.
An evolutionary algorithm for multiobjective optimization
Eng. Optim., 33 (3) (2001), pp. 399-424
Crossref View in Scopus Google Scholar
[150]
Hou Y., Wu Y., Han H.
Multistate-constrained multiobjective differential evolution algorithm with variable neighborhood strategy
IEEE Trans. Cybern. (2022)
Google Scholar
[151]
Yin X., Zhao Z., Yang W.
Optimizing cleaner productions of sustainable energies: A co-design framework for complementary operations of offshore wind and pumped hydro-storages
J. Clean. Prod., 396 (2023), Article 135832
View PDF View article View in Scopus Google Scholar
[152]
Lee L.H., Tan K.C., Ou K., Chew Y.H.
Vehicle capacity planning system: A case study on vehicle routing problem with time windows
IEEE Trans. Syst. Man Cybern. A, 33 (2) (2003), pp. 169-178
View in Scopus Google Scholar
[153]
Rathore A.K., Holtz J., Boller T.
Synchronous optimal pulsewidth modulation for low-switching-frequency control of medium-voltage multilevel inverters
IEEE Trans. Ind. Electron., 57 (7) (2010), pp. 2374-2381
View in Scopus Google Scholar
[154]
Rathore A.K., Holtz J., Boller T.
Optimal pulsewidth modulation of multilevel inverters for low switching frequency control of medium voltage high power industrial AC drives
2010 IEEE Energy Conversion Congress and Exposition, IEEE (2010), pp. 4569-4574
View in Scopus Google Scholar
[155]
Edpuganti A., Rathore A.K.
Optimal pulsewidth modulation for common-mode voltage elimination scheme of medium-voltage modular multilevel converter-fed open-end stator winding induction motor drives
IEEE Trans. Ind. Electron., 64 (1) (2016), pp. 848-856
Google Scholar
[156]
Yang G., Li X., Huo L., Liu Q.
A solving approach for fuzzy multi-objective linear fractional programming and application to an agricultural planting structure optimization problem
Chaos Solitons Fractals, 141 (2020), Article 110352
View PDF View article View in Scopus Google Scholar
[157]
Ozgen D., Gulsun B.
Combining possibilistic linear programming and fuzzy AHP for solving the multi-objective capacitated multi-facility location problem
Inform. Sci., 268 (2014), pp. 185-201
View PDF View article View in Scopus Google Scholar
[158]
Peykani P., Nouri M., Eshghi F., Khamechian M., Farrokhi-Asl H.
A novel mathematical approach for fuzzy multi-period multi-objective portfolio optimization problem under uncertain environment and practical constraints
J. Fuzzy Ext. Appl., 2 (3) (2021), pp. 191-203
View in Scopus Google Scholar
[159]
Mikolov T., Chen K., Corrado G., Dean J.
Efficient estimation of word representations in vector space
(2013)
arXiv preprint arXiv:1301.3781
Google Scholar
[160]
Joulin A., Grave E., Bojanowski P., Mikolov T.
Bag of tricks for efficient text classification
(2016)
arXiv preprint arXiv:1607.01759
Google Scholar
[161]
Ilić S., Marrese-Taylor E., Balazs J.A., Matsuo Y.
Deep contextualized word representations for detecting sarcasm and irony
(2018)
arXiv preprint arXiv:1809.09795
Google Scholar
[162]
Devlin J., Chang M.-W., Lee K., Toutanova K.
Bert: Pre-training of deep bidirectional transformers for language understanding
(2018)
arXiv preprint arXiv:1810.04805
Google Scholar
[163]
Scarselli F., Gori M., Tsoi A.C., Hagenbuchner M., Monfardini G.
The graph neural network model
IEEE Trans. Neural Netw., 20 (1) (2008), pp. 61-80
Google Scholar
[164]
Chen Q., Ding J., Chai T., Pan Q.
Evolutionary optimization under uncertainty: The strategies to handle varied constraints for fluid catalytic cracking operation
IEEE Trans. Cybern., 52 (4) (2020), pp. 2249-2262
View in Scopus Google Scholar
[165]
Yang C., Ding J.
Constrained dynamic multi-objective evolutionary optimization for operational indices of beneficiation process
J. Intell. Manuf., 30 (2019), pp. 2701-2713
Crossref View in Scopus Google Scholar
[166]
Schlegel M., Marquardt W.
Adaptive switching structure detection for the solution of dynamic optimization problems
Ind. Eng. Chem. Res., 45 (24) (2006), pp. 8083-8094
Crossref View in Scopus Google Scholar
[167]
Cai X., Xia C., Zhang Q., Mei Z., Hu H., Wang L., Hu J.
The collaborative local search based on dynamic-constrained decomposition with grids for combinatorial multiobjective optimization
IEEE Trans. Cybern., 51 (5) (2019), pp. 2639-2650
Google Scholar
[168]
Qiao K., Yu K., Qu B., Liang J., Song H., Yue C., Lin H., Tan K.C.
Dynamic auxiliary task-based evolutionary multitasking for constrained multi-objective optimization
IEEE Trans. Evol. Comput. (2022)
Google Scholar
[169]
Azzouz R., Bechikh S., Said L.B., Trabelsi W.
Handling time-varying constraints and objectives in dynamic evolutionary multi-objective optimization
Swarm Evol. Comput., 39 (2018), pp. 222-248
View PDF View article View in Scopus Google Scholar
[170]
Chen Q., Ding J., Yang S., Chai T.
A novel evolutionary algorithm for dynamic constrained multiobjective optimization problems
IEEE Trans. Evol. Comput., 24 (4) (2019), pp. 792-806
Crossref Google Scholar
[171]
Liang Z., Wu T., Ma X., Zhu Z., Yang S.
A dynamic multiobjective evolutionary algorithm based on decision variable classification
IEEE Trans. Cybern., 52 (3) (2020), pp. 1602-1615
Google Scholar
[172]
Deb K., Rao N U.B., Karthik S.
Dynamic multi-objective optimization and decision-making using modified NSGA-II: a case study on hydro-thermal power scheduling
International Conference on Evolutionary Multi-Criterion Optimization, Springer (2007), pp. 803-817
Crossref View in Scopus Google Scholar
[173]
Goh C.-K., Tan K.C.
A competitive-cooperative coevolutionary paradigm for dynamic multiobjective optimization
IEEE Trans. Evol. Comput., 13 (1) (2008), pp. 103-127
Crossref Google Scholar
[174]
Zhou A., Jin Y., Zhang Q.
A population prediction strategy for evolutionary dynamic multiobjective optimization
IEEE Trans. Cybern., 44 (1) (2013), pp. 40-53
View in Scopus Google Scholar
[175]
Muruganantham A., Tan K.C., Vadakkepat P.
Evolutionary dynamic multiobjective optimization via kalman filter prediction
IEEE Trans. Cybern., 46 (12) (2015), pp. 2862-2873
Google Scholar
[176]
Lin J., Liu M., Hao J., Jiang S.
A multi-objective optimization approach for integrated production planning under interval uncertainties in the steel industry
Comput. Oper. Res., 72 (2016), pp. 189-203
View PDF View article View in Scopus Google Scholar
[177]
Zhang Y., Gong D.-w., Zhang J.-h.
Robot path planning in uncertain environment using multi-objective particle swarm optimization
Neurocomputing, 103 (2013), pp. 172-185
View PDF View article View in Scopus Google Scholar
[178]
Liu Y.-J., Zhang W.-G., Wang J.-B.
Multi-period cardinality constrained portfolio selection models with interval coefficients
Ann. Oper. Res., 244 (2016), pp. 545-569
Crossref View in Scopus Google Scholar
[179]
Sun J., Yan X., Heng Z., Zhiping L.
Interval multi-objective programming methods for solving multi-period portfolio selection problems
Control Decis., 35 (3) (2020), pp. 645-650
View in Scopus Google Scholar
[180]
Gong D., Xu B., Zhang Y., Guo Y., Yang S.
A similarity-based cooperative co-evolutionary algorithm for dynamic interval multiobjective optimization problems
IEEE Trans. Evol. Comput., 24 (1) (2019), pp. 142-156
View in Scopus Google Scholar
[181]
Chiang T.-C., Hsu W.-H.
A knowledge-based evolutionary algorithm for the multiobjective vehicle routing problem with time windows
Comput. Oper. Res., 45 (2014), pp. 25-37
View PDF View article View in Scopus Google Scholar
[182]
Xiang Y., Zhou Y., Zheng Z., Li M.
Configuring software product lines by combining many-objective optimization and SAT solvers
ACM Trans. Softw. Eng. Methodol., 26 (4) (2018), pp. 1-46
Google Scholar
[183]
Nicosia G., Rinaudo S., Sciacca E.
An evolutionary algorithm-based approach to robust analog circuit design using constrained multi-objective optimization
International Conference on Innovative Techniques and Applications of Artificial Intelligence, Springer (2007), pp. 7-20
Google Scholar
[184]
Kwag S., Ok S.-Y.
Robust design of seismic isolation system using constrained multi-objective optimization technique
KSCE J. Civ. Eng., 17 (2013), pp. 1051-1063
Crossref View in Scopus Google Scholar
[185]
Zhan D., Xing H.
Expected improvement for expensive optimization: a review
J. Global Optim., 78 (3) (2020), pp. 507-544
Crossref View in Scopus Google Scholar
[186]
Pan J.-S., Liu N., Chu S.-C., Lai T.
An efficient surrogate-assisted hybrid optimization algorithm for expensive optimization problems
Inform. Sci., 561 (2021), pp. 304-325
View PDF View article View in Scopus Google Scholar
[187]
Forrester A.I., Bressloff N.W., Keane A.J.
Optimization using surrogate models and partially converged computational fluid dynamics simulations
Proc. R. Soc. A, 462 (2071) (2006), pp. 2177-2204
Crossref View in Scopus Google Scholar
[188]
Gu Q., Wang Q., Li X., Li X.
A surrogate-assisted multi-objective particle swarm optimization of expensive constrained combinatorial optimization problems
Knowl.-Based Syst., 223 (2021), Article 107049
View PDF View article View in Scopus Google Scholar
[189]
Song Z., Wang H., Xue B., Zhang M., Jin Y.
Balancing objective optimization and constraint satisfaction in expensive constrained evolutionary multi-objective optimization
IEEE Trans. Evol. Comput. (2023)
Google Scholar
[190]
Yang Z., Qiu H., Gao L., Chen L., Liu J.
Surrogate-assisted MOEA/D for expensive constrained multi-objective optimization
Inform. Sci., 639 (2023), Article 119016
View PDF View article Google Scholar
[191]
Wei F.-F., Chen W.-N., Li Q., Jeon S.-W., Zhang J.
Distributed and expensive evolutionary constrained optimization with on-demand evaluation
IEEE Trans. Evol. Comput. (2022)
Google Scholar

