A Review on Constraint Handling Techniques for Population-based Algorithms: from single-objective to multi- objective optimization
基于种群的算法约束处理技术综述：从单目标到多目标优化

Iman Rahimi ${}^1$ ,Amir H. Gandomi ${}^{1\ast }$ ,Fang Chen ${}^1$ ,and Efrén Mezura-Montes ${}^2$
伊曼·拉希米 ${}^1$ 、阿米尔·H·甘多米 ${}^{1\ast }$ 、陈芳 ${}^1$ 、埃弗伦·梅祖拉-蒙特斯 ${}^2$

${}^1$ Data Science Institute,Faculty of Engineering &Information Technology,University of Technology
${}^1$ 数据科学研究所，工程与信息技术学院，悉尼科技大学

Sydney, Australia; iman83@gmail.com , Gandomi@uts.edu.au, fang.chen@uts.edu.au
澳大利亚悉尼；iman83@gmail.com，Gandomi@uts.edu.au，fang.chen@uts.edu.au

2 Artificial Intelligence Research Institute, University of Veracruz, MEXICO, emezura@uv.mx
2 墨西哥韦拉克鲁斯大学人工智能研究所，eme@uv.mx

Abstract- Most real-world problems involve some type of optimization that is often constrained. Numerous researches have investigated several techniques to deal with constrained single-objective and multi-objective evolutionary optimization in many fields, including theory and application. Also, this presented study provides a novel analysis of scholarly literature on constraint handling techniques for single-objective and multi-objective population-based algorithms according to the most relevant journals, keywords, authors, and articles. The paper reviews the main ideas of the most state-of-the-art constraint handling techniques in multiobjective population-based optimization, and then the study addresses the bibliometric analysis in the field. The extracted papers include research articles, reviews, book/book chapters, and conference papers published between 2000 and 2020 for the analysis. The results indicate that the constraint handling techniques for multiobjective optimization have received much less attention compared with single-objective optimization. The most promising algorithms for such optimization were determined to be genetic algorithms, differential evolutionary algorithms, and particle swarm intelligence. Additionally, "Engineering," "Computer Science," and " Mathematics" were identified as the top three research fields, in which future research work is anticipated to increase.
摘要——大多数现实世界问题都涉及某种通常带有约束的优化。大量研究已探索了处理约束单目标和多目标进化优化的多种技术，涵盖理论与应用等多个领域。本研究还根据最相关的期刊、关键词、作者和文章，对基于种群的单目标和多目标算法约束处理技术的学术文献进行了新颖的分析。本文回顾了多目标种群优化中最先进约束处理技术的核心思想，随后对该领域的文献计量学分析进行了阐述。提取的文献包括 2000 年至 2020 年间发表的研究论文、综述、书籍/书籍章节及会议论文。结果表明，与单目标优化相比，多目标优化的约束处理技术受到的关注要少得多。 经确定，此类优化问题最具前景的算法包括遗传算法、差分进化算法和粒子群智能算法。此外，"工程学"、"计算机科学"和"数学"被列为未来研究工作量有望增长的三大重点研究领域。

Keywords- Constraint handling technique, Evolutionary Computation, Multiobjective optimization, Population-based algorithms.
关键词- 约束处理技术、进化计算、多目标优化、群体智能算法。

Most real-world problems are considered as multi-objective optimization problems (MOOPs). No single solution exists for an MOOP, instead different solutions generate trade-offs for different objectives. Furthermore, MOOPs arise in a natural fashion in most disciplines, and solving them has been a challenging issue for researchers. Although a variety of methods have been developed in Operations Research and other fields to address these problems, there is an urgent need for alternative approaches because of the complexities of their solution [1]-[3]. Evolutionary computation (EC) methods have been identified as more effective methods to handle this limitation and are suitable for solving MOOPs, for which the form of the Pareto-optimal front (discontinuity, nonconvexity, etc.) is not important [4], [5]. Moreover, most multiobjective evolutionary algorithms (MOEAs) use the dominance concept [6], [7].
大多数现实世界的问题都被视为多目标优化问题（MOOPs）。对于 MOOP 而言，不存在单一解，不同的解会为不同目标生成权衡方案。此外，MOOPs 在大多数学科中自然出现，解决它们一直是研究人员面临的挑战性问题。尽管运筹学等领域已开发出多种方法来应对这些问题，但由于解决方案的复杂性，迫切需要替代方法[1]-[3]。进化计算（EC）方法已被证明是更有效处理这一局限性的方法，适用于求解 MOOPs，其中帕累托最优前沿的形式（不连续性、非凸性等）并不重要[4][5]。此外，大多数多目标进化算法（MOEAs）都采用支配关系概念[6][7]。

To solve the constrained optimization of all real-world problems, constrained evolutionary algorithm optimization (CEAO) implements an evolutionary algorithm (EA) combining with a constraint handling technique (CHT). In a work by [8], an infeasible individual will be divided into different categories based on their distances to the feasible region, and ranking will be conducted according to the classes. [9] introduced an approach that assigns high and low priorities to constraints and objective functions, respectively. The authors of [10] proposed a CHT that only considers the inequality constraints, wherein the algorithm uses tournament selection that has better convergence properties in comparison to the proportionate selection operator [11]. However, the latter algorithm employs niche count for all populations that may increase the complexity of the computation.
为解决现实世界中的所有约束优化问题，约束进化算法优化（CEAO）通过将进化算法（EA）与约束处理技术（CHT）相结合来实现。在文献[8]的工作中，不可行个体将根据其与可行区域的距离被划分为不同类别，并依据类别进行排序。[9]提出了一种方法，分别为约束条件和目标函数分配高优先级和低优先级。[10]的作者提出了一种仅考虑不等式约束的 CHT，其中算法采用具有比比例选择算子[11]更好收敛特性的锦标赛选择。然而，后一种算法对所有种群采用小生境计数，这可能会增加计算的复杂性。

The authors of [43] Used a hybrid of PSO and GA to improve the balance between exploration and exploitation by using genetic operators, namely crossover and mutation in PSO. A few years later, [44] extended the parameter-less CHT so as to provide a balance between the feasible and infeasible solutions in a GA population. The authors of [45] proposed a new approach, known as boundary update (BU) technique, which is able to handle constraints directly (i.e. updating variable bounds) and tested the proposed method on several constrained optimization problem. The method proposed by [45] possesses the potential to couple with MOEA.
文献[43]的作者通过将遗传算子（即交叉和变异）引入粒子群优化(PSO)，提出了一种 PSO 与遗传算法(GA)的混合方法，以改善探索与开发之间的平衡。几年后，文献[44]扩展了无参数约束处理技术(CHT)，旨在实现遗传算法种群中可行解与不可行解之间的平衡。文献[45]的作者提出了一种称为边界更新(BU)的新方法，该方法能直接处理约束条件（即更新变量边界），并在多个约束优化问题上验证了所提方法的有效性。文献[45]提出的方法具有与多目标进化算法(MOEA)耦合的潜力。

It is noteworthy to mention that the majority of the mentioned studies focused on CHTs for single objective optimization with little attention towards multiobjective optimization. This is attributed to fact that most constraint handling methods developed for single objective optimization could be modified for multiobjective optimization as well [30].
值得注意的是，上述大多数研究集中于单目标优化的约束处理技术(CHTs)，对多目标优化的关注较少。这归因于为单目标优化开发的大多数约束处理方法经过修改后同样适用于多目标优化[30]。

To attain a better understanding of the research field and to provide new insights from relevant publications, this work aimed to answer the following questions:
为了更深入地理解该研究领域并从相关文献中获取新见解，本研究旨在回答以下问题：

The reminder of the study is as follows. Section 2 describes the research methodology. Section 3 presents the CHTs in EAs. Section 4 describes the CHTs in nature-inspired algorithms. Section 5 addresses the CHTs in multiobjective genetic algorithms. Section 6 presents a summary of novel approaches between 2020 and 2021. Section 7 discusses the scientometric analysis. Section 8 provides a summary of the study along with recommendations for future research. Concluding remarks are offered in the last section.
本研究的后续安排如下：第 2 节阐述研究方法论；第 3 节介绍 EA 中的 CHT 技术；第 4 节描述自然启发算法中的 CHT；第 5 节探讨多目标遗传算法中的 CHT；第 6 节汇总 2020 至 2021 年间的新颖研究方法；第 7 节进行科学计量分析；第 8 节总结研究成果并提出未来研究方向建议；末节给出结论性评述。

The research procedure in this work was divided into five stages (Figure 1). In the first stage, documents from databases were gathered from Scopus and Web of Science (WOS). For this aim, the authors used special keywords, namely (TITLE-ABS-KEY (constrained AND multi AND objective AND evol utionary AND optimization) OR TITLE-ABS-KEY (constraint AND handling AND multi AND o bjective AND evolutionary AND optimization) OR TITLE-ABS-
本研究的工作流程分为五个阶段（图 1）。第一阶段从 Scopus 和 Web of Science（WOS）数据库中收集文献资料。为此，作者使用了特定关键词组合，即(TITLE-ABS-KEY（constrained AND multi AND objective AND evolutionary AND optimization）或 TITLE-ABS-KEY（constraint AND handling AND multi AND objective AND evolutionary AND optimization）或 TITLE-ABS-

KEY (constrained AND multi AND objective AND swarm AND optimization) OR TITLE-ABS-KEY (constraint AND handling AND multi AND o bjective AND swarm AND optimization) to find the related articles published as of May 4, 2021. Supplementary $\mathrm{A}$ and $\mathrm{B}$ present the data extracted from Scopus and WOS, respectively. Since some of the articles were duplicates, they were identified and removed from the library in stage 2 using Mendeley as a powerful reference manager. Also, in stage 2, some research questions for this study were designed. An overview was initiated in stage 3 with a general illustration of the basic concepts of CHTs and comparison of methods. In stage 4, a social network analysis was performed to provide a scientometric analysis of the documents using VOSviewer [46], [47] and RStudio, which have been identified as powerful tools for scientometric analysis. Also, some interesting analytical features, such as number of pages and authors per article, were conducted in this stage. Moreover, stage 4 required several steps, including co-occurrence, co-authorship, citation, and citation network analyses. In the last stage, the results were obtained to formulate a discussion to answer the proposed research questions. Stage 5 required preparing the findings, identifying important gaps, and determining future research directions.
KEY（constrained AND multi AND objective AND swarm AND optimization）或 TITLE-ABS-KEY（constraint AND handling AND multi AND objective AND swarm AND optimization）用于查找截至 2021 年 5 月 4 日发表的相关文章。补充材料 $\mathrm{A}$ 和 $\mathrm{B}$ 分别展示了从 Scopus 和 WOS 提取的数据。由于部分文章存在重复，在第二阶段使用 Mendeley 作为强大的参考文献管理器进行了识别并从库中移除。同时，在第二阶段设计了本研究的一些研究问题。第三阶段启动了一个概述，对 CHTs 的基本概念进行了总体说明并比较了方法。第四阶段进行了社交网络分析，使用 VOSviewer[46]、[47]和 RStudio 对文档进行了科学计量分析，这些工具已被认为是科学计量分析的强大工具。此外，此阶段还进行了一些有趣的分析特征，如每篇文章的页数和作者数量。此外，第四阶段需要几个步骤，包括共现分析、合著分析、引用分析和引用网络分析。 在最后阶段，我们获得了相关结果以展开讨论，从而回答所提出的研究问题。第五阶段需要整理研究发现、识别重要空白领域并确定未来的研究方向。

Figure 1 Research Procedure
图 1 研究流程

Almost all real-world problems are considered as constraint problems. A general form of a constrained multi-objective optimization problem(CMOOP) is described as follows (Equations 1-3):
几乎所有现实世界问题都被视为约束问题。一个典型的约束多目标优化问题（CMOOP）的一般形式描述如下（公式 1-3）：

Maximize (Minimize)(1)  最大化（最小化）(1)

s.t.  约束条件

Where $\mathrm{F}\left(\mathrm{x}\right)$ is the objective vector; and $\mathrm{t},\mathrm{n}$ and $\mathrm{m}$ are the number of objective function, equality and inequality constraints, respectively. There is no single solution for a MOOP that simultaneously optimizes each objective, instead there exists a number of Pareto optimal solutions. A Pareto front of possible solutions is called optimal or nondominated if improving anyone's objective further would lead to a decrease in other objectives. According to previous surveys [17], [31], a simple taxonomy of the constraint handling methods in nature-inspired optimization algorithms is as follows:
其中 $\mathrm{F}\left(\mathrm{x}\right)$ 是目标向量； $\mathrm{t},\mathrm{n}$ 和 $\mathrm{m}$ 分别表示目标函数、等式约束和不等式约束的数量。多目标优化问题（MOOP）不存在能同时优化所有目标的单一解，而是存在一系列帕累托最优解。若一个解集的帕累托前沿在进一步优化任一目标时会导致其他目标劣化，则称该解集为最优或非支配解集。根据既往综述[17][31]，自然启发式优化算法中约束处理方法的简单分类如下：

1- Penalty functions methods
1- 罚函数法

2- Decoders  2-解码器

3- Special operators  3-特殊运算符

The first and fourth techniques are discussed in details later in the paper. As an example of decoders,[48] proposed a homomorphous mapping (HM) method between an $n$ -dimensional cube and feasible space. The feasible region can be mapped onto a sample space where a population-based algorithm could run a comparative performance [48]-[51]. However, this method requires high computational costs. A special operator is used to preserve the feasibility of a solution or move within a special region [52]-[54]. Nevertheless, this method is hindered by the initialization of feasible solutions in the initial population, which is challenging with highly-constrained optimization problems. In addition, the authors of [55] presented a taxonomy of CHTs in MOEA as follows (Figure 2):
第一和第四种技术将在后文详细讨论。以解码器为例，文献[48]提出了一种 $n$ 维立方体与可行空间之间的同态映射（HM）方法。可行域可映射到样本空间，基于种群的算法可在其中进行性能比较[48]-[51]。但该方法计算成本较高。特殊算子用于保持解的可行性或在特定区域内移动[52]-[54]，然而该方法受限于初始种群中可行解的生成，这在高度约束的优化问题中具有挑战性。此外，文献[55]提出了 MOEA 中约束处理技术（CHTs）的分类如下（图 2）：

method  方法

infeasible  不可行

solutions  解决方案

Figure 2 Taxonomy of different constraint handling methods in MOEA
图 2 MOEA 中各类约束处理方法的分类体系

