A subspace strategy based coevolutionary framework for constrained multimodal multiobjective optimization problems

doi:10.1016/j.swevo.2025.101941

Swarm and Evolutionary Computation
蜂群和进化计算

Volume 95, June 2025, 101941
第 95 卷，2025 年 6 月，第 101941 期

https://doi.org/10.1016/j.swevo.2025.101941 Get rights and content 获取权利和内容

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Abstract 摘要

Constrained multimodal multiobjective optimization problems (CMMOPs) consist of multiple equivalent constrained Pareto sets (CPSs) that have the identical constrained Pareto front (CPF). The key to solving CMMOPs lies in how to locate and retain CPSs and CPF in search spaces. Thus, this paper proposes a subspace strategy based coevolutionary framework for CMMOPs, named SCCMMO. Firstly, the subspace generation and maintenance strategy is proposed to efficiently locate multiple CPSs within the decision space. Secondly, the subspace-type perception strategy is used to exploit the feasible and infeasible information in subspaces. Finally, a coevolutionary framework is introduced to improve search efficiency. To prove the effectiveness of the algorithm, the proposed method is compared with ten state-of-the-art algorithms on seventeen benchmarks. The results demonstrate the superiority of SCCMMO in solving CMMOPs. Moreover, SCCMMO also achieves better performance on the real-world problem.
约束多模式多目标优化问题（CMMOPs）由多个等效约束帕累托集（CPSs）组成，这些约束帕累托集具有相同的约束帕累托前沿（CPF）。解决 CMMOPs 的关键在于如何在搜索空间中找到并保留 CPS 和 CPF。因此，本文提出了一种基于子空间策略的 CMMOPs 协同演化框架，命名为 SCCMMO。首先，本文提出了子空间生成和维护策略，以在决策空间中有效定位多个 CPS。其次，采用子空间型感知策略来利用子空间中的可行和不可行信息。最后，引入协同进化框架来提高搜索效率。为了证明算法的有效性，我们在 17 个基准上将所提出的方法与 10 种最先进的算法进行了比较。结果表明，SCCMMO 在求解 CMMOP 方面更具优势。此外，SCCMMO 在实际问题上也取得了更好的性能。

Keywords 关键词

Constrained multimodal multiobjective optimization

Subspace strategy

Coevolution

Evolution algorithms

受限多模态多目标优化子空间策略进化进化算法

1. Introduction 1.导言

Constrained multiobjective optimization problems (CMOPs) [1] are a common type of optimization problem in scientific research and engineering practices. These problems involve optimizing multiple conflicting objectives, and the solutions need to satisfy the complex constraints. CMOPs can be described as follows:
约束多目标优化问题（CMOPs）[1] 是科学研究和工程实践中常见的一种优化问题。这些问题涉及对多个相互冲突的目标进行优化，其解决方案需要满足复杂的约束条件。CMOP 可描述如下：(1)

m i n F (x) = {(f_{1} (x), f_{2} (x), \dots, f_{m} (x))}^{T}

subject to 受(2)

\{\begin{matrix} g_{j} (x) \leq 0, j = 1, . . ., p \\ h_{k} (x) = 0, k = 1, . . ., q \end{matrix}

where

x = (x_{1}, \dots, x_{D})

is a

D

-dimensional decision vector, and

F (x)

is a

M

-dimensional objective vector.

g_{j} (x)

is the

j

th inequality constraint, and

h_{k} (x)

is the

k

th equality constraint. To judge whether the solution vector satisfies the constraint condition, the constraint violation function is introduced, and its value is called the constraint violation (CV) degree. It can be described as follows:
其中

x = (x_{1}, \dots, x_{D})

是

D

维决策向量，

F (x)

是

M

维目标向量。

g_{j} (x)

是第

j

个不等式约束条件，

h_{k} (x)

是第

k

个等式约束条件。为了判断解向量是否满足约束条件，引入了约束违反函数，其值称为约束违反（CV）度。其描述如下(3)

C V_{i} (x) = \{\begin{matrix} m a x (0, g_{i} (x)), i = 1, . . ., p \\ m a x (0, | h_{i} (f x) | - ξ), i = p + 1, . . ., q \end{matrix}

(4)