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[1] [1]
Yu Q., Yang C., Dai G., Peng L., Chen X.
Synchronous wireless sensor and sink placement method using dual-population coevolutionary constrained multi-objective optimization algorithm
IEEE Trans. Ind. Inform. (2022)
Google Scholar

[2] [2]
Sorkhabi S.Y.D., Romero D.A., Beck J.C., Amon C.H.
Constrained multi-objective wind farm layout optimization: Novel constraint handling approach based on constraint programming
Renew. Energy, 126 (2018), pp. 341-353
Google Scholar

[3] [3]
Lwin K., Qu R., Kendall G.
A learning-guided multi-objective evolutionary algorithm for constrained portfolio optimization
Appl. Soft Comput., 24 (2014), pp. 757-772
View PDF View article View in Scopus Google Scholar

[4] [4]
Wang Y., Zixing C., Yuren Z., Chixin X.
Constrained optimization evolutionary algorithms
J. Softw., 20 (1) (2009), pp. 11-29
Google Scholar

[5] [5]
Liu Z.-Z., Wang Y., Wang B.-C.
Indicator-based constrained multiobjective evolutionary algorithms
IEEE Trans. Syst. Man. Cybern. Syst., 51 (9) (2021), pp. 5414-5426, 10.1109/TSMC.2019.2954491
View in Scopus Google Scholar

[6] [6]
Li W., Gong W., Ming F., Wang L.
Constrained multi-objective evolutionary algorithm with an improved two-archive strategy
Knowl.-Based Syst., 246 (2022), Article 108732
View PDF View article View in Scopus Google Scholar

[7] [7]
Xia M., Dong M.
A novel two-archive evolutionary algorithm for constrained multi-objective optimization with small feasible regions
Knowl.-Based Syst., 237 (2022), Article 107693
View PDF View article View in Scopus Google Scholar

[8] [8]
Ma Z., Wang Y.
Evolutionary constrained multiobjective optimization: Test suite construction and performance comparisons
IEEE Trans. Evol. Comput., 23 (6) (2019), pp. 972-986
Crossref View in Scopus Google Scholar

[9] [9]
Fan Z., Li W., Cai X., Li H., Wei C., Zhang Q., Deb K., Goodman E.
Difficulty adjustable and scalable constrained multiobjective test problem toolkit
Evol. Comput., 28 (3) (2020), pp. 339-378
Crossref Google Scholar

[10] [10]
Fan Z., Li W., Cai X., Huang H., Fang Y., You Y., Mo J., Wei C., Goodman E.
An improved epsilon constraint-handling method in MOEA/D for CMOPs with large infeasible regions
Soft Comput., 23 (2019), pp. 12491-12510
Crossref View in Scopus Google Scholar

[11] [11]
Hsiao Y.-T., Chiang H.-D., Liu C.-C., Chen Y.-L.
A computer package for optimal multi-objective VAR planning in large scale power systems
IEEE Trans. Power Syst., 9 (2) (1994), pp. 668-676
View in Scopus Google Scholar

[12] [12]
Fu L., Fan J., Yuanchun L., Yantao T., Yisong D.
Study and application of a constrained multi-objective optimization algorithm
Proceedings of the IEEE International Vehicle Electronics Conference (IVEC’99)(Cat. No. 99EX257), IEEE (1999), pp. 305-307
View in Scopus Google Scholar

[13] [13]
Coello C.A.C., Brambila S.G., Gamboa J.F., Tapia M.G.C.
Multi-objective evolutionary algorithms: Past, present, and future
Pardalos P.M., Rasskazova V., Vrahatis M.N. (Eds.), Black Box Optimization, Machine Learning, and No-Free Lunch Theorems, Springer International Publishing, Cham (2021), pp. 137-162
Crossref View in Scopus Google Scholar

[14] [14]
Liang J., Ban X., Yu K., Qu B., Qiao K., Yue C., Chen K., Tan K.C.
A survey on evolutionary constrained multiobjective optimization
IEEE Trans. Evol. Comput., 27 (2) (2022), pp. 201-221
Crossref Google Scholar

[15] [15]
Mavrotas G.
Effective implementation of the $ɛ$ -constraint method in multi-objective mathematical programming problems
Appl. Math. Comput., 213 (2) (2009), pp. 455-465, 10.1016/j.amc.2009.03.037
URL: https://www.sciencedirect.com/science/article/pii/S0096300309002574
View PDF View article View in Scopus Google Scholar

[16] [16]
Yang G., Li X., Huo L., Liu Q.
A solving approach for fuzzy multi-objective linear fractional programming and application to an agricultural planting structure optimization problem
Chaos Solitons Fractals, 141 (2020), Article 110352
View PDF View article View in Scopus Google Scholar

[17] [17]
Tian Y., Zhang T., Xiao J., Zhang X., Jin Y.
A coevolutionary framework for constrained multiobjective optimization problems
IEEE Trans. Evol. Comput., 25 (1) (2020), pp. 102-116
Google Scholar