Generally, penalty function techniques are one of the most simple CHTs. There are several types of penalty functions used with EAs, which the most important ones include [56]:
通常而言，惩罚函数技术是最简单的约束处理技术（CHTs）之一。进化算法中使用的惩罚函数主要有以下几种重要类型[56]：

Details regarding the penalty function methods will be discussed in the next section.
关于惩罚函数方法的具体细节将在下一节讨论。

3.1 Penalty function approach
3.1 惩罚函数法

One of the easiest and most common ways to handle constraints in multiobjective evolutionary algorithms is the penalty function method.
处理多目标进化算法中约束条件最简便且常用的方法之一就是惩罚函数法。

From a mathematical point-of-view, two types of penalty functions could be considered as follows:
从数学角度来看，可以考虑以下两种类型的惩罚函数：

The first type of penalty functions, interior methods, the penalty factor is selected such that the value will be small away from the constraint boundaries and need an initial feasible solution
第一类惩罚函数（内点法）的惩罚因子选择原则是：在远离约束边界时函数值较小，且需要初始可行解

[55], whereas exterior methods do not need an initial feasible solution [5]. Also, it should be noted that some of the infeasible solutions should be retained in the populations so that they are able to converge to a solution, which lies in the boundary between the feasible and infeasible regions [57].
而外点法则不需要初始可行解[5]。还需注意的是，应保留部分不可行解，使种群能够收敛到位于可行域与不可行域边界上的解[57]。

In the penalty function method, any infeasible solution is ignored [58]. First, all constraints should be normalized, and for each solution, the constraint violations are calculated as follows (Equation 4):
惩罚函数法会直接忽略任何不可行解[58]。首先需对所有约束进行归一化处理，并按下式（公式 4）计算每个解的约束违反量：

where $\overline{g}\left(x^i\right)$ refer to the normalized values for a given constraint ${\overline{g}}_j\left(x^i\right)\ge 0,j=1,\dots ,J$ . Once the violations for the constraints are calculated, the values are added to determine the overall violation as follows (Equation 5):
其中 $\overline{g}\left(x^i\right)$ 表示给定约束 ${\overline{g}}_j\left(x^i\right)\ge 0,j=1,\dots ,J$ 的归一化值。计算完各约束的违反值后，通过以下方式将这些值相加以确定总体违反程度（公式 5）：

Also, a penalty parameter is multiplied to the sum of constraint violations and then added to the objective function values. If a proper penalty parameter is selected, MOEAs will work well; otherwise, a set of infeasible solutions or a poor distribution of solution is possible.
此外，约束违反值的总和会乘以一个惩罚参数，然后加到目标函数值中。若选择合适的惩罚参数，多目标进化算法（MOEAs）将表现良好；否则可能导致不可行解集或解分布不佳的情况。

In static penalty proposed by [59], the penalty coefficient increases as a higher level of violation is reached. In fact, penalty functions do not change, a static penalty function is suggested, and several levels of violation are introduced in which the static penalty parameter is changed in case higher levels of violation are achieved [60]. In the static penalty function, the expanded objective function is (Equation 6):
在[59]提出的静态罚函数中，罚系数随违反程度增加而递增。实际上罚函数本身不变，而是建议采用静态罚函数，并引入多个违反等级——当达到更高违反等级时，静态罚参数会相应调整[60]。静态罚函数中的扩展目标函数表达式为（公式 6）：

Where $G_j=max{\{0,g_j\left(x\right)\}}^{\beta }$ ; and $\mathrm{k}=1,\dots ,1$ where 1 presents the number of levels of violation.
其中 $G_j=max{\{0,g_j\left(x\right)\}}^{\beta }$ ；而 $\mathrm{k}=1,\dots ,1$ 中的 1 表示违规层级数。

3.1.2 Dynamic penalty functions
3.1.2 动态惩罚函数

In this category, functions are changed based on the iteration number. [61] Proposed the following dynamic penalty function, in which the penalty increased when the iteration number increases.
本类函数会随迭代次数变化而调整。[61]提出以下动态惩罚函数，其惩罚力度随迭代次数增加而递增。

Dynamic multiobjective optimization problems (DMOOPs) involve the simultaneous optimization of different objectives subject to a number of given constraints, where the objective functions, constraints, and/or dimensions of the objective space could change over time. EAs have acquired great attention among researchers for solving the above-mentioned problems.
动态多目标优化问题（DMOOPs）需在满足若干约束条件下同时优化多个目标，其目标函数、约束条件和/或目标空间维度可能随时间变化。进化算法在解决此类问题方面已获得研究者的广泛关注。

3.1.3 Adaptive penalty functions In this category, infeasible individuals are penalized according to the feedback taken from the search process [62]. [63] proposed a CHT based on the adaptive penalty function and distance measure, which both change as the objective function value and constraint violations of an individual varies.
3.1.3 自适应惩罚函数 此类方法根据搜索过程的反馈对不可行个体施加惩罚[62]。[63]提出了一种基于自适应惩罚函数和距离度量的约束处理技术，二者会随着个体的目标函数值及约束违反程度动态调整。

Penalty-based constraint handing for multiobjective optimization is similar to single-objective problems in which a penalty factor is added to all the objectives. [63] Proposed a self-adaptive penalty function, which is suitable for solving constraint multiobjective optimization problems using evolutionary algorithms. In the self-adaptive penalty function method, the amount of penalty added to infeasible individuals are identified by tracking the number of feasible individuals. Also, the method uses improved objective values instead of the original objective function values [64].
多目标优化中的基于惩罚的约束处理方法与单目标问题类似，即在所有目标函数中添加惩罚因子。[63]提出了一种适用于进化算法求解约束多目标优化问题的自适应惩罚函数。该方法通过追踪可行解数量来确定对不可行个体的惩罚量，并使用改进后的目标函数值替代原始目标函数值[64]。

3.1.4 Annealing-based penalty
3.1.4 基于退火的惩罚

functions  函数

[65] Introduced a multiplicative penalty function based on simulated annealing. In this type of penalty function, the temperature is decreased when the iteration number increases, which leads to an increased penalty.
[65] 引入了一种基于模拟退火的乘法惩罚函数。在此类惩罚函数中，温度随迭代次数增加而降低，从而导致惩罚力度增强。

3.1.5 Co-evolutionary-based penalty functions
3.1.5 基于协同进化的惩罚函数

[66] Proposed a co-evolutionary approach in which the population is partitioned into two subpopulations. The first population evolves solutions, and the second population evolves penalty factors. In this approach, the penalty function considers information taken from the amount of constraint violations and a number of violations.
[66]提出了一种协同进化方法，将种群划分为两个子群。第一子群进化解决方案，第二子群进化惩罚因子。该方法中，惩罚函数综合考虑了约束违反程度和违反次数的信息。

There are other types of penalty function methods, which table 1 presents a summary and critique of the techniques for constraint handling.
还存在其他类型的惩罚函数方法，表 1 对这些约束处理技术进行了总结与评述。

The next section provides further details about this type of constraint handling method.
下一节将详细介绍此类约束处理方法的更多细节。

3.2.1 Constraint dominance principle In constraint dominance principle (CDP), three feasibility rules are applied to compare any two solutions. If $x^1$ is feasible and $x^2$ is infeasible, then $x^1$ would be better than $x^2$ . If both solutions are infeasible, then the solution with a smaller constraint violation is better. If both are feasible, then the one dominating the other is better. [16] Adopted CDP to handle constraints in NSGAII (NSGAII-CDP), in which the population is divided into feasible and infeasible subpopulations. NSGAII-CDP first selects offspring from the feasible solutions and then selects solutions from the infeasible solutions.[80] Also adopted CDP to handle constraints in the MOEA/D framework.
3.2.1 约束支配原则 在约束支配原则（CDP）中，采用三条可行性规则比较任意两个解：若解 $x^1$ 可行而解 $x^2$ 不可行，则解 $x^1$ 优于解 $x^2$ ；若两解均不可行，则约束违反程度较小的解更优；若两解均可行，则支配解更优。[16] 该研究将 CDP 应用于 NSGAII 算法的约束处理（NSGAII-CDP），将种群划分为可行与不可行子群，优先从可行解中选择后代，再从不可行解中补充选择。[80] 另有研究将 CDP 整合至 MOEA/D 框架处理约束。

The basic principle of the $\epsilon$ -constrained method, first introduced by [81], is similar to the superiority of feasible solution (SF) proposed by [82] (Eqs. 12-13). The epsilon value is updated until the parameter k reaches the control generations $T_c$ . [83] Embedded the epsilon CHT in MOEA/D so that the epsilon value is set adaptively of $r$ comparison. Also,the violation threshold is based on the type of constraints, size of the feasible space, and the search outcome. In the method proposed by [83], the infeasible solutions with violations less than threshold are identified (Eqs. 7-8).
ε约束方法的基本原理最早由[81]提出，其思想类似于[82]提出的可行解优越性（SF）准则（见公式 12-13）。该算法持续更新ε值直至参数 k 达到控制代数 $T_c$ 。[83]将ε约束处理技术嵌入 MOEA/D 框架，使ε值能根据 $r$ 比较自适应调整。此外，违界阈值设置综合考虑约束类型、可行空间大小及搜索状态。在[83]提出的方法中，会筛选出违界值低于阈值的不可行解（参见公式 7-8）。

$\epsilon \left(k\right)$(8)

where $x_{\theta }$ presents the top $\theta$ th individual at initialization; and the cp parameter is selected between $\left[2,10\right]\left[84\right]$ .
其中 $x_{\theta }$ 表示初始化阶段排名前 $\theta$ 位的个体；cp 参数取值区间为 $\left[2,10\right]\left[84\right]$ 。

3.2.3 Feasibility rules  3.2.3 可行性规则

The popularity of this method depends on its ability to be coupled to a range of algorithms without announcing new parameters (factors) [30].
该方法的普及性取决于其能与多种算法耦合且无需引入新参数（因子）的能力[30]。

The feasibility rules proposed by [13] are simple, could be integrated into a variety of algorithms without adding new parameters, and thus, are largely used in the research field. [85]-[87] developed feasibility rules for the selection process, which have been adopted by different evolutionary algorithms such as DE, PSO, and GA. According to the number of feasible solutions, the search space could be divided into three phases as follows [84]:
文献[13]提出的可行性规则简单易行，可集成到多种算法中且无需新增参数，因此在研究领域被广泛采用。[85]-[87]开发了用于选择过程的可行性规则，这些规则已被差分进化算法(DE)、粒子群优化(PSO)和遗传算法(GA)等不同进化算法所采纳。根据可行解的数量，搜索空间可分为以下三个阶段[84]：

The feasibility rules used in multiobjective optimization, also known as the superiority of feasible solution (SF), are addressed as follows [84] (Equation 9):
多目标优化中使用的可行性规则，即可行解优越性（SF），按如下方式处理[84]（公式 9）：

${\text{fitness}}_m\left(x\right)=\{\begin{array}{cc}{f}_m\left(x\right)& \text{ if }x\text{ is feasible }\\ f_{\text{worst }}^m+v\left(x\right)& \text{ otherwise }\end{array}$(9)

where $f_{\text{worst }}^m$ and $\mathrm{v}\left(\mathrm{x}\right)$ show the mth objective value of the worst feasible solution and the overall constraint violation, respectively.
其中 $f_{\text{worst }}^m$ 和 $\mathrm{v}\left(\mathrm{x}\right)$ 分别表示最差可行解的第 m 个目标值和总体约束违反程度。

3.2.3.1 Feasibility rules in Differential Evolution (DE)
3.2.3.1 差分进化算法中的可行性规则

Although the feasibility rules introduced by [13] have also been widely used by other researchers in DE [85], [86], [88]-[94], they have been rarely used in multiobjective differential evolution. Particularly, [95] used Pareto dominance in constrained space instead of the sum of constraint violations. Later, [96] adopted the Pareto dominance in Generalized Differential Evolution (GDE), but encountered difficulties when there exist more than three constraints and/or objective functions.
尽管文献[13]提出的可行性规则已被其他研究者广泛应用于差分进化算法[85][86][88]-[94]，但在多目标差分进化中鲜少使用。特别是文献[95]采用约束空间中的帕累托支配关系替代约束违反值之和。随后文献[96]在广义差分进化(GDE)中采用了帕累托支配，但当存在超过三个约束和/或目标函数时遇到了困难。

[97] Proposed a scheme for partitioning the objective space using the conflict information for multiobjective optimization. [98] introduced an operational efficient model based on Data Envelopment Analysis (DEA) and introduced DE along with the feasibility rules to optimize the mentioned model. [99] proposed a combined constraint handling framework, known as CCHF, for solving constrained optimization problems, in which the features of two well-known CHTs (i.e. feasibility rules and multiobjective optimization) were addressed in three different situations (feasible situation, infeasible situation, and semi-feasible situation).
[97]提出了一种利用冲突信息对多目标优化的目标空间进行划分的方案。[98]引入了一种基于数据包络分析(DEA)的高效操作模型，并结合可行性规则引入了 DE 来优化上述模型。[99]提出了一个名为 CCHF 的组合约束处理框架，用于求解约束优化问题，该框架在三种不同情境（可行情境、不可行情境和半可行情境）中整合了两种著名 CHT（即可行性规则和多目标优化）的特性。

3.2.3.2 Feasibility rules in PSO [100] Employed feasibility rules as a constraint handling technique to recognize the most competitive PSO variant when solving constrained numerical optimization problems (CNOPs). In the research by [100], local-best was identified to be better than global best PSO. [101] [102] adapted an artificial bee colony algorithm (ABC) to solve CNOPs by using feasibility rules by modifying the probability assignment for the roulette wheel selection.[103] Compared different GA variants using the feasibility rules as the constraint handling method. [104] Introduced a hybrid version of PSO to solve constrained optimization problems and found that the swarm at each generation is split into several sub-swarms. Also, the hybrid version applied the feasibility rules to compare particles in the swarm.
3.2.3.2 PSO 中的可行性规则 [100]采用可行性规则作为约束处理技术，用于在求解约束数值优化问题（CNOPs）时识别最具竞争力的 PSO 变体。在[100]的研究中，发现局部最优 PSO 优于全局最优 PSO。[101][102]通过修改轮盘赌选择的概率分配，采用人工蜂群算法（ABC）结合可行性规则来求解 CNOPs。[103]使用可行性规则作为约束处理方法，比较了不同遗传算法（GA）变体的性能。[104]提出了一种混合 PSO 版本用于求解约束优化问题，发现每一代的群体被分割成若干子群，且该混合版本应用可行性规则来比较群体中的粒子。