C V (x) = \sum_{i = 1}^{q} C V_{i} (x) = 0

where

ξ

denotes tolerance value applied to relax equality constraints. If the solution

x

meets the constraint condition in Eq. (4), indicating that

x

is a feasible solution. Otherwise,

x

is an infeasible solution. The set of feasible Pareto optimal solutions is called the constrained Pareto optimal solution set (CPS), and the set of Pareto optimal solutions without considering constraints is called the unconstrained Pareto optimal solution set (UPS). CPS and UPS are mapped to object space form constrained Pareto front (CPF) and unconstrained Pareto front (UPF) [2].
其中

ξ

表示用于放松相等约束的容许值。如果解

x

满足公式 (4) 中的约束条件，说明

x

是可行解。否则，

x

为不可行解。可行的帕累托最优解集称为受约束帕累托最优解集（CPS），不考虑约束条件的帕累托最优解集称为无约束帕累托最优解集（UPS）。CPS 和 UPS 被映射到对象空间，形成约束帕累托前沿（CPF）和无约束帕累托前沿（UPF）[2]。

Recently, theoretical research on CMOPs has developed relatively mature. However, a single Pareto optimal solution set (PS) often fails to satisfy the needs of decision makers in practical applications, it is necessary to provide rich choices [3]. Therefore, the constrained multimodal multiobjective optimization problems (CMMOPs) that can satisfy the constraints and multimodality features are more in line with practical engineering applications.
近年来，关于 CMOP 的理论研究已经发展得相对成熟。然而，在实际应用中，单一的帕累托最优解集（PS）往往无法满足决策者的需求，有必要提供丰富的选择[3]。因此，能满足约束条件和多模态特征的约束多模态多目标优化问题（CMMOPs）更符合实际工程应用的需要。

The diagram of CMMOP is illustrated in Fig. 1, where the shadow part represents feasible region. In decision space, there are two equivalent UPSs (including

{UPS}_{1}

and

{UPS}_{2}

) and two equivalent CPSs (including

{CPS}_{1}

and

{CPS}_{2}

), which correspond to UPF and CPF in objective space respectively. Compared with CMOPs, the introduction of multimodality features in the CMMOPs increases the difficulty of the problem. Specifically,

{CPS}_{1}

and

{CPS}_{2}

in the decision space may have different shapes, distributions and dispersions, resulting in unbalanced convergence difficulty. Therefore, CMMOPs have higher requirements and challenges for the diversity, convergence and feasibility of the solutions in decision space.
图 1 是 CMMOP 的示意图，阴影部分代表可行区域。在决策空间中，有两个等价的 UPS（包括

{UPS}_{1}

和

{UPS}_{2}

）和两个等价的 CPS（包括

{CPS}_{1}

和

{CPS}_{2}

），分别对应目标空间中的 UPF 和 CPF。与 CMOPs 相比，CMMOPs 中多模态特征的引入增加了问题的难度。具体来说，决策空间中的

{CPS}_{1}

和

{CPS}_{2}

可能具有不同的形状、分布和分散性，导致收敛难度不均衡。因此，CMMOPs 对决策空间解的多样性、收敛性和可行性提出了更高的要求和挑战。

Evolutionary algorithms demonstrate the effectiveness in solving a range of optimization problems [4], [5]. At present, evolutionary algorithms mainly focus on solving CMOPs (ignoring multimodality features) and multimodal multiobjective optimization problems (MMOPs) [5]. While in face of CMMOPs (containing constraints and multimodality features), the performance of the algorithm will decline sharply. On the one hand, due to the lack of diversity in decision space, the constrained multiobjective evolutionary algorithm (CMOEA) only locates one CPS or a part of multiple CPSs and cannot locate multiple complete CPSs simultaneously. On the other hand, due to the lack of constraint handling technology (CHT), the multimodal multiobjective evolutionary algorithm (MMOEA) only locates multiple UPSs but cannot converge to multiple CPSs in the feasible region.
进化算法在解决一系列优化问题方面显示出了其有效性[4], [5]。目前，进化算法主要集中于解决 CMOPs（忽略多模态特征）和多模态多目标优化问题（MMOPs）[5]。而面对 CMMOPs（包含约束和多模态特征），算法的性能会急剧下降。一方面，由于决策空间缺乏多样性，约束多目标进化算法（CMOEA）只能定位一个 CPS 或多个 CPS 的一部分，无法同时定位多个完整的 CPS。另一方面，由于缺乏约束处理技术（CHT），多模式多目标进化算法（MMOEA）只能定位多个不间断电源，但无法在可行区域内收敛到多个 CPS。