[18] [18]
Fan Z., Li W., Cai X., Li H., Wei C., Zhang Q., Deb K., Goodman E.
Push and pull search for solving constrained multi-objective optimization problems
Swarm Evol. Comput., 44 (2019), pp. 665-679
View PDF View article View in Scopus Google Scholar

[19] [19]
Zuo M., Gong D., Wang Y., Ye X., Zeng B., Meng F.
Process knowledge-guided autonomous evolutionary optimization for constrained multiobjective problems
IEEE Trans. Evol. Comput. (2023)
Google Scholar

[20] [20]
Ming F., Gong W., Li S., Wang L., Liao Z.
Handling constrained many-objective optimization problems via determinantal point processes
Inform. Sci. (2023), Article 119260
View PDF View article View in Scopus Google Scholar

[21] [21]
Dantzig G.B.
Linear programming
Oper. Res., 50 (1) (2002), pp. 42-47
View in Scopus Google Scholar

[22] [22]
Ehrgott M., Ruzika S.
Improved $ɛ$ -constraint method for multiobjective programming
J. Optim. Theory Appl., 138 (3) (2008), pp. 375-396
Crossref View in Scopus Google Scholar

[23] [23]
Lodwick W.A., Kacprzyk J.
Fuzzy Optimization: Recent Advances and Applications, Springer (2010)
Google Scholar

[24] [24]
Liu B., Fernandez F.V., Gao P., Gielen G.
A fuzzy selection based constraint handling method for multi-objective optimization of analog cells
2009 European Conference on Circuit Theory and Design, IEEE (2009), pp. 611-614
View in Scopus Google Scholar

[25] [25]
Saha C., Das S., Pal K., Mukherjee S.
A fuzzy rule-based penalty function approach for constrained evolutionary optimization
IEEE Trans. Cybern., 46 (12) (2014), pp. 2953-2965
Google Scholar

[26] [26]
Li Z., Chen S., Zhang S., Jiang S., Gu Y., Nouioua M.
FSB-EA: Fuzzy search bias guided constraint handling technique for evolutionary algorithm
Expert Syst. Appl., 119 (2019), pp. 20-35
View PDF View article Crossref Google Scholar

[27] [27]
Cocchi G., Lapucci M.
An augmented Lagrangian algorithm for multi-objective optimization
Comput. Optim. Appl., 77 (1) (2020), pp. 29-56
Crossref View in Scopus Google Scholar

[28] [28]
Ansary M.A.T., Panda G.
A sequential quadratic programming method for constrained multi-objective optimization problems
J. Appl. Math. Comput., 64 (1) (2020), pp. 379-397
Crossref View in Scopus Google Scholar

[29] [29]
Yang X., Leng Z., Xu S., Yang C., Yang L., Liu K., Song Y., Zhang L.
Multi-objective optimal scheduling for CCHP microgrids considering peak-load reduction by augmented $ɛ$ -constraint method
Renew. Energy, 172 (2021), pp. 408-423
View PDF View article View in Scopus Google Scholar

[30] [30]
Han D., Du W., Jin Y., Du W., Yu G.
A fuzzy constraint handling technique for decomposition-based constrained multi-and many-objective optimization
Inf. Sci., 597 (2022), pp. 318-340
View PDF View article View in Scopus Google Scholar

[31] [31]
Lai K.K., Maurya J.K., Mishra S.K.
Multiobjective approximate gradient projection method for constrained vector optimization: Sequential optimality conditions without constraint qualifications
J. Comput. Appl. Math., 410 (2022), Article 114122
View PDF View article View in Scopus Google Scholar

[32] [32]
Pérez-Cañedo B., Verdegay J.L., Concepción-Morales E.R., et al.
An $ɛ$ -constraint method for multiobjective linear programming in intuitionistic fuzzy environment
Int. J. Intell. Syst., 2023 (2023)
Google Scholar

[33] [33]
Sharma K., Singh V.P., Ebrahimnejad A., Chakraborty D.
Solving a multi-objective chance constrained hierarchical optimization problem under intuitionistic fuzzy environment with its application
Expert Syst. Appl., 217 (2023), Article 119595
View PDF View article View in Scopus Google Scholar

[34] [34]
Alouane B., Boulif M.
Fuzzy constraint prioritization to solve heavily constrained problems with the genetic algorithm
Eng. Appl. Artif. Intell., 119 (2023), Article 105768
View PDF View article View in Scopus Google Scholar

[35] [35]
Ming M., Trivedi A., Wang R., Srinivasan D., Zhang T.
A dual-population-based evolutionary algorithm for constrained multiobjective optimization
IEEE Trans. Evol. Comput., 25 (4) (2021), pp. 739-753
Crossref View in Scopus Google Scholar

[36] [36]
Liu Z.-Z., Wang B.-C., Tang K.
Handling constrained multiobjective optimization problems via bidirectional coevolution
IEEE Trans. Cybern., 52 (10) (2021), pp. 10163-10176
Google Scholar

[37] [37]
Jiao R., Xue B., Zhang M.
A multiform optimization framework for constrained multiobjective optimization
IEEE Trans. Cybern. (2022)
Google Scholar

[38] [38]
Qiao K., Liang J., Yu K., Wang M., Qu B., Yue C., Guo Y.
A self-adaptive evolutionary multi-task based constrained multi-objective evolutionary algorithm
IEEE Trans. Emerg. Top. Computat. Intell. (2023)
Google Scholar

[39] [39]
Li L., He C., Lin J., Pan L.
Two-archive-based competitive swarm optimizer for constrained multiobjective optimization
2023 5th International Conference on Data-Driven Optimization of Complex Systems, DOCS (2023), pp. 1-6, 10.1109/DOCS60977.2023.10294727
View in Scopus Google Scholar

[40] [40]
Yang Y., Liu J., Tan S.
A multi-objective evolutionary algorithm for steady-state constrained multi-objective optimization problems
Appl. Soft Comput., 101 (2021), Article 107042
View PDF View article View in Scopus Google Scholar

[41] [41]
Li M., Yang S., Liu X.
Pareto or non-Pareto: Bi-criterion evolution in multiobjective optimization
IEEE Trans. Evol. Comput., 20 (5) (2015), pp. 645-665
View in Scopus Google Scholar

[42] [42]
Wang J., Li Y., Zhang Q., Zhang Z., Gao S.
Cooperative multiobjective evolutionary algorithm with propulsive population for constrained multiobjective optimization
IEEE Trans. Syst. Man. Cybern. Syst., 52 (6) (2021), pp. 3476-3491
Crossref View in Scopus Google Scholar

[43] [43]
Qiao K., Yu K., Qu B., Liang J., Song H., Yue C.
An evolutionary multitasking optimization framework for constrained multiobjective optimization problems
IEEE Trans. Evol. Comput., 26 (2) (2022), pp. 263-277
Crossref View in Scopus Google Scholar

[44] [44]
Ming F., Gong W., Wang L., Lu C.
A tri-population based co-evolutionary framework for constrained multi-objective optimization problems
Swarm Evol. Comput., 70 (2022), Article 101055
View PDF View article View in Scopus Google Scholar

[45] [45]
Li Y., Feng X., Yu H.
A constrained multiobjective evolutionary algorithm with the two-archive weak cooperation
Inform. Sci., 615 (2022), pp. 415-430
View PDF View article Google Scholar

[46] [46]
Kong X., Yang Y., Lv Z., Zhao J., Fu R.
A dynamic dual-population co-evolution multi-objective evolutionary algorithm for constrained multi-objective optimization problems
Appl. Soft Comput., 141 (2023), Article 110311
View PDF View article View in Scopus Google Scholar

[47] [47]
Cao J., Yan Z., Chen Z., Zhang J.
A coevolutionary constrained multi-objective algorithm with a learning constraint boundary
Appl. Soft Comput., 148 (2023), Article 110845, 10.1016/j.asoc.2023.110845
URL: https://www.sciencedirect.com/science/article/pii/S1568494623008633
View PDF View article View in Scopus Google Scholar