3.2.3.3 Feasibility rules in GA [105] Proposed a GA with a new multi-parent crossover for solving constraint optimization problems, in which the feasibility rules were added to handle the constraints. The latter authors also solved constraint numerical optimization problems by using different GA variants along with feasibility rules and found that all GAs perform equally. [22] Introduced a two-phase framework for solving constraint optimization problems. Specifically, the first phase ignores the objective function and the genetic algorithm minimizes the violation of the solutions, while the second phase optimizes bi-objective functions, including the original objective function and constraints satisfaction. Moreover, feasibility rules is applied to assign fitness values to the individuals.
3.2.3.3 遗传算法中的可行性规则 [105]提出了一种采用新型多亲代交叉的遗传算法来解决约束优化问题，其中加入可行性规则以处理约束条件。后者还通过结合不同遗传算法变体与可行性规则求解约束数值优化问题，发现所有遗传算法表现相当。[22]提出了解决约束优化问题的两阶段框架：第一阶段忽略目标函数，遗传算法专注于最小化解的违反程度；第二阶段优化双目标函数（包括原始目标函数和约束满足度）。此外，应用可行性规则为个体分配适应度值。

3.2.3.4 Feasibility rules in other population-based algorithms
3.2.3.4 其他群体智能算法中的可行性规则

Feasibility rules have been adapted to other population-based algorithms, such as artificial immune systems [106]-[111], organizational evolutionary algorithm [112], biogeography-based optimization [113], and bacterial foraging optimization [114].
可行性规则已被应用于人工免疫系统[106]-[111]、组织进化算法[112]、生物地理学优化[113]和细菌觅食优化[114]等群体智能算法。

3.3 Retaining infeasible solutions in the population
3.3 在种群中保留不可行解

Another CHT is used to retain the infeasible individuals in the population. In other words, a constraint multiobjective optimization problem with $\mathrm{m}$ objective is transformed to an optimization problem with $\mathrm{m}+1$ objectives,which could save the infeasible solution during the evolution process [28], [115]. [116] proposed a constraint handling technique so that the individuals that possess low Pareto rank and low constraint violation will be chosen.
另一种约束处理技术（CHT）用于保留种群中的不可行个体。换言之，通过将带有 $\mathrm{m}$ 目标的约束多目标优化问题转化为具有 $\mathrm{m}+1$ 目标的优化问题，可以在进化过程中保留不可行解[28][115]。文献[116]提出了一种约束处理技术，使得具有低帕累托等级和低约束违反的个体被优先选择。

3.4 Hybrid methods  3.4 混合方法

Hybrid methods combine several CHTs to handle constraints. [117] addressed four different hybrid methods in their work: 1) the ensemble of constraint handling method [84] separates the population into three sub-populations; 2) the adaptive trade-off model [118] contains two different CHTs; 3) in the push and pull search [119], the population is pushed to the unconstrained Pareto front (push) then the population is pulled back to the Pareto front (pull); 4) the two-phase framework (ToP) [120] solves a constraint multiobjective optimization problem by first converting the objective functions into a single objective function via the weighting method then, in the second phase, a constrained MOEA is adopted to attain the Pareto feasible solutions. The disadvantages for each category in Figure 1 are summarized in Figure3. The authors of [30] Presented a state-of-the-art taxonomy of CHTs, which is illustrated in Figure 4. Figure 5 presents the different state-of-the-art CHTs that have been used since 2000. As mentioned, [13] first applied feasibility rules to the genetic algorithm. [121] introduced stochastic ranking, which employs a user-defined parameter instead of using penalty factors and is able to control the infeasible solutions based on the sum of constraint violation and objective function values. [33] Proposed the epsilon-constraint method that transforms the constraint optimization problem into an unconstrained problem. [33] Addressed a multi-constrained optimization problem based on the KS function. [122] proposed a boundary search approach inspired by the ant colony metaphor based on conducting a boundary search between a feasible and infeasible solution. [28] Proposed an additional objective to solve a bi-objective optimization problem, where the first objective is the original problem and the second objective is the constraint violation measure. [36] Addressed a combination of four CHTs, namely feasibility rules, stochastic ranking, self-adaptive penalty function and the epsilon-constraint method to solve constraint numerical optimization problems. The first and second techniques are discussed in detail in the next sections, while the other methods presented in Figure 3 have been addressed previously in this study.
混合方法结合了多种约束处理技术（CHT）来处理约束条件。[117]在其工作中探讨了四种不同的混合方法：1）约束处理方法集成[84]将种群划分为三个子种群；2）自适应权衡模型[118]包含两种不同的 CHT；3）在推拉搜索[119]中，种群先被推向无约束帕累托前沿（推阶段），随后被拉回至约束帕累托前沿（拉阶段）；4）两阶段框架（ToP）[120]通过权重法首先将多目标函数转化为单目标函数，然后在第二阶段采用约束多目标进化算法（MOEA）获取帕累托可行解。图 1 中各类别的缺点总结于图 3。[30]的作者提出了当前最先进的 CHT 分类法，如图 4 所示。图 5 展示了自 2000 年以来使用的各类前沿 CHT 技术。如前所述，[13]首次将可行性规则应用于遗传算法。 [121]引入了随机排序法，该方法采用用户定义参数而非惩罚因子，并能基于约束违反与目标函数值的总和控制不可行解。[33]提出了ε-约束方法，将约束优化问题转化为无约束问题。[33]基于 KS 函数处理了多约束优化问题。[122]提出受蚁群隐喻启发的边界搜索方法，通过在可行解与不可行解之间进行边界搜索来实现。[28]通过添加第二目标（约束违反度量）来解决双目标优化问题，其中第一目标为原始问题。[36]综合运用四种约束处理技术（可行性规则、随机排序、自适应惩罚函数及ε-约束法）求解约束数值优化问题。前两种技术将在后续章节详细讨论，图 3 所示的其他方法已在本研究前期予以阐述。

Figure 5 Timeline of different state-of-the-art CHTs
图 5 不同最先进 CHTs 技术的时间线

[121] proposed the stochastic ranking (SR) approach to balance between the objective and penalty functions stochastically. The method was tested using a strategy evolution on several benchmarks, and the results showed that the method is able to improve the search performance with a user-defined parameter without introducing complicated variation operators. SR has also been coupled with other population-based algorithms, such as ant colony optimization (ACO) [123], [124], differential evolutionary (DE) [125]- [127], and evolutionary programming (EP) [128].
[121]提出了随机排序（SR）方法，以随机方式平衡目标函数与罚函数。该方法通过策略进化在多个基准测试上进行验证，结果表明该方法能够通过用户定义的参数提升搜索性能，且无需引入复杂的变异算子。SR 还与其他群体智能算法相结合，如蚁群优化（ACO）[123][124]、差分进化（DE）[125]-[127]以及进化规划（EP）[128]。

3.6 Ensemble techniques  3.6 集成技术

Ensemble CHTs provide a new research platform to tackle constrained multiobjective optimization problems. Combining several CHTs could improve the capability of an approach compared with a single CHTs [30], [129]. For instance, [130] proposed a combination of four CHTs, namely nondominated sorting, constrained-domination principle, multiple constraint ranking, and dynamic penalty function, and incorporated the proposed technique into an MOEA based on NSGAII. Some other ensemble CHTs have been reported [36], [131], [132]. Although the ensemble CHT has a competetive performance, it suffers from being parameter-dependent.
集成约束处理技术（CHTs）为解决约束多目标优化问题提供了新的研究平台。相较于单一 CHT[30][129]，组合多种 CHT 能显著提升算法性能。例如文献[130]提出将非支配排序、约束支配原则、多重约束排序和动态罚函数四种 CHT 技术相结合，并将该技术集成到基于 NSGAII 的多目标进化算法中。其他集成 CHT 方案也见诸报道[36][131][132]。尽管集成 CHT 具有竞争优势，但其性能仍受参数依赖性制约。

3.7 Multiobjective concept
3.7 多目标概念

Based on the multiobjective optimization concept, a constraint single-objective optimization problem is transferred to an unconstrained multiobjective optimization problem [5]. The multiobjective version of the optimization problem possesses an extra objective function, which presents the sum of constraint violation [28], [133]-[135].
基于多目标优化理念，约束单目标优化问题可转化为无约束多目标优化问题[5]。该多目标版本优化问题新增了一个目标函数，用于表征约束违反量总和[28][133]-[135]。

The authors of [136] presented a taxonomy for constraint handling strategies in multiobjective GA, which include:
文献[136]作者提出了多目标遗传算法中约束处理策略的分类体系，主要包括：

Among these strategies, the penalty function method is not straightforward in multiobjective GA since the fitness assignment is based on the non-dominance rank of a solution rather than its objective function values [137]. Yet, the penalty function method is one of the most popular CHTs in constraint multiobjective optimization. Whenever the multiobjective function and constraint violation for each constraint are assessed, the sum of violations is added to each objective function value considering the multiplication of the penalty parameter [138], [139].[19] Proposed two approaches, namely OEGADO and OSGADO. OEGADO runs several GAs in parallel so that each GA optimizes one objective, whereas OSGADO runs each objective sequentially with a common population for all objectives.
在这些策略中，罚函数法在多目标遗传算法中并不直观，因为适应度分配是基于解的非支配等级而非其目标函数值[137]。然而，罚函数法是约束多目标优化中最流行的约束处理技术之一。每当评估多目标函数和每个约束的约束违反时，将违反总和乘以罚参数后叠加到各目标函数值上[138][139]。[19]提出了两种方法：OEGADO 和 OSGADO。OEGADO 并行运行多个遗传算法，每个算法优化一个目标；而 OSGADO 则采用共享种群的方式依次优化各目标。

There are several techniques used as repair algorithms, in which the search space is reduced (since only feasible individuals are considered):
修复算法采用多种技术手段，通过仅考虑可行个体来缩减搜索空间：

Figure 8 Combo chart of number of documents vs. total citations (Scopus)
图 8 文献数量与总引用量的组合图表（Scopus 数据）

Among the document types, including article, proceedings paper, review, and other items indexed by WOS, a total of 735 publications on constraint handling multiobjective population-based optimization algorithms was found (Table 4). From the search, articles were the most popular document type,comprising a total of 522 articles (71.02% of 735 documents) with 2.60 authors per publication (APP). Also, articles as the document type had the highest CPP2020 of 84.10, followed by proceedings papers with TP of 220 (29.93% of contributions and
在 Web of Science 收录的文献类型中（包括期刊论文、会议论文、综述及其他类型），共发现 735 篇关于约束处理多目标群体优化算法的出版物（表 4）。检索结果显示，期刊论文是最主要的文献类型，共计 522 篇（占 735 篇文献的 71.02%），篇均作者数为 2.60 人。其中期刊论文的 2020 年篇均被引次数最高（CPP2020=84.10），其次是会议论文（总发文量 TP=220 篇，占贡献总量的 29.93%）

Table 4 Citations analysis based on document type APP=2.13). Moreover, there is a significant difference between the TC2020 of article and that of proceedings paper.
表 4 基于文献类型的引用分析（APP=2.13）。此外，论文与会议论文的 TC2020 值存在显著差异。

Figure 9 presents the distribution of documents based on different types, according to WOS. It is clear from the figure that conference papers have the most contributions before 2010 followed by articles. However, since 2010, articles have the most contributions in the field. It is also interesting to note that book/book chapters have been published since 2000, however, the most number of book/book chapters have been published after 2010. 6.3 Publication statistics based on journal
图 9 展示了根据 WOS 统计的不同类型文献的分布情况。从图中可以明显看出，在 2010 年之前，会议论文的贡献量最大，其次是期刊文章。然而自 2010 年起，期刊文章成为该领域最主要的贡献来源。值得注意的是，书籍/书籍章节自 2000 年起开始出版，但绝大多数书籍/书籍章节的出版集中在 2010 年之后。6.3 基于期刊的出版统计

TP, AU, APP, TC2020, and CPP2020 present total number of articles; total number of authors; total number of authors for each publication; total citations from WOS since publication year to the end of 2020; total citations for each paper, respectively; Other items: early access and letters [246].
TP（总论文数）、AU（总作者数）、APP（各出版物作者数）、TC2020（自发表年至 2020 年底的 WOS 总引用数）和 CPP2020（单篇论文引用数）分别表示：文章总数；作者总数；各出版物作者总数；从发表年份至 2020 年底的 Web of Science 总引用次数；各论文的引用次数；其他项目包括：早期访问文献和信件[246]。

Figure 9 Type of research outputs
图 9 研究成果类型

Table 5 presents the top 20 journals that have published the greatest number of constraint handling multiobjective population-based algorithms papers based on Scopus. Accordingly, Lecture Notes In Computer Science (117), Applied Soft Computing Journal (57), and Swarm and Evolutionary Computation (30) are most are the most utilized, which predominate in the field of optimization and evolutionary computations.
表 5 展示了基于 Scopus 数据库发表约束处理多目标群体算法论文数量最多的前 20 种期刊。其中，《计算机科学讲义》(117 篇)、《应用软计算杂志》(57 篇)和《群体与进化计算》(30 篇)是该领域最常用的三大期刊，在优化与进化计算领域占据主导地位。

A total of 735 articles were published in 399 journals, which are classified among the 51 WOS categories in SCI-EXPANDED. Table 6 lists the 10 most productive WOS categories. A total of 271 articles (36.87% of 735 articles) were published in the first category (Computer Science Artificial Intelligence),of which $83.39\mathrm{\%}$ were published in Engineering Electrical Electronic (23.26%) and Computer Science Theory Methods (23.26%). Comparing the top 10 categories, the highest CPP2020 of articles published in the Computer Science Theory Methods category is 190.011 , which includes the paper entitled: "A fast and elitist multiobjective genetic algorithm: NSGA-II" by [16], and the highest APP for articles published in the Energy Fuels category is 2.97 . each paper, respectively ; Other items: early access and letters [247]
共有 735 篇文章发表在 399 种期刊上，这些期刊分属 SCI-EXPANDED 数据库的 51 个 WOS 类别。表 6 列出了产出量最高的 10 个 WOS 类别。其中第一大类（计算机科学人工智能）发表了 271 篇文章（占 735 篇的 36.87%），其中 $83.39\mathrm{\%}$ 篇发表于工程电子电气（23.26%）和计算机科学理论方法（23.26%）类别。对比前 10 大类，计算机科学理论方法类别所发表文章的 CPP2020 值最高，达 190.011，包含文献[16]所著题为《一种快速精英多目标遗传算法：NSGA-II》的论文；而能源燃料类别文章的 APP 值最高，为 2.97/篇。其他条目包括：早期访问文献及信件[247]