There are few relevant studies about CMMOPs, mainly including CMMODE [6] and CMMOCEA [7]. The former proposed the differential evolution based on speciation to solve CMMOPs, however, the optimization performance deteriorates in the face of complex problems. The latter focuses on dealing with constraints and multimodality, resulting in poor convergence and distribution of the solutions in objective space. In summary, the difficulty of the CMMOPs is that there are multiple CPSs in decision space, which will inevitably aggravate the difficulty of locating feasible region. Therefore, how to locate and retain CPSs and CPF in search space is the key to solving CMMOPs.
关于 CMMOPs 的相关研究较少，主要包括 CMMODE [6] 和 CMMOCEA [7]。前者提出了基于标化的微分进化来求解 CMMOPs，但面对复杂问题时优化性能下降。后者侧重于处理约束条件和多模态问题，结果收敛性差，解在目标空间的分布也不均匀。综上所述，CMMOPs 的难点在于决策空间中存在多个 CPS，这势必会加剧可行区域定位的难度。因此，如何在搜索空间中定位并保留 CPS 和 CPF 是求解 CMMOP 的关键。

In view of the above problems, this paper proposes a subspace strategy based coevolutionary framework for CMMOPs, named SCCMMO, which includes two evolutionary stages. In the first stage, the proposed subspace generation and maintenance strategies can help the population separately evolve in each subspace and thus locate more constrained or unconstrained Pareto solutions. Besides, the coevolutionary framework is involved, which enables the information exchange between constrained and unconstrained populations and helps the constrained population across the infeasible regions and find more feasible solutions. However, in the later stage of the first phase, as the population gradually converges, the useful information provided by the unconstrained population is limited. Relying solely on the simple interaction makes it difficult to further improve the quality of the CPS. Moreover, during this period, without any knowledge guidance the search efficiency of both the constrained and unconstrained populations will also reduce. Therefore, when the populations evolve to a certain degree, another evolution stage is introduced to fully utilize the knowledge accumulated during the previous evolution stage and continuously improve the interaction efficiency. Specifically, the subspace-type perception and subspace-type-based coevolution strategies are designed and employed in this stage. Based on the knowledge accumulated in the first stage, the relationships between constrained and unconstrained populations in the subspaces are identified. Then, based on their types, the subspace-type-based coevolution strategy dispatches suitable evolution strategies for the unconstrained population. In this way, the unconstrained population can continuously provide useful information to the constrained population, helping enhance its diversity and improve its convergence.
针对上述问题，本文提出了一种基于子空间策略的 CMMOPs 协同演化框架，命名为 SCCMMO，包括两个演化阶段。在第一阶段，所提出的子空间生成和维护策略可以帮助种群分别在每个子空间中进化，从而找到更多受约束或无约束的帕累托解决方案。此外，协同进化框架也参与其中，实现了受约束种群与非受约束种群之间的信息交流，帮助受约束种群跨越不可行区域，找到更多可行解。然而，在第一阶段的后期，随着种群逐渐收敛，无约束种群提供的有用信息有限。仅仅依靠简单的交互作用很难进一步提高 CPS 的质量。此外，在此期间，由于没有任何知识指导，受限种群和非受限种群的搜索效率也会降低。因此，当种群进化到一定程度时，需要引入另一个进化阶段，充分利用前一个进化阶段积累的知识，不断提高交互效率。具体来说，这一阶段设计并采用了子空间类型感知和基于子空间类型的协同进化策略。根据第一阶段积累的知识，确定子空间中受限种群和非受限种群之间的关系。然后，根据它们的类型，基于子空间类型的协同进化策略为非受限种群调度合适的进化策略。这样，无约束种群就能不断为受约束种群提供有用的信息，帮助增强其多样性并改善其收敛性。

The main contributions of this paper are as follows:
本文的主要贡献如下：

•
To localize multiple CPSs in decision space, the subspace generation and maintenance strategy is designed, which employs the grid division method to divide the decision space into multiple subspaces and generate multiple subpopulations. Each subpopulation evolves independently in corresponding subspace. In this way, the difficulty of unbalanced convergence caused by multiple CPSs is alleviated while ensuring the diversity of solutions through the subspace generation and maintenance strategy.
为了在决策空间中定位多个 CPS，设计了子空间生成和维护策略，该策略采用网格划分法将决策空间划分为多个子空间，并生成多个子群。每个子群在相应的子空间中独立演化。这样，既缓解了多个 CPS 带来的不平衡收敛困难，又通过子空间生成和维护策略确保了解的多样性。
•
To utilize the feasible and infeasible information in subspaces, the subspace-type perception strategy is proposed. By judging the relationship between CPS and UPS in each subspace, all subspaces are divided into seven categories. According to the obtained relationship types, the proposed method designs suitable evolutionary strategies for the unconstrained population to provide information and help obtain the final CPSs.
为了利用子空间中的可行和不可行信息，提出了子空间型感知策略。通过判断每个子空间中 CPS 和 UPS 的关系，所有子空间被分为七类。根据所获得的关系类型，所提出的方法为无约束群体设计了合适的进化策略，以提供信息并帮助获得最终的 CPS。