[48] [48]
Song S., Zhang K., Zhang L., Wu N.
A dual-population algorithm based on self-adaptive epsilon method for constrained multi-objective optimization
Inform. Sci., 655 (2024), Article 119906
View PDF View article View in Scopus Google Scholar

[49] [49]
Jiao R., Zeng S., Li C.
A feasible-ratio control technique for constrained optimization
Inform. Sci., 502 (2019), pp. 201-217
View PDF View article View in Scopus Google Scholar

[50] [50]
Fan Z., Wang Z., Li W., Yuan Y., You Y., Yang Z., Sun F., Ruan J.
Push and pull search embedded in an M2M framework for solving constrained multi-objective optimization problems
Swarm Evol. Comput., 54 (2020), Article 100651
View PDF View article View in Scopus Google Scholar

[51] [51]
Yu K., Liang J., Qu B., Yue C.
Purpose-directed two-phase multiobjective differential evolution for constrained multiobjective optimization
Swarm Evol. Comput., 60 (2021), Article 100799
View PDF View article View in Scopus Google Scholar

[52] [52]
Tian Y., Zhang Y., Su Y., Zhang X., Tan K.C., Jin Y.
Balancing objective optimization and constraint satisfaction in constrained evolutionary multiobjective optimization
IEEE Trans. Cybern., 52 (9) (2021), pp. 9559-9572
Google Scholar

[53] [53]
Yu K., Liang J., Qu B., Luo Y., Yue C.
Dynamic selection preference-assisted constrained multiobjective differential evolution
IEEE Trans. Syst. Man. Cybern. Syst., 52 (5) (2021), pp. 2954-2965
Crossref View in Scopus Google Scholar

[54] [54]
Ming F., Gong W., Zhen H., Li S., Wang L., Liao Z.
A simple two-stage evolutionary algorithm for constrained multi-objective optimization
Knowl.-Based Syst., 228 (2021), Article 107263
View PDF View article View in Scopus Google Scholar

[55] [55]
Xu H., Cai X., Li Z., Fan Z.
A two-phase constrained multi-objective evolutionary algorithm based on the constrained decomposition approach
2021 IEEE 7th International Conference on Cloud Computing and Intelligent Systems, CCIS, IEEE (2021), pp. 154-158
Crossref View in Scopus Google Scholar

[56] [56]
Ma H., Wei H., Tian Y., Cheng R., Zhang X.
A multi-stage evolutionary algorithm for multi-objective optimization with complex constraints
Inform. Sci., 560 (2021), pp. 68-91
View PDF View article Crossref View in Scopus Google Scholar

[57] [57]
Jiao R., Zeng S., Li C., Ong Y.-S.
Two-type weight adjustments in MOEA/D for highly constrained many-objective optimization
Inform. Sci., 578 (2021), pp. 592-614
View PDF View article View in Scopus Google Scholar

[58] [58]
Feng X., Pan A., Ren Z., Fan Z.
Hybrid driven strategy for constrained evolutionary multi-objective optimization
Inform. Sci., 585 (2022), pp. 344-365
View PDF View article View in Scopus Google Scholar

[59] [59]
Zhang K., Xu Z., Yen G.G., Zhang L.
Two-stage multi-objective evolution strategy for constrained multi-objective optimization
IEEE Trans. Evol. Comput. (2022)
Google Scholar

[60] [60]
Sun R., Zou J., Liu Y., Yang S., Zheng J.
A multi-stage algorithm for solving multi-objective optimization problems with multi-constraints
IEEE Trans. Evol. Comput. (2022)
Google Scholar

[61] [61]
Song E., Han G., Zhang G.
A two-stage differential evolution for constrained multi-objective optimization problems
2022 18th International Conference on Computational Intelligence and Security, CIS, IEEE (2022), pp. 12-17
Crossref View in Scopus Google Scholar

[62] [62]
Liang J., Qiao K., Yu K., Qu B., Yue C., Guo W., Wang L.
Utilizing the relationship between unconstrained and constrained pareto fronts for constrained multiobjective optimization
IEEE Trans. Cybern. (2022)
Google Scholar

[63] [63]
Liu Y., Liu J., Jin Y., Li F., Zheng T.
A surrogate-assisted two-stage differential evolution for expensive constrained optimization
IEEE Trans. Emerg. Top. Computat. Intell. (2023)
Google Scholar

[64] [64]
Zou J., Luo J., Liu Y., Yang S., Zheng J.
A flexible two-stage constrained multi-objective evolutionary algorithm based on automatic regulation
Inform. Sci., 634 (2023), pp. 227-243
View PDF View article View in Scopus Google Scholar

[65] [65]
Xia M., Chong Q., Dong M.
A constrained multi-objective evolutionary algorithm with two-stage resources allocation
Swarm Evol. Comput., 79 (2023), Article 101313
View PDF View article View in Scopus Google Scholar

[66] [66]
Feng J., Liu S., Yang S., Zheng J., Liu J.
An adaptive tradeoff evolutionary algorithm with composite differential evolution for constrained multi-objective optimization
Swarm Evol. Comput., 83 (2023), Article 101386
View PDF View article View in Scopus Google Scholar

[67] [67]
Chen G., Guo Y., Wang Y., Liang J., Gong D., Yang S.
Evolutionary dynamic constrained multiobjective optimization: Test suite and algorithm
IEEE Trans. Evol. Comput. (2023)
Google Scholar

[68] [68]
Huang W., Zou J., Liu Y., Yang S., Zheng J.
Global and local feasible solution search for solving constrained multi-objective optimization
Inform. Sci., 649 (2023), Article 119467
View PDF View article View in Scopus Google Scholar

[69] [69]
Yang D., Li G., Cheng G.
On the efficiency of chaos optimization algorithms for global optimization
Chaos Solitons Fractals, 34 (4) (2007), pp. 1366-1375
View PDF View article View in Scopus Google Scholar

[70] [70]
Aslimani N., El-Ghazali T., Ellaia R.
A new chaotic-based approach for multi-objective optimization
Algorithms, 13 (9) (2020), p. 204
Crossref View in Scopus Google Scholar

[71] [71]
Li Q., Liu L., Yuan X., et al.
Multiobjective parallel chaos optimization algorithm with crossover and merging operation
Math. Probl. Eng., 2020 (2020)
Google Scholar

[72] [72]
Zhang L., Yu R., Liu J., Xu Y., Jiang H., Xu P., Lin P., Luo Y.
A constrained multi-objective evolutionary algorithm based on weak cooperation framework and multi-chaotic operators
2022 Global Conference on Robotics, Artificial Intelligence and Information Technology, GCRAIT (2022), pp. 102-106, 10.1109/GCRAIT55928.2022.00030
View in Scopus Google Scholar

[73] [73]
Takubo Y., Kanazaki M.
Robust constrained multi-objective evolutionary algorithm based on polynomial chaos expansion for trajectory optimization
2022 IEEE Congress on Evol. Comput., CEC, IEEE (2022), pp. 1-9
Crossref Google Scholar

[74] [74]
Yacoubi S., Manita G., Chhabra A., Korbaa O., Mirjalili S.
A multi-objective chaos game optimization algorithm based on decomposition and random learning mechanisms for numerical optimization
Appl. Soft Comput., 144 (2023), Article 110525
View PDF View article View in Scopus Google Scholar

[75] [75]
Rauf H.T., Gao J., Almadhor A., Haider A., Zhang Y.-D., Al-Turjman F.
Multi population-based chaotic differential evolution for multi-modal and multi-objective optimization problems
Appl. Soft Comput., 132 (2023), Article 109909
View PDF View article View in Scopus Google Scholar

[76] [76]
Tian Y., Peng S., Rodemann T., Zhang X., Jin Y.
Automated selection of evolutionary multi-objective optimization algorithms
2019 IEEE Symposium Series on Computational Intelligence, SSCI, IEEE (2019), pp. 3225-3232
Crossref View in Scopus Google Scholar