Table 5 The top 10 sources that have published the greatest number of constraint handling on GA papers (Scopus)
表 5 发表遗传算法约束处理相关论文数量最多的前 10 种来源（Scopus 数据库）

Figure 11 presents the distribution of documents by the top most active countries in both databases. It is apparent that China, India, and the USA are the top three active countries in the field according to Scopus (respectively), while China, the USA, and India are the top 3 active territories in the field based on WOS, respectively. It is pertinent to mention that the USA is ranked second based on WOS, but India is ranked second according to Scopus. Also, it can be seen that there is a significant difference between the first rank (China) and second rank (India) based on the number of publications indexed by Scopus. Moreover, Figure 12 presents the collaboration among countries, where the links across the circles depict the collaborations, and the size of the circles represents the activities of the countries in the field. The green and yellow colors present the keywords that have been used recently, while the dark blue color indicates those that were used earlier (around 2008).
图 11 呈现了两大数据库中最高产国家的文献分布情况。显然，根据 Scopus 数据，中国、印度和美国是该领域活跃度前三的国家（按此顺序）；而基于 WOS 统计，中国、美国和印度则分别位列前三。值得注意的是，美国在 WOS 排名中位居第二，而印度则在 Scopus 排名中位列次席。此外可以看出，根据 Scopus 收录的出版物数量，第一名（中国）与第二名（印度）之间存在显著差距。图 12 进一步展示了国家间的合作网络，其中圆圈间的连线表征合作关系，圆圈大小反映各国在该领域的活跃程度。绿色与黄色代表近期使用的关键词，深蓝色则标识较早时期（约 2008 年）使用的术语。

Figure 11 Research output of top 10 most productive countries across all databases
图 11 各数据库中最高产的前 10 个国家的研究成果

Table 9 Top 5 research fields in 2019 and 2020
表 9 2019 年和 2020 年排名前 5 的研究领域

Constraint population-based optimization involves the use of a population-based algorithm in combining with a CHT to solve a constraint optimization problem. This paper presents an analysis and evaluation of the CHTs on multiobjective optimization population-based algorithms, which support evolutionary and swarm intelligence algorithms. To the best of our knowledge, this study is the first analysis of relevant journals evaluated over the most relevant journals, keywords, authors, and articles in this field. All related papers, including research articles, reviews, book/book chapters, conference papers, etc., as of writing this paper, were extracted and analyzed. Publication statistics by year, journal, country, affiliation, author, number of pages, number of authors, and keywords are discussed in this paper as follows:
基于种群的约束优化涉及将种群算法与约束处理技术（CHT）结合以解决约束优化问题。本文对多目标优化种群算法中的 CHT 进行了分析与评估，这些算法支持进化算法和群体智能算法。据我们所知，本研究首次对该领域最具相关性的期刊、关键词、作者和文章进行了系统分析。截至本文撰写时，所有相关论文（包括研究文章、综述、书籍/书籍章节、会议论文等）均被提取并分析。本文按年份、期刊、国家、机构、作者、页数、作者数量和关键词等维度讨论了发表统计数据，具体如下：

"Engineering Electrical Electronic," "Computer Science Theory Methods," and "Computer Science Interdisciplinary Applications" were the top 4 productive WOS categories in the field.
"工程电气电子"、"计算机科学理论与方法"和"计算机科学跨学科应用"是该领域 Web of Science 数据库中最高产的 4 个类别。

Author Contributions: "Conceptualization, Iman Rahimi. and Amir H. Gandomi.; methodology, Iman Rahimi; software, Iman Rahimi; validation, Amir H. Gandomi, Fang Chen. and Efrén Mezura-Montes; formal analysis, Iman Rahimi; data curation, Iman Rahimi; writing-original draft preparation, Iman Rahimi-review and editing, Amir H. Gandomi, Fang Chen. and Efrén Mezura-Montes ;supervision, Amir H. Gandomi, Fang Chen;
作者贡献："概念设计，Iman Rahimi 与 Amir H. Gandomi；方法论，Iman Rahimi；软件实现，Iman Rahimi；验证，Amir H. Gandomi、Fang Chen 及 Efrén Mezura-Montes；形式分析，Iman Rahimi；数据整理，Iman Rahimi；初稿撰写，Iman Rahimi；审阅与修改，Amir H. Gandomi、Fang Chen 及 Efrén Mezura-Montes；项目监督，Amir H. Gandomi、Fang Chen；"

Funding: "This research received no external funding".
资助声明："本研究未接受任何外部资金支持"。

Conflicts of Interest: "The authors declare no conflict of interest."
利益冲突声明：'作者声明无利益冲突。'

[1] R. Behmanesh, I. Rahimi, and A. H. Gandomi, "Evolutionary many-objective algorithms for combinatorial optimization problems: a comparative study," Archives of Computational Methods in Engineering, vol. 28, no. 2, pp. 673-688, 2021.
[1] R. Behmanesh, I. Rahimi 与 A. H. Gandomi 合著，《组合优化问题的进化多目标算法：一项比较研究》，载《工程计算方法档案》，2021 年第 28 卷第 2 期，第 673-688 页。

[2] K. Deb, "Multi-objective optimization. Search methodologies," Search Methodologies. New York: Springer, 2014.
[2] K. Deb，《多目标优化。搜索方法论》，收录于《搜索方法论》。纽约：Springer 出版社，2014 年。

[3] R. T. Marler and J. S. Arora, "Survey of multi-objective optimization methods for engineering," Structural and multidisciplinary optimization, vol. 26, no. 6, pp. 369- 395, 2004.
[3] R. T. Marler 与 J. S. Arora 合著，《工程多目标优化方法综述》，载《结构与多学科优化》，2004 年第 26 卷第 6 期，第 369-395 页。

[4] K. Deb, "Multi-objective optimisation using evolutionary algorithms: an introduction," in Multi-objective evolutionary optimisation for product design and manufacturing, Springer, 2011, pp. 3-34.
[4] K. Deb，《多目标进化优化算法导论》，载于《产品设计与制造中的多目标进化优化》，Springer 出版社，2011 年，第 3-34 页。

[5] C. A. C. Coello, G. B. Lamont, D. A. Van Veldhuizen, and others, Evolutionary algorithms for solving multi-objective problems, vol. 5. Springer, 2007.
[5] C. A. C. Coello, G. B. Lamont, D. A. Van Veldhuizen 等，《求解多目标问题的进化算法》第 5 卷，Springer 出版社，2007 年。

[6] J. Horn, N. Nafpliotis, and D. E. Goldberg, "A niched Pareto genetic algorithm for multiobjective optimization," in Proceedings of the first IEEE conference on evolutionary computation. IEEE world congress on computational intelligence, 1994, pp. 82-87.
[6] J. Horn, N. Nafpliotis, D. E. Goldberg，《基于生态位帕累托遗传算法的多目标优化》，载于《第一届 IEEE 进化计算会议论文集》（IEEE 世界计算智能大会），1994 年，第 82-87 页。

[7] D. W. Corne, J. D. Knowles, and M. J. Oates, "The Pareto envelope-based selection algorithm for multiobjective optimization," in International conference on parallel problem solving from nature, 2000, pp. 839- 848.
[7] D. W. Corne, J. D. Knowles, M. J. Oates，《基于帕累托包络选择的多目标优化算法》，载于《自然并行问题求解国际会议》，2000 年，第 839-848 页。

[8] T. T. Binh and U. Korn, "MOBES: A multiobjective evolution strategy for constrained optimization problems," in The third international conference on genetic algorithms (Mendel 97), 1997, vol. 25, p. 27.
[8] T. T. Binh 与 U. Korn，《MOBES：一种用于约束优化问题的多目标进化策略》，发表于第三届遗传算法国际会议（Mendel 97），1997 年，第 25 卷，第 27 页。

[9] C. M. Fonseca and P. J. Fleming, "Multiobjective optimization and multiple constraint handling with evolutionary algorithms. I. A unified formulation," IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, vol. 28, no. 1, pp. 26-37, 1998.
[9] C. M. Fonseca 与 P. J. Fleming，《多目标优化及多约束处理的进化算法研究 I：统一表述》，刊载于《IEEE 系统、人与控制论汇刊-A 辑：系统与人类》，1998 年，第 28 卷第 1 期，第 26-37 页。

[10] F. Jiménez, J. L. Verdegay, and others, "Evolutionary techniques for constrained multiobjective optimization problems," 1999.
[10] F. Jiménez、J. L. Verdegay 等，《约束多目标优化问题的进化技术》，1999 年。

[11] D. E. Goldberg and K. Deb, "A comparative analysis of selection schemes used in genetic algorithms," in Foundations of genetic algorithms, vol. 1, Elsevier, 1991, pp. 69-93.
[11] D. E. Goldberg 与 K. Deb，《遗传算法中选择机制的对比分析》，收录于《遗传算法基础》第 1 卷，Elsevier 出版社，1991 年，第 69-93 页。

[12] C. A. Coello Coello and A. D. Christiansen, "MOSES: A multiobjective optimization tool for engineering design," Engineering Optimization, vol. 31, no. 3, pp. 337-368, 1999.
[12] C. A. 科埃略·科埃略与 A. D. 克里斯蒂安森，《MOSES：一种用于工程设计的多元优化工具》，《工程优化》，第 31 卷第 3 期，第 337-368 页，1999 年。

[13] K. Deb, "An efficient constraint handling method for genetic algorithms," Comput Methods Appl Mech Eng, vol. 186, no. 2-4, pp. 311-338, 2000.
[13] K. Deb，《遗传算法中一种高效的约束处理方法》，《计算机方法应用力学工程》，第 186 卷，第 2-4 期，第 311-338 页，2000 年。

[14] T. Ray, K. Tai, and C. Seow, "An evolutionary algorithm for multiobjective optimization," Eng. Optim, vol. 33, no. 3, pp. 399-424, 2001.
[14] T. 雷、K. 泰与 C. 萧，《一种用于多元优化的进化算法》，《工程优化》，第 33 卷第 3 期，第 399-424 页，2001 年。

[15] F. Jimenez, A. F. Gómez-Skarmeta, G. Sánchez, and K. Deb, "An evolutionary algorithm for constrained multi-objective optimization," in Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No. 02TH8600), 2002, vol. 2, pp. 1133-1138.
[15] F. 希门尼斯、A. F. 戈麦斯-斯卡梅塔、G. 桑切斯与 K. 德布，《一种用于约束多元优化的进化算法》，收录于《2002 年进化计算大会论文集》。CEC'02（分类号 02TH8600），2002 年，第 2 卷，第 1133-1138 页。

[16] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, "A fast and elitist multiobjective genetic algorithm: NSGA-II," IEEE transactions on evolutionary computation, vol. 6, no. 2, pp. 182-197, 2002.
[16] K. Deb, A. Pratap, S. Agarwal 和 T. Meyarivan，《一种快速精英多目标遗传算法：NSGA-II》，《IEEE 进化计算汇刊》，第 6 卷第 2 期，第 182-197 页，2002 年。

[17] C. A. C. Coello, "Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art," Comput Methods Appl Mech Eng, vol. 191, no. 11-12, pp. 1245-1287, 2002.
[17] C. A. C. Coello，《进化算法中理论与数值约束处理技术综述：最新进展》，《计算机方法应用力学工程》，第 191 卷第 11-12 期，第 1245-1287 页，2002 年。

[18] A. Angantyr, J. Andersson, and J.-O. Aidanpaa, "Constrained optimization based on a multiobjective evolutionary algorithm," in The 2003 Congress on Evolutionary Computation, 2003. CEC'03., 2003, vol. 3, pp. 1560-1567.
[18] A. Angantyr, J. Andersson 和 J.-O. Aidanpaa，《基于多目标进化算法的约束优化》，《2003 年进化计算大会论文集》，2003 年，第 3 卷，第 1560-1567 页。

[19] D. Chafekar, J. Xuan, and K. Rasheed, "Constrained multi-objective optimization using steady state genetic algorithms," in Genetic and Evolutionary Computation Conference, 2003, pp. 813-824.
[19] D. Chafekar, J. Xuan 和 K. Rasheed，《采用稳态遗传算法的约束多目标优化》，《遗传与进化计算会议论文集》，2003 年，第 813-824 页。

[20] X.-F. Zou, M.-Z. Liu, Z.-J. Wu, and L.-S. Kang, "A robust evolutionary algorithm for constrained multi-objective optimization problems," Journal of computer research and development, vol. 41, no. 6, pp. 985-990, 2004.
[20] 邹晓峰、刘明哲、吴志坚、康立山，"一种求解约束多目标优化问题的鲁棒进化算法"，《计算机研究与发展》，第 41 卷第 6 期，第 985-990 页，2004 年。

[21] N. Young, "Blended ranking to cross infeasible regions in constrainedmultiobjective problems," in International Conference on Computational Intelligence for Modelling, Control and Automation and International Conference on Intelligent Agents, Web Technologies and Internet Commerce (CIMCA-IAWTIC'06), 2005, vol. 2, pp. 191-196.
[21] N. Young，"混合排序法跨越约束多目标问题中的不可行区域"，载于《国际计算智能建模、控制与自动化会议及智能代理、网络技术与互联网商务国际会议论文集》(CIMCA-IAWTIC'06)，2005 年，第 2 卷，第 191-196 页。

[22] S. Venkatraman and G. G. Yen, "A generic framework for constrained optimization using genetic algorithms," IEEE Transactions on Evolutionary Computation, vol. 9, no. 4, pp. 424-435, 2005.
[22] S. Venkatraman、G. G. Yen，"基于遗传算法的约束优化通用框架"，《IEEE 进化计算汇刊》，第 9 卷第 4 期，第 424-435 页，2005 年。

[23] Z. Cai and Y. Wang, "A multiobjective optimization-based evolutionary algorithm for constrained optimization," IEEE Transactions on evolutionary computation, vol. 10, no. 6, pp. 658-675, 2006.
[23] 蔡志强、王勇，"基于多目标优化的约束优化进化算法"，《IEEE 进化计算汇刊》，第 10 卷第 6 期，第 658-675 页，2006 年。