The organization of this paper is structured as follows. Section 2 reviews the related work. Section 3 provides a detailed description of the proposed SCCMMO. Experimental studies are carried out in Section 4. Finally, Section 5 concludes this paper.
本文的结构安排如下。第 2 节回顾了相关工作。第 3 节详细介绍了拟议的 SCCMMO。第 4 节进行了实验研究。最后，第 5 节对本文进行总结。

2. Related work 2.相关工作

In this section, this paper reviews related studies focusing on three important research areas, namely constrained multiobjective optimization (CMO), multimodal multiobjective optimization (MMO) and constrained multimodal multiobjective optimization (CMMO).
在本节中，本文将回顾相关研究，重点关注三个重要研究领域，即约束多目标优化（CMO）、多模态多目标优化（MMO）和约束多模态多目标优化（CMMO）。

2.1. Constrained multiobjective optimization
2.1.约束多目标优化

In recent years, there have been many studies on CMO [8], [9]. According to the different ways of dealing with the problem constraints [10], existing CMOEAs can be broadly categorized into four distinct groups.
近年来，已有许多关于 CMO 的研究[8]、[9]。根据处理问题约束的不同方式[10]，现有的 CMOEA 大致可分为四类。

(1) Constraints and objectives separation. By comparing the CV and dominance relationship between individuals, the algorithm can retain the feasible solution and quickly converge to the feasible region. The related methods mainly include the constraint dominance principle (CDP) [11],

ϵ

constrained method [12], [13], and stochastic ranking (SR) [14]. Fan et al. [15] designed a new angle-based CHT, named ACDP, which integrates angle information into CDP and further uses the information of infeasible solutions. Further, Fan et al. [13] designed a strategy to adaptively adjust the

ϵ

parameter according to the proportion of feasible solutions.
(1) 约束与目标分离。通过比较个体间的 CV 和支配关系，算法可以保留可行解，并快速收敛到可行区域。相关方法主要包括约束支配原理（CDP）[11]、

ϵ

约束法[12]、[13]和随机排序法（SR）[14]。Fan 等人[15] 设计了一种新的基于角度的 CHT，命名为 ACDP，它将角度信息集成到 CDP 中，并进一步利用了不可行解的信息。此外，Fan 等人[13] 设计了一种根据可行解比例自适应调整

ϵ

参数的策略。

(2) Penalty function. This method transforms the constrained optimization problem into an unconstrained optimization problem by adding a penalty term to the objective function [16]. For example, Jiao et al. [17] proposed a feasible-guiding strategy, which uses CV to modify the objective function to obtain a new fitness. To achieve a balance between objectives and constraints, the penalty parameter is adjusted by the feasibility proportion in the current population. Fan et al. [18] proposed a learning-guided parameter setting method, which can adaptively generate penalty parameters. The penalty function combines constraint violation degree, objective function values and current generation to design an exponential decay model, which is embedded into the push and pull search framework to solve CMOPs [19].
(2) 惩罚函数。这种方法通过在目标函数中加入惩罚项，将约束优化问题转化为无约束优化问题 [16]。例如，Jiao 等人[17] 提出了一种可行引导策略，利用 CV 修改目标函数以获得新的适应度。为了实现目标和约束之间的平衡，惩罚参数由当前种群中的可行性比例进行调整。Fan 等人[18]提出了一种学习引导下的参数设置方法，它可以自适应地生成惩罚参数。该惩罚函数结合了约束违反程度、目标函数值和当前生成量，设计了一个指数衰减模型，并将其嵌入到推拉搜索框架中，用于求解 CMOPs [19]。