[77] [77]
Qiao K., Yu K., Qu B., Liang J., Yue C., Ban X.
Feature extraction for recommendation of constrained multi-objective evolutionary algorithms
IEEE Trans. Evol. Comput. (2022)
Google Scholar

[78] [78]
Han Z., Yang Y., Ye H.
A deep reinforcement learning based multiple meta-heuristic methods approach for resource constrained multi-project scheduling problem
2022 7th International Conference on Intelligent Computing and Signal Processing, ICSP, IEEE (2022), pp. 26-29
Crossref View in Scopus Google Scholar

[79] [79]
Wang Y., Zuo M., Gong D.
Migration-based algorithm library enrichment for constrained multi-objective optimization and applications in algorithm selection
Inform. Sci., 649 (2023), Article 119593
View PDF View article View in Scopus Google Scholar

[80] [80]
Ming F., Gong W., Gao L.
Adaptive auxiliary task selection for multitasking-assisted constrained multi-objective optimization [feature]
IEEE Comput. Intell. Mag., 18 (2) (2023), pp. 18-30
Crossref View in Scopus Google Scholar

[81] [81]
Fallahi A., Bani E.A., Niaki S.T.A.
A constrained multi-item EOQ inventory model for reusable items: Reinforcement learning-based differential evolution and particle swarm optimization
Expert Syst. Appl., 207 (2022), Article 118018
View PDF View article View in Scopus Google Scholar

[82] [82]
Yu X., Xu P., Wang F., Wang X.
Reinforcement learning-based differential evolution algorithm for constrained multi-objective optimization problems
Eng. Appl. Artif. Intell., 131 (2024), Article 107817
View PDF View article View in Scopus Google Scholar

[83] [83]
Ma L., Li N., Guo Y., Wang X., Yang S., Huang M., Zhang H.
Learning to optimize: reference vector reinforcement learning adaption to constrained many-objective optimization of industrial copper burdening system
IEEE Trans. Cybern. (2021)
Google Scholar

[84] [84]
Wang C., Huang H., Zhang X.
Growing neural gas network for offspring generation in evolutionary constrained multi-objective optimization
IEEE Trans. Emerg. Top. Computat. Intell. (2023)
Google Scholar

[85] [85]
Kulesza A., Taskar B., et al.
Determinantal point processes for machine learning
Found. Trends Mach. Learn., 5 (2–3) (2012), pp. 123-286
Crossref View in Scopus Google Scholar

[86] [86]
Hu Z., Gong W., Pedrycz W., Li Y.
Deep reinforcement learning assisted co-evolutionary differential evolution for constrained optimization
Swarm Evol. Comput., 83 (2023), Article 101387
View PDF View article View in Scopus Google Scholar

[87] [87]
Liu Z., Zhao P., Cao J., Zhang J., Chen Z.
A constrained multi-objective evolutionary algorithm with Pareto estimation via neural network
Expert Syst. Appl., 237 (2024), Article 121718
View PDF View article View in Scopus Google Scholar

[88] [88]
Wolpert D.H., Macready W.G.
No free lunch theorems for optimization
IEEE Trans. Evol. Comput., 1 (1) (1997), pp. 67-82
View in Scopus Google Scholar

[89] [89]
Liu F., Lin X., Wang Z., Yao S., Tong X., Yuan M., Zhang Q.
Large language model for multi-objective evolutionary optimization
(2023)
arXiv preprint arXiv:2310.12541
Google Scholar

[90] [90]
Ray T., Tai K., Seow K.C.
Multiobjective design optimization by an evolutionary algorithm
Eng. Optim., 33 (4) (2001), pp. 399-424
Crossref View in Scopus Google Scholar

[91] [91]
Bao Q., Wang M., Dai G., Chen X., Song Z., Li S.
An archive-based two-stage evolutionary algorithm for constrained multi-objective optimization problems
Swarm Evol. Comput., 75 (2022), Article 101161
View PDF View article View in Scopus Google Scholar

[92] [92]
Kocis G.R., Grossmann I.E.
Global optimization of nonconvex mixed-integer nonlinear programming (MINLP) problems in process synthesis
Ind. Eng. Chem. Res., 27 (8) (1988), pp. 1407-1421
Crossref View in Scopus Google Scholar

[93] [93]
Xiang Y., Yang X., Zhou Y., Huang H.
Enhancing decomposition-based algorithms by estimation of distribution for constrained optimal software product selection
IEEE Trans. Evol. Comput., 24 (2) (2019), pp. 245-259
Crossref View in Scopus Google Scholar

[94] [94]
Xiang Y., Yang X., Huang H., Wang J.
Balancing constraints and objectives by considering problem types in constrained multiobjective optimization
IEEE Trans. Cybern. (2021)
Google Scholar

[95] [95]
Santana-Quintero L.V., Hernández-Díaz A.G., Molina J., Coello C.A.C., Caballero R.
DEMORS: A hybrid multi-objective optimization algorithm using differential evolution and rough set theory for constrained problems
Comput. Oper. Res., 37 (3) (2010), pp. 470-480
View PDF View article View in Scopus Google Scholar

[96] [96]
Qin C., Ming F., Gong W., Gu Q.
Constrained multi-objective optimization via two archives assisted push–pull evolutionary algorithm
Swarm Evol. Comput., 75 (2022), Article 101178
View PDF View article View in Scopus Google Scholar

[97] [97]
Kohli M., Arora S.
Chaotic grey wolf optimization algorithm for constrained optimization problems
J. Comput. Des. Eng., 5 (4) (2018), pp. 458-472
View PDF View article Crossref View in Scopus Google Scholar

[98] [98]
Vallozzi L., Vandendriessche W., Rogier H., Hertleer C., Scarpello M.
Design of a protective garment GPS antenna
Microw. Opt. Technol. Lett., 51 (6) (2009), pp. 1504-1508
Crossref View in Scopus Google Scholar

[99] [99]
Saravanan R., Ramabalan S., Ebenezer N.G.R., Dharmaraja C.
Evolutionary multi criteria design optimization of robot grippers
Appl. Soft Comput., 9 (1) (2009), pp. 159-172
View PDF View article View in Scopus Google Scholar

[100] [100]
Yang Y., Liu J., Tan S., Wang H.
Application of constrained multi-objective evolutionary algorithm in multi-source compressed-air pipeline optimization problems
IFAC-PapersOnLine, 51 (18) (2018), pp. 168-173
View PDF View article Google Scholar

[101] [101]
Yang Y., Liu J., Tan S., Liu Y.
A multi-objective differential evolution algorithm based on domination and constraint-handling switching
Inform. Sci., 579 (2021), pp. 796-813
View PDF View article View in Scopus Google Scholar

[102] [102]
Yang Y., Liu J., Tan S.
A partition-based constrained multi-objective evolutionary algorithm
Swarm Evol. Comput., 66 (2021), Article 100940
View PDF View article View in Scopus Google Scholar

[103] [103]
Zhang Q., Wang Q., Li G.
Switched system identification based on the constrained multi-objective optimization problem with application to the servo turntable
Int. J. Control Autom. Syst., 14 (5) (2016), pp. 1153-1159
Crossref View in Scopus Google Scholar

[104] [104]
Mala-Jetmarova H., Sultanova N., Savic D.
Lost in optimisation of water distribution systems? A literature review of system operation
Environ. Model. Softw., 93 (2017), pp. 209-254
View PDF View article View in Scopus Google Scholar

[105] [105]
Ouyang Q., Lu W., Hou Z., Zhang Y., Li S., Luo J.
Chance-constrained multi-objective optimization of groundwater remediation design at DNAPLs-contaminated sites using a multi-algorithm genetically adaptive method
J. Contam. Hydrol., 200 (2017), pp. 15-23
View PDF View article View in Scopus Google Scholar