[24] H.-Q. Min, Y. Zhou, Y. Lu, and J. Jiang, "An evolutionary algorithm for constrained multi-objective optimization problems," in 2006 IEEE Asia-Pacific Conference on Services Computing (APSCC'06), 2006, pp. 667-670.
[24] 闵华清、周永录、卢炎生、江建慧，《约束多目标优化问题的进化算法》，载于《2006 年 IEEE 亚太服务计算会议（APSCC'06）》，2006 年，第 667-670 页。

[25] K. Harada, J. Sakuma, I. Ono, and S. Kobayashi, "Constraint-handling method for multi-objective function optimization: Pareto descent repair operator," in International Conference on Evolutionary Multi-Criterion Optimization, 2007, pp. 156-170.
[25] 原田香、佐久间淳、小野功、小林重信，《多目标函数优化的约束处理方法：帕累托下降修复算子》，载于《国际进化多准则优化会议》，2007 年，第 156-170 页。

[26] Y. Wang, Z. Cai, Y. Zhou, and W. Zeng, "An adaptive tradeoff model for constrained evolutionary optimization," IEEE Transactions on Evolutionary Computation, vol. 12, no. 1, pp. 80-92, 2008.
[26] 王耀、蔡之华、周永录、曾文华，《约束进化优化的自适应权衡模型》，《IEEE 进化计算汇刊》，第 12 卷第 1 期，2008 年，第 80-92 页。

[27] A. Kaveh and S. Talatahari, "Particle swarm optimizer, ant colony strategy and harmony search scheme hybridized for optimization of truss structures," Computers & Structures, vol. 87, no. 5-6, pp. 267-283, 2009.
[27] 阿里·卡维、萨塔尔·塔拉塔哈里，《粒子群优化器、蚁群策略与和声搜索方案混合用于桁架结构优化》，《计算机与结构》，第 87 卷第 5-6 期，2009 年，第 267-283 页。

[28] T. Ray, H. K. Singh, A. Isaacs, and W. Smith, "Infeasibility driven evolutionary algorithm for constrained optimization," in Constraint-handling in evolutionary optimization, Springer, 2009, pp. 145-165.
[28] T. Ray, H. K. Singh, A. Isaacs, 和 W. Smith, "用于约束优化的不可行驱动进化算法," 载于《进化优化中的约束处理》, Springer, 2009 年, 第 145-165 页。

[29] R. Tulshyan, R. Arora, K. Deb, and J. Dutta, "Investigating EA solutions for approximate KKT conditions in smooth problems," in Proceedings of the 12th annual conference on Genetic and evolutionary computation, 2010, pp. 689-696.
[29] R. Tulshyan, R. Arora, K. Deb, 和 J. Dutta, "研究光滑问题中近似 KKT 条件的 EA 解," 载于《第 12 届遗传与进化计算年会论文集》, 2010 年, 第 689-696 页。

[30] E. Mezura-Montes and C. A. C. Coello, "Constraint-handling in nature-inspired numerical optimization: past, present and future," Swarm and Evolutionary Computation, vol. 1, no. 4, pp. 173-194, 2011.
[30] E. Mezura-Montes 和 C. A. C. Coello, "自然启发数值优化中的约束处理：过去、现在与未来," 《群体与进化计算》, 第 1 卷第 4 期, 第 173-194 页, 2011 年。

[31] Z. Michalewicz and M. Schoenauer, "Evolutionary algorithms for constrained parameter optimization problems," Evol Comput, vol. 4, no. 1, pp. 1-32, 1996.
[31] Z. Michalewicz 和 M. Schoenauer, "约束参数优化问题的进化算法," 《进化计算》, 第 4 卷第 1 期, 第 1-32 页, 1996 年。

[32] T. Takahama, S. Sakai, and N. Iwane, "Constrained optimization by the $\mathrm{\$}\mathrm{\setminus }$ varepsilon $\mathrm{\$}$ constrained hybrid algorithm of particle swarm optimization and genetic algorithm," in Australasian Joint Conference on Artificial Intelligence, 2005, pp. 389-400.
[32] T. Takahama、S. Sakai 与 N. Iwane，《基于粒子群优化与遗传算法混合的ε约束算法求解约束优化问题》，载《Australasian 人工智能联合会议论文集》，2005 年，第 389-400 页。

[33] J. Xiao, J. Xu, Z. Shao, C. Jiang, and L. Pan, "A genetic algorithm for solving multi-constrained function optimization problems based on KS function," in 2007 IEEE Congress on Evolutionary Computation, 2007, pp. 4497-4501.
[33] 肖剑、徐杰、邵志峰、蒋超与潘磊，《基于 KS 函数求解多约束函数优化问题的遗传算法》，载《2007 年 IEEE 进化计算大会论文集》，2007 年，第 4497-4501 页。

[34] K. Deb and R. Datta, "A fast and accurate solution of constrained optimization problems using a hybrid bi-objective and penalty function approach," in IEEE Congress on Evolutionary Computation, 2010, pp. 1-8.
[34] K. Deb 与 R. Datta，《采用混合双目标与罚函数方法快速精确求解约束优化问题》，载《IEEE 进化计算大会论文集》，2010 年，第 1-8 页。

[35] D. H. Wolpert and W. G. Macready, "No free lunch theorems for optimization," IEEE transactions on evolutionary computation, vol. 1, no. 1, pp. 67-82, 1997.
[35] D. H. Wolpert 与 W. G. Macready，《优化问题中的无免费午餐定理》，载《IEEE 进化计算汇刊》，第 1 卷第 1 期，1997 年，第 67-82 页。

[36] R. Mallipeddi and P. N. Suganthan, "Ensemble of constraint handling techniques," IEEE Transactions on Evolutionary Computation, vol. 14, no. 4, pp. 561-579, 2010.
[36] R. Mallipeddi 与 P. N. Suganthan 合著，《约束处理技术集成》，刊于《IEEE 进化计算汇刊》，第 14 卷第 4 期，第 561-579 页，2010 年。

[37] A. H. Gandomi and X.-S. Yang, "Evolutionary boundary constraint handling scheme," Neural Computing and Applications, vol. 21, no. 6, pp. 1449-1462, 2012.
[37] A. H. Gandomi 与 X.-S. Yang 合著，《进化边界约束处理方案》，刊于《神经计算与应用》，第 21 卷第 6 期，第 1449-1462 页，2012 年。

[38] H. Eskandar, A. Sadollah, A. Bahreininejad, and M. Hamdi, "Water cycle algorithm--A novel metaheuristic optimization method for solving constrained engineering optimization problems," Computers & Structures, vol. 110, pp. 151-166, 2012.
[38] H. Eskandar、A. Sadollah、A. Bahreininejad 和 M. Hamdi 合著，《水循环算法——一种解决约束工程优化问题的新型元启发式优化方法》，刊于《计算机与结构》，第 110 卷，第 151-166 页，2012 年。

[39] A. Sadollah, A. Bahreininejad, H. Eskandar, and M. Hamdi, "Mine blast algorithm: A new population based algorithm for solving constrained engineering optimization problems," Applied Soft Computing, vol. 13, no. 5, pp. 2592-2612, 2013.
[39] A. Sadollah、A. Bahreininejad、H. Eskandar 和 M. Hamdi 合著，《矿爆算法：一种解决约束工程优化问题的新型群体智能算法》，刊于《应用软计算》，第 13 卷第 5 期，第 2592-2612 页，2013 年。

[40] N. M. Hamza, R. A. Sarker, D. L. Essam, K. Deb, and S. M. Elsayed, "A constraint consensus memetic algorithm for solving constrained optimization problems," Engineering Optimization, vol. 46, no. 11, pp. 1447- 1464, 2014.
[40] N. M. Hamza, R. A. Sarker, D. L. Essam, K. Deb, 和 S. M. Elsayed, 《一种用于求解约束优化问题的约束共识文化基因算法》,《工程优化》, 卷 46, 期 11, 页 1447-1464, 2014 年。

[41] L. Jiao, J. Luo, R. Shang, and F. Liu, "A modified objective function method with feasible-guiding strategy to solve constrained multi-objective optimization problems," Applied Soft Computing, vol. 14, pp. 363- 380, 2014.
[41] L. Jiao, J. Luo, R. Shang, 和 F. Liu, 《采用可行引导策略的改进目标函数法求解约束多目标优化问题》,《应用软计算》, 卷 14, 页 363-380, 2014 年。

[72] A. Ben Hadj-Alouane and J. C. Bean, "A genetic algorithm for the multiple-choice integer program," Oper Res, vol. 45, no. 1, pp. 92-101, 1997.
[72] A. Ben Hadj-Alouane 和 J. C. Bean 发表，《多选择整数规划的遗传算法》，《运筹学》期刊，第 45 卷第 1 期，第 92-101 页，1997 年。

[73] A. E. Eiben and J. K. Van Der Hauw, "Adaptive penalties for evolutionary graph coloring," in European Conference on Artificial Evolution, 1997, pp. 95-106.
[73] A. E. Eiben 与 J. K. Van Der Hauw 合著，《进化图着色的自适应惩罚策略》，收录于《欧洲人工进化会议论文集》，1997 年，第 95-106 页。

[74] K. Watanabe and M. M. A. Hashem, "Evolutionary optimization of constrained problems," in Evolutionary computations, Springer, 2004, pp. 53-64.
[74] K. Watanabe 和 M. M. A. Hashem 著，《约束问题的进化优化》，载于《进化计算》，Springer 出版社，2004 年，第 53-64 页。

[75] C. A. C. Coello, "Self-adaptive penalties for GA-based optimization," in Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406), 1999, vol. 1, pp. 573-580.
[75] C. A. C. Coello 著，《基于遗传算法的自适应惩罚优化》，载于《1999 年进化计算大会论文集-CEC99》（分类号 99TH8406），1999 年，第 1 卷，第 573-580 页。

[76] C. A. C. Coello, "Use of a self-adaptive penalty approach for engineering optimization problems," Computers in Industry, vol. 41, no. 2, pp. 113-127, 2000.
[76] C. A. C. Coello 著，《自适应性惩罚方法在工程优化问题中的应用》，《工业计算机》，第 41 卷第 2 期，2000 年，第 113-127 页。

[77] R. Le Riche, C. Knopf-Lenoir, and R. T. Haftka, "A Segregated Genetic Algorithm for Constrained Structural Optimization.," in ICGA, 1995, pp. 558-565.
[77] R. Le Riche、C. Knopf-Lenoir 和 R. T. Haftka 著，《一种用于约束结构优化的隔离遗传算法》，载于《ICGA》，1995 年，第 558-565 页。

[78] D. Powell and M. M. Skolnick, "Using genetic algorithms in engineering design optimization with nonlinear constraints," in Proceedings of the 5th International conference on Genetic Algorithms, 1993, pp. 424-431.
[78] D. Powell 与 M. M. Skolnick，《在非线性约束条件下使用遗传算法进行工程设计优化》，发表于《第五届国际遗传算法会议论文集》，1993 年，第 424-431 页。

[79] R. Hinterding and Z. Michalewicz, "Your brains and my beauty: parent matching for constrained optimisation," in 1998 IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence (Cat. No. 98TH8360), 1998, pp. 810-815.
[79] R. Hinterding 与 Z. Michalewicz，《你的智慧与我的美貌：约束优化中的亲本匹配》，收录于《1998 年 IEEE 进化计算国际会议论文集——IEEE 世界计算智能大会》（分类号 98TH8360），1998 年，第 810-815 页。

[80] M. A. Jan and R. A. Khanum, "A study of two penalty-parameterless constraint handling techniques in the framework of MOEA/D," Applied Soft Computing, vol. 13, no. 1, pp. 128-148, 2013.
[80] M. A. Jan 与 R. A. Khanum，《MOEA/D 框架下两种无惩罚参数约束处理技术的研究》，《应用软计算》第 13 卷第 1 期，2013 年，第 128-148 页。

[81] T. Takahama and S. Sakai, "Constrained optimization by $varepsilon$ constrained particle swarm optimizer with $varepsilon$-level control," in Soft computing as transdisciplinary science and technology, Springer, 2005, pp. 1019-1029.
[81] T. Takahama 与 S. Sakai，《采用 $varepsilon$ 约束粒子群优化器与 $varepsilon$ 级控制的约束优化》，载于《跨学科科学与技术中的软计算》，Springer 出版社，2005 年，第 1019-1029 页。

[82] D. Powell and M. M. Skolnick, "Using genetic algorithms in engineering design optimization with nonlinear constraints," in Proceedings of the 5th International conference on Genetic Algorithms, 1993, pp. 424-431.
[82] D. Powell 与 M. M. Skolnick，《在非线性约束条件下使用遗传算法进行工程设计优化》，发表于《第五届国际遗传算法会议论文集》，1993 年，第 424-431 页。

[83] M. Asafuddoula, T. Ray, R. Sarker, and K. Alam, "An adaptive constraint handling approach embedded MOEA/D," in 2012 IEEE Congress on Evolutionary Computation, 2012, pp. 1-8.
[83] M. Asafuddoula、T. Ray、R. Sarker 与 K. Alam，《MOEA/D 框架中嵌入的自适应约束处理方法》，发表于《2012 年 IEEE 进化计算大会》，2012 年，第 1-8 页。

[84] B. Y. Qu and P. N. Suganthan, "Constrained multi-objective optimization algorithm with an ensemble of constraint handling methods," Engineering Optimization, vol. 43, no. 4, pp. 403-416, 2011.
[84] 屈博雅与 P. N. Suganthan，《集成多种约束处理方法的约束多目标优化算法》，《工程优化》，第 43 卷第 4 期，2011 年，第 403-416 页。

[85] E. Mezura-Montes, J. Velázquez-Reyes, and C. A. Coello Coello, "Promising infeasibility and multiple offspring incorporated to differential evolution for constrained optimization," in Proceedings of the 7th annual conference on Genetic and evolutionary computation, 2005, pp. 225-232.
[85] E. Mezura-Montes、J. Velázquez-Reyes 与 C. A. Coello Coello，《结合有希望不可行解与多子代策略的差分进化约束优化方法》，发表于《第七届遗传与进化计算年会论文集》，2005 年，第 225-232 页。

[86] E. Mezura-Montes, J. Velázquez-Reyes, and C. A. C. Coello, "Modified differential evolution for constrained
[86] E. Mezura-Montes、J. Velázquez-Reyes 和 C. A. C. Coello 合著的《Modified differential evolution for constrained》

optimization," in 2006 IEEE International Conference on Evolutionary Computation, 2006, pp. 25-32.
优化," in 2006 IEEE International Conference on Evolutionary Computation, 2006, pp. 25-32.