[106] [106]
Marti J., Geissbühler L., Becattini V., Haselbacher A., Steinfeld A.
Constrained multi-objective optimization of thermocline packed-bed thermal-energy storage
Appl. Energy, 216 (2018), pp. 694-708
View PDF View article View in Scopus Google Scholar

[107] [107]
Su Z., Zhang G., Yue F., Zhan D., Li M., Li B., Yao X.
Enhanced constraint handling for reliability-constrained multiobjective testing resource allocation
IEEE Trans. Evol. Comput., 25 (3) (2021), pp. 537-551
Crossref View in Scopus Google Scholar

[108] [108]
Dhiman G., Kumar V.
Multi-objective spotted hyena optimizer: a multi-objective optimization algorithm for engineering problems
Knowl.-Based Syst., 150 (2018), pp. 175-197
View PDF View article View in Scopus Google Scholar

[109] [109]
Ming F., Gong W., Wang L., Gao L.
A constrained many-objective optimization evolutionary algorithm with enhanced mating and environmental selections
IEEE Trans. Cybern. (2022)
Google Scholar

[110] [110]
Shi J., Chen M., Ma Y., Qiao F.
A new boredom-aware dual-resource constrained flexible job shop scheduling problem using a two-stage multi-objective particle swarm optimization algorithm
Inform. Sci., 643 (2023), Article 119141
View PDF View article View in Scopus Google Scholar

[111] [111]
Hou Y., Zhang Y., Wu N., Zhu Q.
Constrained multi-objective optimization of short-term crude oil scheduling with dual pipelines and charging tank maintenance requirement
Inform. Sci., 588 (2022), pp. 381-404
View PDF View article Google Scholar

[112] [112]
Zuo X., Chen C., Tan W., Zhou M.
Vehicle scheduling of an urban bus line via an improved multiobjective genetic algorithm
IEEE Trans. Intell. Transp. Syst., 16 (2) (2014), pp. 1030-1041
Google Scholar

[113] [113]
Ma Z., Wang Y.
Shift-based penalty for evolutionary constrained multiobjective optimization and its application
IEEE Trans. Cybern. (2021)
Google Scholar

[114] [114]
Güvenc U., Sönmez Y., Duman S., Yörükeren N.
Combined economic and emission dispatch solution using gravitational search algorithm
Sci. Iran., 19 (6) (2012), pp. 1754-1762
View PDF View article View in Scopus Google Scholar

[115] [115]
Chen M.-R., Zeng G.-Q., Lu K.-D.
Constrained multi-objective population extremal optimization based economic-emission dispatch incorporating renewable energy resources
Renew. Energy, 143 (2019), pp. 277-294
View PDF View article View in Scopus Google Scholar

[116] [116]
El-Shorbagy M.A., Mousa A.
Constrained multiobjective equilibrium optimizer algorithm for solving combined economic emission dispatch problem
Complexity, 2021 (2021), pp. 1-14
Google Scholar

[117] [117]
Yu X., Yu X., Lu Y., Yen G.G., Cai M.
Differential evolution mutation operators for constrained multi-objective optimization
Appl. Soft Comput., 67 (2018), pp. 452-466
View PDF View article View in Scopus Google Scholar

[118] [118]
Mohseni-Bonab S.M., Rabiee A., Mohammadi-Ivatloo B.
Voltage stability constrained multi-objective optimal reactive power dispatch under load and wind power uncertainties: A stochastic approach
Renew. Energy, 85 (2016), pp. 598-609
View PDF View article View in Scopus Google Scholar

[119] [119]
Peng C., Qiu S.
A decomposition-based constrained multi-objective evolutionary algorithm with a local infeasibility utilization mechanism for UAV path planning
Appl. Soft Comput., 118 (2022), Article 108495
View PDF View article View in Scopus Google Scholar

[120] [120]
Wang X., Zhao J., Cheng C., Qi M.
A multi-objective fuzzy facility location problem with congestion and priority for drone-based emergency deliveries
Comput. Ind. Eng., 179 (2023), Article 109167
View PDF View article Google Scholar

[121] [121]
Latpate R., Kurade S.S.
Multi-objective multi-index transportation model for crude oil using fuzzy NSGA-II
IEEE Trans. Intell. Transp. Syst., 23 (2) (2020), pp. 1347-1356
Google Scholar

[122] [122]
Chen C.-T., Pham H.-V.
Trajectory planning in parallel kinematic manipulators using a constrained multi-objective evolutionary algorithm
Nonlinear Dynam., 67 (2012), pp. 1669-1681
Crossref View in Scopus Google Scholar

[123] [123]
Maity S., Roy A., Maiti M.
A rough multi-objective genetic algorithm for uncertain constrained multi-objective solid travelling salesman problem
Granul. Comput., 4 (1) (2019), pp. 125-142
Crossref View in Scopus Google Scholar

[124] [124]
Kocis G., Grossmann I.
A modelling and decomposition strategy for the minlp optimization of process flowsheets
Comput. Chem. Eng., 13 (7) (1989), pp. 797-819, 10.1016/0098-1354(89)85053-7
URL: https://www.sciencedirect.com/science/article/pii/0098135489850537
View PDF View article View in Scopus Google Scholar

[125] [125]
Zou J., Sun R., Yang S., Zheng J.
A dual-population algorithm based on alternative evolution and degeneration for solving constrained multi-objective optimization problems
Inform. Sci., 579 (2021), pp. 89-102
View PDF View article View in Scopus Google Scholar

[126] [126]
Zou J., Sun R., Liu Y., Hu Y., Yang S., Zheng J., Li K.
A multi-population evolutionary algorithm using new cooperative mechanism for solving multi-objective problems with multi-constraint
IEEE Trans. Evol. Comput. (2023)
Google Scholar

[127] [127]
Zhu X., Jiang C., Hao K., Wang R.
High-dimensional interactive adaptive RVEA for multi-objective optimization of polyester polymerization process
Inform. Sci., 650 (2023), Article 119707
View PDF View article View in Scopus Google Scholar

[128] [128]
Kumar A., Wu G., Ali M.Z., Luo Q., Mallipeddi R., Suganthan P.N., Das S.
A benchmark-suite of real-world constrained multi-objective optimization problems and some baseline results
Swarm Evol. Comput., 67 (2021), Article 100961
View PDF View article View in Scopus Google Scholar

[129] [129]
Kumar A., Wu G., Ali M.Z., Mallipeddi R., Suganthan P.N., Das S.
A test-suite of non-convex constrained optimization problems from the real-world and some baseline results
Swarm Evol. Comput., 56 (2020), Article 100693
View PDF View article View in Scopus Google Scholar

[130] [130]
Tawhid M.A., Savsani V.
A novel multi-objective optimization algorithm based on artificial algae for multi-objective engineering design problems
Appl. Intell., 48 (2018), pp. 3762-3781
Crossref View in Scopus Google Scholar

[131] [131]
Kotinis M.
A particle swarm optimizer for constrained multi-objective engineering design problems
Eng. Optim., 42 (10) (2010), pp. 907-926
Crossref View in Scopus Google Scholar

[132] [132]
Chiandussi G., Codegone M., Ferrero S., Varesio F.E.
Comparison of multi-objective optimization methodologies for engineering applications
Comput. Math. Appl., 63 (5) (2012), pp. 912-942
View PDF View article View in Scopus Google Scholar

[133] [133]
Jain H., Deb K.
An evolutionary many-objective optimization algorithm using reference-point based nondominated sorting approach, part II: Handling constraints and extending to an adaptive approach
IEEE Trans. Evol. Comput., 18 (4) (2013), pp. 602-622
Google Scholar

[134] [134]
He C., Zhang Y., Gong D., Ji X.
A review of surrogate-assisted evolutionary algorithms for expensive optimization problems
Expert Syst. Appl. (2023), Article 119495
View PDF View article View in Scopus Google Scholar

[135] [135]
Huang H.-Z., Gu Y.-K., Du X.
An interactive fuzzy multi-objective optimization method for engineering design
Eng. Appl. Artif. Intell., 19 (5) (2006), pp. 451-460
View PDF View article View in Scopus Google Scholar