[87] E. Mezura-Montes, C. A. C. Coello, and E. I. Tun-Morales, "Simple feasibility rules and differential evolution for constrained optimization," in Mexican International Conference on Artificial Intelligence, 2004, pp. 707-716.
[87] E. Mezura-Montes, C. A. C. Coello, 和 E. I. Tun-Morales, "用于约束优化的简单可行性规则与差分进化算法," in 墨西哥人工智能国际会议, 2004, pp. 707-716.

[88] J. Lampinen, "A constraint handling approach for the differential evolution algorithm," in Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No. 02TH8600), 2002, vol. 2, pp. 1468-1473.
[88] J. Lampinen, "差分进化算法的一种约束处理方法," in 《2002 年进化计算大会论文集》. CEC'02 (Cat. No. 02TH8600), 2002, vol. 2, pp. 1468-1473.

[89] S. Kukkonen and J. Lampinen, "Constrained real-parameter optimization with generalized differential evolution," in 2006 IEEE International Conference on Evolutionary Computation, 2006, pp. 207-214.
[89] S. Kukkonen 与 J. Lampinen，《广义差分进化算法在约束实数参数优化中的应用》，收录于《2006 年 IEEE 进化计算国际会议论文集》，2006 年，第 207-214 页。

[90] A. L. Jaimes, C. A. C. Coello, H. Aguirre, and K. Tanaka, "Adaptive objective space partitioning using conflict information for many-objective optimization," in International Conference on Evolutionary Multi-Criterion Optimization, 2011, pp. 151-165.
[90] A. L. Jaimes、C. A. C. Coello、H. Aguirre 和 K. Tanaka，《基于冲突信息的多目标优化目标空间自适应划分方法》，收录于《进化多准则优化国际会议论文集》，2011 年，第 151-165 页。

[91] E. Mezura-Montes and A. G. Palomeque-Ortiz, "Parameter control in differential evolution for constrained optimization," in 2009 IEEE Congress on Evolutionary Computation, 2009, pp. 1375-1382.
[91] E. Mezura-Montes 与 A. G. Palomeque-Ortiz，《差分进化算法参数控制在约束优化中的应用》，收录于《2009 年 IEEE 进化计算大会论文集》，2009 年，第 1375-1382 页。

[92] K. Zielinski, X. Wang, and R. Laur, "Comparison of adaptive approaches for differential evolution," in International Conference on Parallel Problem Solving from Nature, 2008, pp. 641-650.
[92] K. Zielinski、X. Wang 和 R. Laur，《差分进化算法自适应方法的比较研究》，收录于《自然并行问题求解国际会议论文集》，2008 年，第 641-650 页。

[93] K. Zielinski, S. P. Vudathu, and R. Laur, "Influence of different deviations allowed for equality constraints on particle swarm optimization and differential evolution," in Nature Inspired Cooperative Strategies for Optimization (NICSO 2007), Springer, 2008, pp. 249- 259.
[93] K. Zielinski、S. P. Vudathu 和 R. Laur，《允许等式约束存在不同偏差对粒子群优化和差分进化的影响》，载于《自然启发的协同优化策略（NICSO 2007）》，Springer，2008 年，第 249-259 页。

[94] E. Mezura-Montes and A. G. Palomeque-Ortiz, "Self-adaptive and deterministic parameter control in differential evolution for constrained optimization," in Constraint-Handling in Evolutionary Optimization, Springer, 2009, pp. 95-120.
[94] E. Mezura-Montes 和 A. G. Palomeque-Ortiz，《差分进化中约束优化的自适应与确定性参数控制》，载于《进化优化中的约束处理》，Springer，2009 年，第 95-120 页。

[95] J. Lampinen, "A constraint handling approach for the differential evolution algorithm," in Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No. 02TH8600), 2002, vol. 2, pp. 1468-1473.
[95] J. Lampinen，《差分进化算法的约束处理方法》，载于《2002 年进化计算大会论文集（CEC'02）》（分类号 02TH8600），2002 年，第 2 卷，第 1468-1473 页。

[96] S. Kukkonen and J. Lampinen, "Constrained real-parameter optimization with generalized differential evolution," in 2006 IEEE International Conference on Evolutionary Computation, 2006, pp. 207-214.
[96] S. Kukkonen 和 J. Lampinen，《基于广义差分进化的约束实数参数优化》，载于《2006 年 IEEE 国际进化计算会议》，2006 年，第 207-214 页。

[97] A. L. Jaimes, C. A. C. Coello, H. Aguirre, and K. Tanaka, "Adaptive objective space partitioning using conflict information for many-objective optimization," in International Conference on Evolutionary Multi-Criterion Optimization, 2011, pp. 151-165.
[97] A. L. 海梅斯, C. A. C. 科埃略, H. 阿吉雷, 与 K. 田中, "基于冲突信息的多目标优化自适应目标空间划分方法," 发表于《国际进化多准则优化会议》, 2011 年, 第 151-165 页.

[98] H. Jiang, "Efficiency measurement and improvement of housing provident fund in China," International Journal of Wireless and Mobile Computing, vol. 12, no. 3, pp. 259-269, 2017.
[98] 江华, "中国住房公积金效率测度与改进研究," 《国际无线与移动计算杂志》, 第 12 卷第 3 期, 第 259-269 页, 2017 年.

[99] C. Si, J. Hu, T. Lan, L. Wang, and Q. Wu, "A combined constraint handling framework: an empirical study," Memetic Computing, vol. 9, no. 1, pp. 69-88, 2017.
[99] C. Si、J. Hu、T. Lan、L. Wang 和 Q. Wu，《组合约束处理框架的实证研究》，《Memetic Computing》第 9 卷第 1 期，第 69-88 页，2017 年。

[100] E. Mezura-Montes and J. I. Flores-Mendoza, "Improved particle swarm optimization in constrained numerical
[100] E. 梅祖拉-蒙特斯 与 J. I. 弗洛雷斯-门多萨, "约束数值优化中改进的粒子群算法"

robust layout synthesis of MEMS components," IEEE Transactions on Industrial Electronics, vol. 56, no. 4, pp. 937-948, 2008.
MEMS 元件鲁棒布局综合，』《IEEE 工业电子学报》，第 56 卷第 4 期，第 937-948 页，2008 年。

[128] R. Mallipeddi, P. N. Suganthan, and B.-Y. Qu, "Diversity enhanced adaptive evolutionary programming for solving single objective constrained problems," in 2009 IEEE Congress on Evolutionary Computation, 2009, pp. 2106-2113.
[128] R. Mallipeddi、P. N. Suganthan 和曲博洋，『多样性增强的自适应进化规划求解单目标约束问题，』收录于《2009 年 IEEE 进化计算大会》，2009 年，第 2106-2113 页。

[129] A. Vodopija, A. Oyama, and B. Filipič, "Ensemble-based constraint handling in multiobjective optimization," in Proceedings of the Genetic and Evolutionary Computation Conference Companion, 2019, pp. 2072-2075.
[129] A. Vodopija、A. Oyama 和 B. Filipič，『多目标优化中基于集成学习的约束处理，』收录于《遗传与进化计算会议论文集》，2019 年，第 2072-2075 页。

[130] A. Vodopija, A. Oyama, and B. Filipič, "Ensemble-based constraint handling in multiobjective optimization," in Proceedings of the Genetic and Evolutionary Computation Conference Companion, 2019, pp. 2072-2075.
[130] A. Vodopija, A. Oyama, 与 B. Filipič, 《多目标优化中基于集成学习的约束处理》, 收录于《遗传与进化计算会议论文集》, 2019 年, 第 2072-2075 页。

[131] M. F. Tasgetiren, P. N. Suganthan, Q.-K. Pan, R. Mallipeddi, and S. Sarman, "An ensemble of differential evolution algorithms for constrained function optimization," in IEEE congress on evolutionary computation, 2010, pp. 1-8.
[131] M. F. Tasgetiren, P. N. Suganthan, 潘庆凯, R. Mallipeddi, 与 S. Sarman, 《约束函数优化的差分进化算法集成》, 收录于《IEEE 进化计算大会》, 2010 年, 第 1-8 页。

[132] R. Mallipeddi and P. N. Suganthan, "Differential evolution with ensemble of constraint handling techniques for solving CEC 2010 benchmark problems," in IEEE congress on evolutionary computation, 2010, pp. $1-8$ .
[132] R. Mallipeddi 与 P. N. Suganthan, 《采用约束处理技术集成的差分进化算法求解 CEC 2010 基准问题》, 收录于《IEEE 进化计算大会》, 2010 年, 第 $1-8$ 页。

[133] G. Reynoso-Meza, X. Blasco, J. Sanchis, and M. Mart\'\inez, "Multiobjective optimization algorithm for solving constrained single objective problems," in IEEE congress on evolutionary computation, 2010, pp. 1-7.
[133] G. Reynoso-Meza, X. Blasco, J. Sanchis, 与 M. Martínez, 《求解约束单目标问题的多目标优化算法》, 收录于《IEEE 进化计算大会》, 2010 年, 第 1-7 页。

[134] Y. Wang, H. Liu, Z. Cai, and Y. Zhou, "An orthogonal design based constrained evolutionary optimization algorithm," Engineering Optimization, vol. 39, no. 6, pp. 715-736, 2007.
[134] 王勇、刘辉、蔡志强、周毅，《基于正交设计的约束进化优化算法》，《工程优化》，第 39 卷第 6 期，第 715-736 页，2007 年。

[135] L. D. Li, X. Li, and X. Yu, "A multi-objective constraint-handling method with PSO algorithm for constrained engineering optimization problems," in 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence), 2008, pp. 1528-1535.
[135] 李立东、李旭、余翔，《基于 PSO 算法的多目标约束处理方法在约束工程优化问题中的应用》，收录于《2008 年 IEEE 进化计算大会（IEEE 世界计算智能大会）》，2008 年，第 1528-1535 页。

[136] F. Samanipour and J. Jelovica, "Adaptive repair method for constraint handling in multi-objective genetic algorithm based on relationship between constraints and variables," Applied Soft Computing, vol. 90, p. 106143, 2020.
[136] 法扎德·萨马尼普尔、贾斯敏·耶洛维察，《基于约束与变量关系的多目标遗传算法自适应约束处理方法》，《应用软计算》，第 90 卷，第 106143 页，2020 年。

[137] Q. Long, "A constraint handling technique for constrained multi-objective genetic algorithm," Swarm and Evolutionary Computation, vol. 15, pp. 66-79, 2014.
[137] 龙强，《约束多目标遗传算法的约束处理技术》，《群体与进化计算》，第 15 卷，第 66-79 页，2014 年。

[138] N. Srinivas and K. Deb, "Muiltiobjective optimization using nondominated sorting in genetic algorithms," Evol Comput, vol. 2, no. 3, pp. 221-248, 1994.
[138] N. Srinivas 和 K. Deb，《遗传算法中基于非支配排序的多目标优化方法》，载《演化计算》第 2 卷第 3 期，第 221-248 页，1994 年。

[139] K. Deb, "Multi-objective genetic algorithms: Problem difficulties and construction of test problems," Evol Comput, vol. 7, no. 3, pp. 205-230, 1999.
[139] K. Deb，《多目标遗传算法：问题难点与测试问题构建》，Evol Comput，第 7 卷，第 3 期，第 205-230 页，1999 年。

[140] T. Bäck, D. B. Fogel, and Z. Michalewicz, Evolutionary computation 1: Basic algorithms and operators. CRC press, 2018.
[140] T. Bäck、D. B. Fogel 和 Z. Michalewicz，《进化计算 1：基础算法与算子》，CRC 出版社，2018 年。

[141] P. W. Poon and J. N. Carter, "Genetic algorithm crossover operators for ordering applications,"
[141] P. W. Poon 和 J. N. Carter，《面向排序应用的遗传算法交叉算子》，

Computers & Operations Research, vol. 22, no. 1, pp. 135-147, 1995.
《计算机与运筹学》第 22 卷第 1 期，第 135-147 页，1995 年。

[142] C. Y. Ngo and V. O. K. Li, "Fixed channel assignment in cellular radio networks using a modified genetic algorithm," IEEE Transactions on Vehicular Technology, vol. 47, no. 1, pp. 163-172, 1998.
[142] C. Y. Ngo 与 V. O. K. Li，《采用改进遗传算法的蜂窝无线电网络固定信道分配》，《IEEE 车载技术汇刊》第 47 卷第 1 期，第 163-172 页，1998 年。

[143] S. Salcedo-Sanz, G. Camps-Valls, F. Pérez-Cruz, J. Sepúlveda-Sanchis, and C. Bousoño-Calzón, "Enhancing genetic feature selection through restricted search and Walsh analysis," IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), vol. 34, no. 4, pp. 398-406, 2004.
[143] S. Salcedo-Sanz、G. Camps-Valls、F. Pérez-Cruz、J. Sepúlveda-Sanchis 与 C. Bousoño-Calzón，《通过受限搜索和沃尔什分析增强遗传特征选择》，《IEEE 系统、人与控制论汇刊》（C 辑：应用与综述）第 34 卷第 4 期，第 398-406 页，2004 年。

[144] B. Dengiz, F. Altiparmak, and A. E. Smith, "Local search genetic algorithm for optimal design of reliable networks," IEEE transactions on Evolutionary Computation, vol. 1, no. 3, pp. 179-188, 1997.
[144] B. Dengiz、F. Altiparmak 与 A. E. Smith，《可靠网络优化设计的局部搜索遗传算法》，《IEEE 进化计算汇刊》第 1 卷第 3 期，第 179-188 页，1997 年。

[145] L. Kou, G. Markowsky, and L. Berman, "A fast algorithm for Steiner trees," Acta Inform, vol. 15, no. 2, pp. 141-145, 1981.
[145] L. Kou、G. Markowsky 和 L. Berman，《斯坦纳树问题的快速算法》，载《信息学报》第 15 卷第 2 期，第 141-145 页，1981 年。

[146] E. Falkenauer, "The grouping genetic algorithms: widening the scope of the GA's," JORBEL-Belgian Journal of Operations Research, Statistics, and Computer Science, vol. 33, no. 1-2, pp. 79-102, 1993.
[146] E. Falkenauer，《分组遗传算法：扩展 GA 的应用范围》，《JORBEL-比利时运筹学、统计与计算机科学杂志》，第 33 卷第 1-2 期，第 79-102 页，1993 年。

[147] L. E. Agust\'\in-Blas, S. Salcedo-Sanz, E. G. Ortiz-Garc\'\ia, A. Portilla-Figueras, and Á. M. Pérez-Bellido, "A hybrid grouping genetic algorithm for assigning students to preferred laboratory groups," Expert Systems with Applications, vol. 36, no. 3, pp. 7234-7241, 2009.
[147] L. E. Agustín-Blas、S. Salcedo-Sanz、E. G. Ortiz-García、A. Portilla-Figueras 和 Á. M. Pérez-Bellido，《一种用于将学生分配到首选实验室组的混合分组遗传算法》，《专家系统及其应用》，第 36 卷第 3 期，第 7234-7241 页，2009 年。

[148] N. Krasnogor and J. Smith, "A tutorial for competent memetic algorithms: model, taxonomy, and design issues," IEEE Transactions on Evolutionary Computation, vol. 9, no. 5, pp. 474-488, 2005.
[148] N. Krasnogor 和 J. Smith，《高效模因算法教程：模型、分类与设计问题》，《IEEE 进化计算汇刊》，第 9 卷第 5 期，第 474-488 页，2005 年。

[149] S. Salcedo-Sanz, "A survey of repair methods used as constraint handling techniques in evolutionary algorithms," Comput Sci Rev, vol. 3, no. 3, pp. 175-192, 2009.
[149] S. Salcedo-Sanz，《进化算法中作为约束处理技术的修复方法综述》，计算机科学评论，第 3 卷第 3 期，第 175-192 页，2009 年。

[150] E.-G. Talbi, "Combining metaheuristics with mathematical programming, constraint programming and machine learning," Annals of Operations Research, vol. 240, no. 1, pp. 171-215, 2016.
[150] E.-G. Talbi，《元启发式与数学规划、约束规划及机器学习的结合》，运筹学年刊，第 240 卷第 1 期，第 171-215 页，2016 年。

[151] J.-Y. Suh and D. Van Gucht, "Incorporating heuristic information into genetic search," 1987.
[151] J.-Y. Suh 与 D. Van Gucht，《将启发式信息融入遗传搜索》，1987 年。

[152] S. Salcedo-Sanz, "A survey of repair methods used as constraint handling techniques in evolutionary algorithms," Comput Sci Rev, vol. 3, no. 3, pp. 175-192, 2009.
[152] S. Salcedo-Sanz，《进化算法中作为约束处理技术的修复方法综述》，计算机科学评论，第 3 卷第 3 期，第 175-192 页，2009 年。

[153] W. K. Lai and G. G. Coghill, "Channel assignment through evolutionary optimization," IEEE Transactions on Vehicular Technology, vol. 45, no. 1, pp. 91-96, 1996.
[153] W. K. Lai 与 G. G. Coghill，《通过进化优化实现信道分配》，《IEEE 车载技术汇刊》，第 45 卷第 1 期，第 91-96 页，1996 年。

[154] P. W. Poon and J. N. Carter, "Genetic algorithm crossover operators for ordering applications," Computers & Operations Research, vol. 22, no. 1, pp. 135-147, 1995.
[154] P. W. Poon 与 J. N. Carter，《用于排序应用的遗传算法交叉算子》，《计算机与运筹学研究》，第 22 卷第 1 期，第 135-147 页，1995 年。

[155] C. Y. Ngo and V. O. K. Li, "Centralized broadcast scheduling in packet radio networks via genetic-fix algorithms," IEEE Transactions on Communications, vol. 51, no. 9, pp. 1439-1441, 2003.
[155] C. Y. Ngo 与 V. O. K. Li，《基于遗传修正算法的分组无线网络集中式广播调度》，《IEEE 通信汇刊》，第 51 卷第 9 期，第 1439-1441 页，2003 年。

[156] H. Esbensen, "Computing near-optimal solutions to the Steiner problem in a graph using a genetic algorithm," Networks, vol. 26, no. 4, pp. 173-185, 1995.
[156] H. Esbensen，《利用遗传算法计算图中 Steiner 问题的近优解》，《网络》，第 26 卷第 4 期，第 173-185 页，1995 年。

[184] Q. Long, "A constraint handling technique for constrained multi-objective genetic algorithm," Swarm and Evolutionary Computation, vol. 15, pp. 66-79, 2014.
[184] 龙强，《一种用于约束多目标遗传算法的约束处理技术》，《群体与进化计算》，第 15 卷，第 66-79 页，2014 年。

[185] Y. Zhou, M. Zhu, J. Wang, Z. Zhang, Y. Xiang, and J. Zhang, "Tri-goal evolution framework for constrained many-objective optimization," IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 50, no. 8, pp. 3086-3099, 2018.
[185] 周毅、朱明、王军、张政、向勇和张军，《面向约束多目标优化的三目标进化框架》，《IEEE 系统、人与控制论汇刊：系统》，第 50 卷第 8 期，第 3086-3099 页，2018 年。

[186] D. K. Saxena, T. Ray, K. Deb, and A. Tiwari, "Constrained many-objective optimization: A way forward," in 2009 IEEE Congress on Evolutionary Computation, 2009, pp. 545-552.
[186] D. K. Saxena、T. Ray、K. Deb 和 A. Tiwari，《约束多目标优化：前进之路》，发表于《2009 年 IEEE 进化计算大会》，2009 年，第 545-552 页。

[187] S. Zapotecas Martinez, A. Arias Montano, and C. A. Coello Coello, "Constrained multi-objective aerodynamic shape optimization via swarm intelligence," in Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation, 2014, pp. 81-88.
[187] S. Zapotecas Martinez、A. Arias Montano 和 C. A. Coello Coello，《基于群体智能的约束多目标气动外形优化》，载《2014 年遗传与进化计算年会论文集》，2014 年，第 81-88 页。

[188] R. L. Becerra, C. A. Coello Coello, A. G. Hernández-D\'\iaz, R. Caballero, and J. Molina, "Alternative techniques to solve hard multi-objective optimization problems," in Proceedings of the 9th annual conference on Genetic and evolutionary computation, 2007, pp. 754-757.
[188] R. L. Becerra、C. A. Coello Coello、A. G. Hernández-D\'\iaz、R. Caballero 和 J. Molina，《解决复杂多目标优化问题的替代技术》，载《第 9 届遗传与进化计算年会论文集》，2007 年，第 754-757 页。

[189] Z. Yang, X. Cai, and Z. Fan, "Epsilon constrained method for constrained multiobjective optimization problems: some preliminary results," in Proceedings of the companion publication of the 2014 annual conference on genetic and evolutionary computation, 2014, pp. 1181-1186.
[189] 杨志、蔡翔和范舟，《约束多目标优化问题的ε约束法：初步成果》，载《2014 年遗传与进化计算年会附属出版物》，2014 年，第 1181-1186 页。

[190] S. Z. Martinez and C. A. C. Coello, "A multi-objective evolutionary algorithm based on decomposition for constrained multi-objective optimization," in 2014 IEEE Congress on evolutionary computation (CEC), 2014, pp. 429-436.
[190] S. Z. Martinez 和 C. A. C. Coello，《基于分解的约束多目标进化算法》，载《2014 年 IEEE 进化计算大会（CEC）》，2014 年，第 429-436 页。

[191] Z. Fan et al., "An improved epsilon constraint handling method embedded in MOEA/D for constrained multi-objective optimization problems," in 2016 IEEE Symposium Series on Computational Intelligence (SSCI), 2016, pp. 1-8.
[191] 范志强等, "MOEA/D 中嵌入改进的ε约束处理方法用于约束多目标优化问题," 发表于《2016 年 IEEE 计算智能系列研讨会(SSCI)》, 2016 年, 第 1-8 页。

[192] Y. Yang, J. Liu, S. Tan, and H. Wang, "A multi-objective differential evolutionary algorithm for constrained multi-objective optimization problems with low feasible ratio," Applied Soft Computing, vol. 80, pp. 42-56, 2019.
[192] 杨勇等, "针对低可行率约束多目标优化问题的多目标差分进化算法," 《应用软计算》, 第 80 卷, 第 42-56 页, 2019 年。

[193] S. Zapotecas-Mart\'\inez and A. Ponsich, "Constraint handling within moea/d through an additional scalarizing function," in Proceedings of the 2020 Genetic and Evolutionary Computation Conference, 2020, pp. 595- 602.
[193] S. Zapotecas-Martínez 与 A. Ponsich, "通过附加标量化函数在 MOEA/D 中实现约束处理," 发表于《2020 年遗传与进化计算会议论文集》, 2020 年, 第 595-602 页。

[194] Y. Yang, J. Liu, and S. Tan, "A constrained multi-objective evolutionary algorithm based on decomposition and dynamic constraint-handling mechanism," Applied Soft Computing, vol. 89, p. 106104, 2020.
[194] 杨勇等, "基于分解和动态约束处理机制的约束多目标进化算法," 《应用软计算》, 第 89 卷, 第 106104 页, 2020 年。

[195] Y. Yang, J. Liu, and S. Tan, "A partition-based constrained multi-objective evolutionary algorithm," Swarm and Evolutionary Computation, vol. 66, p. 100940, 2021.
[195] 杨勇、刘健与谭少华，《基于分区的约束多目标进化算法》，《群体与进化计算》，第 66 卷，第 100940 页，2021 年。

[196] B. Liu, H. Ma, X. Zhang, and Y. Zhou, "A memetic coevolutionary differential evolution algorithm for
[196] 刘斌、马浩、张旭与周勇，《一种基于模因协同进化差分算法的约束优化方法》，载于

constrained optimization," in 2007 IEEE Congress on Evolutionary Computation, 2007, pp. 2996-3002.
《2007 年 IEEE 进化计算大会》，2007 年，第 2996-3002 页。

[197] Y. Tian, T. Zhang, J. Xiao, X. Zhang, and Y. Jin, "A coevolutionary framework for constrained multiobjective optimization problems," IEEE Transactions on Evolutionary Computation, vol. 25, no. 1, pp. 102-116, 2020.
[197] 田雨、张涛、肖俊、张旭与金耀，《约束多目标优化问题的协同进化框架》，《IEEE 进化计算汇刊》，第 25 卷第 1 期，第 102-116 页，2020 年。

[198] K. Li, R. Chen, G. Fu, and X. Yao, "Two-archive evolutionary algorithm for constrained multiobjective optimization," IEEE Transactions on Evolutionary Computation, vol. 23, no. 2, pp. 303-315, 2018.
[198] 李康, 陈锐, 傅冠雄, 姚新, '面向约束多目标优化的双档案进化算法,' 《IEEE 进化计算汇刊》, 卷 23, 期 2, 页 303-315, 2018.

[199] J. Wang, Y. Li, Q. Zhang, Z. Zhang, and S. Gao, "Cooperative multiobjective evolutionary algorithm with propulsive population for constrained multiobjective optimization," IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2021.
[199] 王健, 李毅, 张强, 张钊, 高松, '采用推进种群的协作式多目标进化算法求解约束多目标优化问题,' 《IEEE 系统、人与控制论汇刊: 系统》, 2021.

[200] Z.-Z. Liu, B.-C. Wang, and K. Tang, "Handling Constrained Multiobjective Optimization Problems via Bidirectional Coevolution," IEEE Transactions on Cybernetics, 2021.
[200] 刘志强, 王丙春, 唐珂, '基于双向协同进化的约束多目标优化问题处理方法,' 《IEEE 控制论汇刊》, 2021.

[201] Z. Fan et al., "Push and pull search embedded in an $\mathrm{M}2\mathrm{M}$ framework for solving constrained multi-objective optimization problems," Swarm and Evolutionary Computation, vol. 54, p. 100651, 2020.
[201] 范政等, ' $\mathrm{M}2\mathrm{M}$ 框架中嵌入推拉搜索策略求解约束多目标优化问题,' 《群体与进化计算》, 卷 54, 页 100651, 2020.

[202] H. Wang, T. Cai, K. Li, and W. Pedrycz, "Constraint handling technique based on Lebesgue measure for constrained multiobjective particle swarm optimization algorithm," Knowledge-Based Systems, vol. 227, p. 107131, 2021.
[202] 王 H, 蔡 T, 李 K, 和 W. Pedrycz, "基于勒贝格测度的约束多目标粒子群优化算法约束处理技术," 基于知识的系统, 卷 227, 页 107131, 2021 年.

[203] A. Oyama, K. Shimoyama, and K. Fujii, "New constraint-handling method for multi-objective and multi-constraint evolutionary optimization," Trans Jpn Soc Aeronaut Space Sci, vol. 50, no. 167, pp. 56-62, 2007.
[203] 小山 A, 下山 K, 和藤井 K, "多目标多约束进化优化的新约束处理方法," 日本航空宇宙学会汇刊, 卷 50, 期 167, 页 56-62, 2007 年.

[204] F. Jimenez, A. F. Gómez-Skarmeta, G. Sánchez, and K. Deb, "An evolutionary algorithm for constrained multi-objective optimization," in Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No. 02TH8600), 2002, vol. 2, pp. 1133-1138.
[204] F. Jimenez, A. F. Gómez-Skarmeta, G. Sánchez, 和 K. Deb, "一种用于约束多目标优化的进化算法," 收录于 2002 年进化计算大会论文集. CEC'02 (分类号 02TH8600), 2002 年, 卷 2, 页 1133-1138.

[205] F. Caraffini, F. Neri, G. Iacca, and A. Mol, "Parallel memetic structures," Information Sciences, vol. 227, pp. 60-82, 2013.
[205] F. Caraffini, F. Neri, G. Iacca, 和 A. Mol, "并行模因结构," 信息科学, 卷 227, 页 60-82, 2013 年.