[136] [136]
Parsons M.G., Scott R.L.
Formulation of multicriterion design optimization problems for solution with scalar numerical optimization methods
J. Ship Res., 48 (01) (2004), pp. 61-76
Crossref View in Scopus Google Scholar

[137] [137]
Kannan B., Kramer S.N.
An augmented Lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design
(1994)
Google Scholar

[138] [138]
Gujarathi A.M., Purohit S., Srikanth B.
Optimization of reactor network design problem using jumping gene adaptation of differential evolution
Journal of Physics: Conference Series, vol. 622, no. 1, IOP Publishing (2015), Article 012044
Crossref View in Scopus Google Scholar

[139] [139]
Gadagi A., Adake C.
A constrained multi-objective optimization of turning process parameters by genetic algorithm and particle swarm optimization techniques
Mater. Today: Proc., 42 (2021), pp. 1207-1212
View PDF View article Google Scholar

[140] [140]
Yue H., Guo C., Li Q., Zhao L., Hao G.
Milling parameters optimization of Al-Li alloy thin-wall workpieces using response surface methodology and particle swarm optimization
CMES Comput. Model. Eng. Sci., 124 (3) (2020), pp. 937-952
Crossref View in Scopus Google Scholar

[141] [141]
Yang Y., Huang P.-Q., Kong X., Zhao J.
A constrained multi-objective evolutionary algorithm assisted by an additional objective function
Appl. Soft Comput., 132 (2023), Article 109904
View PDF View article View in Scopus Google Scholar

[142] [142]
Gu Q., Bai J., Li X., Xiong N., Lu C.
A constrained multi-objective evolutionary algorithm based on decomposition with improved constrained dominance principle
Swarm Evol. Comput., 75 (2022), Article 101162
View PDF View article View in Scopus Google Scholar

[143] [143]
Yuan J., Liu H.-L., He Z.
A constrained multi-objective evolutionary algorithm using valuable infeasible solutions
Swarm Evol. Comput., 68 (2022), Article 101020
View PDF View article View in Scopus Google Scholar

[144] [144]
Musselman K., Talavage J.
A tradeoff cut approach to multiple objective optimization
Oper. Res., 28 (6) (1980), pp. 1424-1435
Crossref View in Scopus Google Scholar

[145] [145]
Golafshani E.M., Arashpour M., Kashani A.
Green mix design of rubbercrete using machine learning-based ensemble model and constrained multi-objective optimization
J. Clean. Prod., 327 (2021), Article 129518
Google Scholar

[146] [146]
Wang Y., Hu H., Sun X., Zhang Y., Gong D.
Unified operation optimization model of integrated coal mine energy systems and its solutions based on autonomous intelligence
Appl. Energy, 328 (2022), Article 120106
View PDF View article View in Scopus Google Scholar

[147] [147]
Qiao J.-F., Hou Y., Zhang L., Han H.-G.
Adaptive fuzzy neural network control of wastewater treatment process with multiobjective operation
Neurocomputing, 275 (2018), pp. 383-393
View PDF View article View in Scopus Google Scholar

[148] [148]
Ming M., Wang R., Ishibuchi H., Zhang T.
A novel dual-stage dual-population evolutionary algorithm for constrained multiobjective optimization
IEEE Trans. Evol. Comput., 26 (5) (2021), pp. 1129-1143
Google Scholar

[149] [149]
Ray T., Tai K., Seow C.
An evolutionary algorithm for multiobjective optimization
Eng. Optim., 33 (3) (2001), pp. 399-424
Crossref View in Scopus Google Scholar

[150] [150]
Hou Y., Wu Y., Han H.
Multistate-constrained multiobjective differential evolution algorithm with variable neighborhood strategy
IEEE Trans. Cybern. (2022)
Google Scholar

[151] [151]
Yin X., Zhao Z., Yang W.
Optimizing cleaner productions of sustainable energies: A co-design framework for complementary operations of offshore wind and pumped hydro-storages
J. Clean. Prod., 396 (2023), Article 135832
View PDF View article View in Scopus Google Scholar

[152] [152]
Lee L.H., Tan K.C., Ou K., Chew Y.H.
Vehicle capacity planning system: A case study on vehicle routing problem with time windows
IEEE Trans. Syst. Man Cybern. A, 33 (2) (2003), pp. 169-178
View in Scopus Google Scholar

[153] [153]
Rathore A.K., Holtz J., Boller T.
Synchronous optimal pulsewidth modulation for low-switching-frequency control of medium-voltage multilevel inverters
IEEE Trans. Ind. Electron., 57 (7) (2010), pp. 2374-2381
View in Scopus Google Scholar

[154] [154]
Rathore A.K., Holtz J., Boller T.
Optimal pulsewidth modulation of multilevel inverters for low switching frequency control of medium voltage high power industrial AC drives
2010 IEEE Energy Conversion Congress and Exposition, IEEE (2010), pp. 4569-4574
View in Scopus Google Scholar

[155] [155]
Edpuganti A., Rathore A.K.
Optimal pulsewidth modulation for common-mode voltage elimination scheme of medium-voltage modular multilevel converter-fed open-end stator winding induction motor drives
IEEE Trans. Ind. Electron., 64 (1) (2016), pp. 848-856
Google Scholar

[156] [156]
Yang G., Li X., Huo L., Liu Q.
A solving approach for fuzzy multi-objective linear fractional programming and application to an agricultural planting structure optimization problem
Chaos Solitons Fractals, 141 (2020), Article 110352
View PDF View article View in Scopus Google Scholar

[157] [157]
Ozgen D., Gulsun B.
Combining possibilistic linear programming and fuzzy AHP for solving the multi-objective capacitated multi-facility location problem
Inform. Sci., 268 (2014), pp. 185-201
View PDF View article View in Scopus Google Scholar

[158] [158]
Peykani P., Nouri M., Eshghi F., Khamechian M., Farrokhi-Asl H.
A novel mathematical approach for fuzzy multi-period multi-objective portfolio optimization problem under uncertain environment and practical constraints
J. Fuzzy Ext. Appl., 2 (3) (2021), pp. 191-203
View in Scopus Google Scholar

[159] [159]
Mikolov T., Chen K., Corrado G., Dean J.
Efficient estimation of word representations in vector space
(2013)
arXiv preprint arXiv:1301.3781
Google Scholar

[160] [160]
Joulin A., Grave E., Bojanowski P., Mikolov T.
Bag of tricks for efficient text classification
(2016)
arXiv preprint arXiv:1607.01759
Google Scholar

[161] [161]
Ilić S., Marrese-Taylor E., Balazs J.A., Matsuo Y.
Deep contextualized word representations for detecting sarcasm and irony
(2018)
arXiv preprint arXiv:1809.09795
Google Scholar

[162] [162]
Devlin J., Chang M.-W., Lee K., Toutanova K.
Bert: Pre-training of deep bidirectional transformers for language understanding
(2018)
arXiv preprint arXiv:1810.04805
Google Scholar

[163] [163]
Scarselli F., Gori M., Tsoi A.C., Hagenbuchner M., Monfardini G.
The graph neural network model
IEEE Trans. Neural Netw., 20 (1) (2008), pp. 61-80
Google Scholar

[164] [164]
Chen Q., Ding J., Chai T., Pan Q.
Evolutionary optimization under uncertainty: The strategies to handle varied constraints for fluid catalytic cracking operation
IEEE Trans. Cybern., 52 (4) (2020), pp. 2249-2262
View in Scopus Google Scholar

[165] [165]
Yang C., Ding J.
Constrained dynamic multi-objective evolutionary optimization for operational indices of beneficiation process
J. Intell. Manuf., 30 (2019), pp. 2701-2713
Crossref View in Scopus Google Scholar

[166] [166]
Schlegel M., Marquardt W.
Adaptive switching structure detection for the solution of dynamic optimization problems
Ind. Eng. Chem. Res., 45 (24) (2006), pp. 8083-8094
Crossref View in Scopus Google Scholar

[167] [167]
Cai X., Xia C., Zhang Q., Mei Z., Hu H., Wang L., Hu J.
The collaborative local search based on dynamic-constrained decomposition with grids for combinatorial multiobjective optimization
IEEE Trans. Cybern., 51 (5) (2019), pp. 2639-2650
Google Scholar

[168] [168]
Qiao K., Yu K., Qu B., Liang J., Song H., Yue C., Lin H., Tan K.C.
Dynamic auxiliary task-based evolutionary multitasking for constrained multi-objective optimization
IEEE Trans. Evol. Comput. (2022)
Google Scholar

[169] [169]
Azzouz R., Bechikh S., Said L.B., Trabelsi W.
Handling time-varying constraints and objectives in dynamic evolutionary multi-objective optimization
Swarm Evol. Comput., 39 (2018), pp. 222-248
View PDF View article View in Scopus Google Scholar

[170] [170]
Chen Q., Ding J., Yang S., Chai T.
A novel evolutionary algorithm for dynamic constrained multiobjective optimization problems
IEEE Trans. Evol. Comput., 24 (4) (2019), pp. 792-806
Crossref Google Scholar

[171] [171]
Liang Z., Wu T., Ma X., Zhu Z., Yang S.
A dynamic multiobjective evolutionary algorithm based on decision variable classification
IEEE Trans. Cybern., 52 (3) (2020), pp. 1602-1615
Google Scholar

[172] [172]
Deb K., Rao N U.B., Karthik S.
Dynamic multi-objective optimization and decision-making using modified NSGA-II: a case study on hydro-thermal power scheduling
International Conference on Evolutionary Multi-Criterion Optimization, Springer (2007), pp. 803-817
Crossref View in Scopus Google Scholar

[173] [173]
Goh C.-K., Tan K.C.
A competitive-cooperative coevolutionary paradigm for dynamic multiobjective optimization
IEEE Trans. Evol. Comput., 13 (1) (2008), pp. 103-127
Crossref Google Scholar

[174] [174]
Zhou A., Jin Y., Zhang Q.
A population prediction strategy for evolutionary dynamic multiobjective optimization
IEEE Trans. Cybern., 44 (1) (2013), pp. 40-53
View in Scopus Google Scholar

[175] [175]
Muruganantham A., Tan K.C., Vadakkepat P.
Evolutionary dynamic multiobjective optimization via kalman filter prediction
IEEE Trans. Cybern., 46 (12) (2015), pp. 2862-2873
Google Scholar

[176] [176]
Lin J., Liu M., Hao J., Jiang S.
A multi-objective optimization approach for integrated production planning under interval uncertainties in the steel industry
Comput. Oper. Res., 72 (2016), pp. 189-203
View PDF View article View in Scopus Google Scholar

[177] [177]
Zhang Y., Gong D.-w., Zhang J.-h.
Robot path planning in uncertain environment using multi-objective particle swarm optimization
Neurocomputing, 103 (2013), pp. 172-185
View PDF View article View in Scopus Google Scholar

[178] [178]
Liu Y.-J., Zhang W.-G., Wang J.-B.
Multi-period cardinality constrained portfolio selection models with interval coefficients
Ann. Oper. Res., 244 (2016), pp. 545-569
Crossref View in Scopus Google Scholar

[179] [179]
Sun J., Yan X., Heng Z., Zhiping L.
Interval multi-objective programming methods for solving multi-period portfolio selection problems
Control Decis., 35 (3) (2020), pp. 645-650
View in Scopus Google Scholar

[180] [180]
Gong D., Xu B., Zhang Y., Guo Y., Yang S.
A similarity-based cooperative co-evolutionary algorithm for dynamic interval multiobjective optimization problems
IEEE Trans. Evol. Comput., 24 (1) (2019), pp. 142-156
View in Scopus Google Scholar

[181] [181]
Chiang T.-C., Hsu W.-H.
A knowledge-based evolutionary algorithm for the multiobjective vehicle routing problem with time windows
Comput. Oper. Res., 45 (2014), pp. 25-37
View PDF View article View in Scopus Google Scholar

[182] [182]
Xiang Y., Zhou Y., Zheng Z., Li M.
Configuring software product lines by combining many-objective optimization and SAT solvers
ACM Trans. Softw. Eng. Methodol., 26 (4) (2018), pp. 1-46
Google Scholar

[183] [183]
Nicosia G., Rinaudo S., Sciacca E.
An evolutionary algorithm-based approach to robust analog circuit design using constrained multi-objective optimization
International Conference on Innovative Techniques and Applications of Artificial Intelligence, Springer (2007), pp. 7-20
Google Scholar

[184] [184]
Kwag S., Ok S.-Y.
Robust design of seismic isolation system using constrained multi-objective optimization technique
KSCE J. Civ. Eng., 17 (2013), pp. 1051-1063
Crossref View in Scopus Google Scholar

[185] [185]
Zhan D., Xing H.
Expected improvement for expensive optimization: a review
J. Global Optim., 78 (3) (2020), pp. 507-544
Crossref View in Scopus Google Scholar

[186] [186]
Pan J.-S., Liu N., Chu S.-C., Lai T.
An efficient surrogate-assisted hybrid optimization algorithm for expensive optimization problems
Inform. Sci., 561 (2021), pp. 304-325
View PDF View article View in Scopus Google Scholar

[187] [187]
Forrester A.I., Bressloff N.W., Keane A.J.
Optimization using surrogate models and partially converged computational fluid dynamics simulations
Proc. R. Soc. A, 462 (2071) (2006), pp. 2177-2204
Crossref View in Scopus Google Scholar

[188] [188]
Gu Q., Wang Q., Li X., Li X.
A surrogate-assisted multi-objective particle swarm optimization of expensive constrained combinatorial optimization problems
Knowl.-Based Syst., 223 (2021), Article 107049
View PDF View article View in Scopus Google Scholar

[189] [189]
Song Z., Wang H., Xue B., Zhang M., Jin Y.
Balancing objective optimization and constraint satisfaction in expensive constrained evolutionary multi-objective optimization
IEEE Trans. Evol. Comput. (2023)
Google Scholar

[190] [190]
Yang Z., Qiu H., Gao L., Chen L., Liu J.
Surrogate-assisted MOEA/D for expensive constrained multi-objective optimization
Inform. Sci., 639 (2023), Article 119016
View PDF View article Google Scholar

[191] [191]
Wei F.-F., Chen W.-N., Li Q., Jeon S.-W., Zhang J.
Distributed and expensive evolutionary constrained optimization with on-demand evaluation
IEEE Trans. Evol. Comput. (2022)
Google Scholar

Constrained multi-objective optimization problems: Methodologies, algorithms and applications受约束的多目标优化问题：方法、算法和应用

Abstract 摘要

Keywords 关键词

1. Introduction 1.导言

2. State-of-the-art CMOAs2.最先进的 CMOA

2.1. Methods based on classical mathematics2.1.基于经典数学的方法

2.2. Methods based on evolution2.2.基于进化的方法

2.2.1. Methods based on multi-population coevolution2.2.1.基于多种群协同进化的方法

2.2.2. Methods based on multi-stage2.2.2.基于多阶段的方法

2.2.3. Methods based on chaotic optimization2.2.3.基于混沌优化的方法

2.3. Methods based on machine learning2.3.基于机器学习的方法

2.3.1. Algorithm recommendation2.3.1.算法建议

2.3.2. Intelligent parameter adjustment2.3.2.智能参数调整

2.3.3. Prediction generation2.3.3.预测生成

2.4. Discussion 2.4.讨论情况

3. Applications 3.应用

3.1. Engineering design problems3.1.工程设计问题

3.2. Resource optimization problems3.2.资源优化问题

3.3. Scheduling optimization problems3.3.调度优化问题

3.4. Path planning problems3.4.路径规划问题