[206] F. Caraffini, F. Neri, and L. Picinali, "An analysis on separability for memetic computing automatic design," Information Sciences, vol. 265, pp. 1-22, 2014.
[206] F. Caraffini, F. Neri, 与 L. Picinali, 《模因计算自动设计中的可分离性分析》, 《信息科学》, 卷 265, 页 1-22, 2014 年。

[207] S. Datta, A. Ghosh, K. Sanyal, and S. Das, "A radial boundary intersection aided interior point method for multi-objective optimization," Information Sciences, vol. 377, pp. 1-16, 2017.
[207] S. Datta, A. Ghosh, K. Sanyal, 与 S. Das, 《基于径向边界交点辅助内点法的多目标优化方法》, 《信息科学》, 卷 377, 页 1-16, 2017 年。

[208] V. Morovati and L. Pourkarimi, "Extension of Zoutendijk method for solving constrained multiobjective optimization problems," European Journal of Operational Research, vol. 273, no. 1, pp. 44- 57, 2019.
[208] V. Morovati 与 L. Pourkarimi, 《Zoutendijk 方法在约束多目标优化问题求解中的扩展》, 《欧洲运筹学杂志》, 卷 273, 期 1, 页 44-57, 2019 年。

[209] L. Uribe, A. Lara, and O. Schütze, "On the efficient computation and use of multi-objective descent directions within constrained MOEAs," Swarm and Evolutionary Computation, vol. 52, p. 100617, 2020.
[209] L. Uribe, A. Lara, 与 O. Schütze, 《约束多目标进化算法中高效计算与使用多目标下降方向的研究》, 《群体与进化计算》, 卷 52, 页 100617, 2020 年。

[210] K. Harada, J. Sakuma, I. Ono, and S. Kobayashi, "Constraint-handling method for multi-objective function optimization: Pareto descent repair operator," in
[210] 原田香、佐久间淳、小野泉、小林重信，《多目标函数优化的约束处理方法：帕累托下降修复算子》，收录于

International Conference on Evolutionary Multi-Criterion Optimization, 2007, pp. 156-170.
《进化多准则优化国际会议论文集》，2007 年，第 156-170 页。

[211] F. Qian, B. Xu, R. Qi, and H. Tianfield, "Self-adaptive differential evolution algorithm with $\alpha$ -constrained-domination principle for constrained multi-objective optimization," Soft Computing, vol. 16, no. 8, pp. 1353- 1372, 2012.
[211] 钱锋、徐博、齐瑞、田洪野，《基于 $\alpha$ 约束支配原则的自适应差分进化算法求解约束多目标优化问题》，载《软计算》，第 16 卷第 8 期，2012 年，第 1353-1372 页。

[212] X. Yu, X. Yu, Y. Lu, G. G. Yen, and M. Cai, "Differential evolution mutation operators for constrained multi-objective optimization," Applied Soft Computing, vol. 67, pp. 452-466, 2018.
[212] 余翔、余雪峰、卢彦、G. G. Yen、蔡明，《面向约束多目标优化的差分进化变异算子》，载《应用软计算》，第 67 卷，2018 年，第 452-466 页。

[213] B.-Y. Qu and P. N. Suganthan, "Constrained multi-objective optimization algorithm with diversity enhanced differential evolution," in IEEE Congress on Evolutionary Computation, 2010, pp. 1-5.
[213] 屈宝元与 P. N. Suganthan 合著，《基于多样性增强差分进化的约束多目标优化算法》，发表于 IEEE 进化计算大会，2010 年，第 1-5 页。

[214] Y.-N. Wang, L.-H. Wu, and X.-F. Yuan, "Multi-objective self-adaptive differential evolution with elitist archive and crowding entropy-based diversity measure," Soft Computing, vol. 14, no. 3, pp. 193-209, 2010.
[214] 王亚南、吴立辉、袁晓峰合著，《基于精英存档与拥挤熵多样性测度的多目标自适应差分进化算法》，刊于《软计算》第 14 卷第 3 期，2010 年，第 193-209 页。

[215] Y. Liu, X. Li, and Q. Hao, "A new constrained multi-objective optimization problems algorithm based on group-sorting," in Proceedings of the Genetic and Evolutionary Computation Conference Companion, 2019, pp. 221-222.
[215] 刘洋、李翔、郝强合著，《一种基于分组排序的约束多目标优化问题新算法》，发表于遗传与进化计算会议附属会议论文集，2019 年，第 221-222 页。

[216] H. Rosenbrock, "An automatic method for finding the greatest or least value of a function," The Computer Journal, vol. 3, no. 3, pp. 175-184, 1960.
[216] H. Rosenbrock 著，《寻找函数极值的自动方法》，刊于《计算机杂志》第 3 卷第 3 期，1960 年，第 175-184 页。

[217] Z. Michalewicz, "Genetic algorithms, numerical optimization, and constraints," in Proceedings of the sixth international conference on genetic algorithms, 1995, vol. 195, pp. 151-158.
[217] Z. Michalewicz，《遗传算法、数值优化与约束处理》，收录于《第六届遗传算法国际会议论文集》，1995 年，第 195 卷，第 151-158 页。

[218] D. M. Himmelblau, "Applied Nonlinear Programming McGraw-Hill Book Company," New York, 1972.
[218] D. M. Himmelblau，《应用非线性规划》，McGraw-Hill 图书公司，纽约，1972 年。

[219] S. S. Rao, Engineering optimization: theory and practice. John Wiley & Sons, 2019.
[219] S. S. Rao，《工程优化：理论与实践》，John Wiley & Sons 出版社，2019 年。

[220] E. Sandgren, "Nonlinear integer and discrete programming in mechanical design optimization," 1990.
[220] E. Sandgren，《机械设计优化中的非线性整数与离散规划》，1990 年。

[221] J. S. Arora, "Introduction to Optimum Design, McGraw-Hill, New York, 1989".
[221] J. S. Arora，《优化设计导论》，McGraw-Hill 出版社，纽约，1989 年。

[222] J. Golinski, "An adaptive optimization system applied to machine synthesis," Mechanism and Machine Theory, vol. 8, no. 4, pp. 419-436, 1973.
[222] J. Golinski，“应用于机器综合的自适应优化系统”，《机构与机器理论》，第 8 卷第 4 期，第 419-436 页，1973 年。

[223] D. Kvalie, "Optimization of plane elastic grillages," PhD Thesis, Norges Teknisk Naturvitenskapelige Universitet, Norway, 1967.
[223] D. Kvalie，《平面弹性格栅的优化》，博士论文，挪威科技大学，挪威，1967 年。

[224] X.-S. Yang and A. H. Gandomi, "Bat algorithm: a novel approach for global engineering optimization," Eng Comput (Swansea), 2012.
[224] 杨新社与 A. H. Gandomi，“蝙蝠算法：一种全局工程优化的新方法”，《工程计算》（斯旺西），2012 年。

[225] G. Steven, "Evolutionary algorithms for single and multicriteria design optimization. A. Osyczka. Springer Verlag, Berlin, 2002, ISBN 3-7908-1418-01," Structural and Multidisciplinary Optimization, vol. 24, no. 1, pp. $88-89,2002$ .
[225] G. Steven，《单目标与多目标设计优化的进化算法》，A. Osyczka 著，Springer Verlag 出版社，柏林，2002 年，ISBN 3-7908-1418-01，《结构与多学科优化》，第 24 卷第 1 期，pp. $88-89,2002$ 。

[226] W. Changsen and C. Wan, Analysis of rolling element bearings. Wiley-Blackwell, 1991.
[226] W. Changsen 与 C. Wan，《滚动轴承分析》，Wiley-Blackwell 出版社，1991 年。

[227] B. D. Youn and K. K. Choi, "A new response surface methodology for reliability-based design optimization," Comput Struct, vol. 82, no. 2-3, pp. 241-256, 2004.
[227] B. D. Youn 与 K. K. Choi，《基于可靠性的设计优化新响应面方法》，《计算机与结构》，第 82 卷第 2-3 期，页码 241-256，2004 年。

[228] G. Vanderplaats, "Very large scale optimization," in 8th symposium on multidisciplinary analysis and optimization, 2002, p. 4809.
[228] G. Vanderplaats，《超大规模优化》，收录于《第八届多学科分析与优化研讨会论文集》，2002 年，第 4809 页。

[229] T. T. Binh and U. Korn, "MOBES: A multiobjective evolution strategy for constrained optimization problems," in The third international conference on genetic algorithms (Mendel 97), 1997, vol. 25, p. 27.
[229] T. T. Binh 与 U. Korn，《MOBES：一种用于约束优化问题的多目标进化策略》，发表于第三届国际遗传算法会议（Mendel 97），1997 年，第 25 卷，第 27 页。

[230] A. Osyczka and S. Kundu, "A new method to solve generalized multicriteria optimization problems using the simple genetic algorithm," Structural optimization, vol. 10, no. 2, pp. 94-99, 1995.
[230] A. Osyczka 与 S. Kundu，《采用简单遗传算法解决广义多准则优化问题的新方法》，载《结构优化》，1995 年，第 10 卷第 2 期，第 94-99 页。

[231] E. Zitzler, K. Deb, and L. Thiele, "Comparison of multiobjective evolutionary algorithms: Empirical results," Evol Comput, vol. 8, no. 2, pp. 173-195, 2000.
[231] E. Zitzler、K. Deb 和 L. Thiele，《多目标进化算法的比较：实证结果》，《进化计算》，第 8 卷第 2 期，pp. 173-195，2000 年。

[232] H. Li, Q. Zhang, and J. Deng, "Biased multiobjective optimization and decomposition algorithm," IEEE Trans Cybern, vol. 47, no. 1, pp. 52-66, 2016.
[232] 李辉、张强与邓杰，《偏置多目标优化与分解算法》，载《IEEE 控制论汇刊》，2016 年，第 47 卷第 1 期，第 52-66 页。

[233] K. Deb and A. Srinivasan, "Innovization: Innovating design principles through optimization," in Proceedings of the 8th annual conference on Genetic and evolutionary computation, 2006, pp. 1629-1636.
[233] K. Deb 与 A. Srinivasan 合著，《通过优化实现创新设计原则的"创新化"方法》，收录于《第 8 届遗传与进化计算年会论文集》，2006 年，第 1629-1636 页。

[234] H. Jain and K. Deb, "An evolutionary many-objective optimization algorithm using reference-point based nondominated sorting approach, part II: Handling constraints and extending to an adaptive approach," IEEE Transactions on evolutionary computation, vol. 18, no. 4, pp. 602-622, 2013.
[234] H. Jain 与 K. Deb 合著，《基于参考点的非支配排序多目标进化算法第二部分：约束处理与自适应扩展》，刊载于《IEEE 进化计算汇刊》第 18 卷第 4 期，2013 年，第 602-622 页。

[235] S. Huband, P. Hingston, L. Barone, and L. While, "A review of multiobjective test problems and a scalable test problem toolkit," IEEE Transactions on Evolutionary Computation, vol. 10, no. 5, pp. 477-506, 2006.
[235] S. Huband、P. Hingston、L. Barone 及 L. While 合著，《多目标测试问题综述及可扩展测试问题工具包》，刊载于《IEEE 进化计算汇刊》第 10 卷第 5 期，2006 年，第 477-506 页。

[236] K. Deb, L. Thiele, M. Laumanns, and E. Zitzler, "Scalable test problems for evolutionary multiobjective optimization," in Evolutionary multiobjective optimization, Springer, 2005, pp. 105-145.
[236] K. Deb、L. Thiele、M. Laumanns 及 E. Zitzler 合著，《进化多目标优化的可扩展测试问题集》，收录于《进化多目标优化》Springer 出版社，2005 年，第 105-145 页。

[237] L. Leydesdorff and S. Milojević, "Scientometrics," arXiv preprint arXiv:1208.4566, 2012.
[237] L. Leydesdorff 和 S. Milojević，《科学计量学》，arXiv 预印本 arXiv:1208.4566，2012 年。

[238] D. Childress, "Citation tools in academic libraries: Best practices for reference and instruction," Reference & User Services Quarterly, vol. 51, no. 2, p. 143, 2011.
[238] D. 柴尔德里斯，《高校图书馆引文工具：参考咨询与教学的最佳实践》，《参考与用户服务季刊》第 51 卷第 2 期第 143 页，2011 年。

[239] C. A. Estabrooks et al., "The intellectual structure and substance of the knowledge utilization field: A longitudinal author co-citation analysis, 1945 to 2004," Implementation Science, vol. 3, no. 1, p. 49, 2008.
[239] C. A. 埃斯特布鲁克斯等，《知识利用领域的知识结构与实质：1945 至 2004 年的作者共被引纵向分析》，《实施科学》第 3 卷第 1 期第 49 页，2008 年。

[240] I. Rahimi, A. Ahmadi, A. F. Zobaa, A. Emrouznejad, and S. H. E. Abdel Aleem, "Big data optimization in electric power systems: A review," CRC Press, 2017.
[240] I. 拉希米、A. 艾哈迈迪、A. F. 佐巴、A. 埃姆鲁兹贾德与 S. H. E. 阿卜杜勒·阿利姆，《电力系统中的大数据优化：综述》，CRC 出版社，2017 年。

[241] B. Müßigmann, H. von der Gracht, and E. Hartmann, "Blockchain technology in logistics and supply chain management-a bibliometric literature review from 2016 to january 2020," IEEE Transactions on Engineering Management, vol. 67, no. 4, pp. 988-1007, 2020.
[241] B. Müßigmann, H. von der Gracht 和 E. Hartmann，《区块链技术在物流与供应链管理中的应用——2016 年至 2020 年 1 月的文献计量综述》，《IEEE 工程管理汇刊》，第 67 卷第 4 期，第 988-1007 页，2020 年。

[242] S. Neelam and S. K. Sood, "A scientometric review of global research on smart disaster management," IEEE Transactions on Engineering Management, vol. 68, no. 1, pp. 317-329, 2020.
[242] S. Neelam 和 S. K. Sood，《全球智能灾害管理研究的科学计量评述》，《IEEE 工程管理汇刊》，第 68 卷第 1 期，第 317-329 页，2020 年。

[243] A. H. Gandomi, A. Emrouznejad, and I. Rahimi, "Evolutionary Computation in Scheduling: A Scientometric Analysis," Evolutionary Computation in Scheduling, pp. 1-10, 2020.
[243] A. H. Gandomi, A. Emrouznejad 和 I. Rahimi，《调度中的进化计算：科学计量分析》，《调度中的进化计算》，第 1-10 页，2020 年。

[244] S. Das and P. N. Suganthan, "Differential evolution: A survey of the state-of-the-art," IEEE transactions on evolutionary computation, vol. 15, no. 1, pp. 4-31, 2010.
[244] S. Das 和 P. N. Suganthan，《差分进化算法：技术发展现状综述》，《IEEE 进化计算汇刊》，第 15 卷第 1 期，第 4-31 页，2010 年。