(3) Multiobjective method. In the multiobjective methods, the constraints are regarded as one additional objective or multiple additional objectives, the CMOP is transformed into an unconstrained counterpart, and then the MOEAs can be employed to solve the converted problems [20]. Long [21] constructed a new CHT for solving CMOPs, in which the convergence, diversity and feasibility of the obtained solutions are taken as three new objectives. To make CV as a new objective, Peng et al. [22] designed a new CHT based on direct weights to solve the CMOPs, where the two types of weights are distributed in the feasible regions and the infeasible regions for the purpose of guiding the search to the promising regions. Zhou et al. [23] proposed a tri-objective evolutionary framework to solve CMOPs, where the proposed framework transforms constraints into feasibility indicators and then combines them with convergence indicators and diversity indicators to form three new objective.
(3) 多目标方法。在多目标方法中，约束条件被视为一个附加目标或多个附加目标，CMOP 被转化为无约束的对应目标，然后可以采用 MOEAs 解决转化后的问题[20]。Long[21]构建了一种新的用于求解 CMOP 的 CHT，将求解的收敛性、多样性和可行性作为三个新目标。为了将 CV 作为新目标，Peng 等人[22]设计了一种基于直接权值的新 CHT 来求解 CMOPs，其中两种权值分别分布在可行区域和不可行区域，目的是引导搜索到有希望的区域。Zhou 等人[23]提出了一种求解 CMOPs 的三目标演化框架，该框架将约束条件转化为可行性指标，然后将其与收敛性指标和多样性指标相结合，形成三个新目标。

(4) Transforming the CMOPs into other problems. Tian et al. [24] proposed a coevolutionary framework for solving CMOPs, where one population is devoted to solving the original CMOP, and the other population ignores the constraints to find UPF. The two populations assist each other in solving CMOPs. Fan et al. [19] proposed a push and pull search (PPS) framework, where the goal of the push stage is to cross the infeasible region to the UPF. In the pull stage, the improved

ϵ

constrained method is employed to search the CPF. Based on the PPS framework, a CMOP is decomposed into a group of simple subproblems [25], where each subproblem corresponds to a subpopulation, and the PPS framework is applied to each subpopulation to solve the related subproblem.
(4) 将 CMOP 转化为其他问题。Tian 等人[24]提出了一种求解 CMOP 的协同进化框架，其中一个种群致力于求解原始 CMOP，另一个种群则忽略约束条件，寻找 UPF。两个种群在求解 CMOP 时相互协助。Fan 等人[19]提出了推拉搜索（PPS）框架，其中推阶段的目标是穿过不可行区域，找到 UPF。在拉动阶段，采用改进的

ϵ

约束方法搜索 CPF。基于 PPS 框架，CMOP 被分解为一组简单的子问题[25]，其中每个子问题对应一个子群，PPS 框架应用于每个子群以求解相关的子问题。

2.2. Multimodal multiobjective optimization
2.2.多模式多目标优化

According to the characteristics of different algorithms, MMOEA can be divided into the following categories:
根据不同算法的特点，MMOEA 可分为以下几类：

(1) Multimodal multiobjective evolutionary algorithm based on crowding distance and non-dominated sorting [26], [27]. The Omni-optimizer proposed by Deb et al. [26] is a general optimizer based on NSGA-II. It uses Latin hypercube sampling to initialize the population and introduces the crowding distance in the decision space. By combining with the non-dominated sorting method, the proposed method select the better individuals as the next generation of evolutionary populations.
(1) 基于拥挤距离和非支配排序的多模式多目标进化算法[26]，[27]。Deb 等人[26]提出的 Omni-optimizer 是一种基于 NSGA-II 的通用优化算法。它使用拉丁超立方采样来初始化种群，并在决策空间中引入拥挤距离。通过与非优势排序法相结合，该方法可以选择较好的个体作为下一代进化种群。

(2) Multimodal multiobjective evolutionary algorithm based on niche technology [28], [29]. Its core idea is to divide the population into multiple niches and maintain the diversity of the population through the independent evolution of each niche. M. Preuss et al. [30] propose Niching-CMA to enhance the diversity of decision space, the proposed method firstly uses a niche strategy with a dynamic peak recognition method in environment selection. Then the number and the radius of niches are automatically assigned according to the evolution of the population. Finally, the better individuals in each niche are retained to the next generation.
(2) 基于生态位技术的多模式多目标进化算法[28]，[29]。其核心思想是将种群划分为多个壁龛，通过每个壁龛的独立进化来维持种群的多样性。M. Preuss 等人[30]提出了 Niching-CMA 来增强决策空间的多样性，所提方法首先在环境选择中使用了具有动态峰值识别方法的利基策略。然后，根据种群的进化自动分配壁龛的数量和半径。最后，每个生态位中的优秀个体会被保留到下一代。

(3) Multimodal multiobjective evolutionary algorithm based on new evolutionary paradigm [31], [32] . Liang [31] proposed a self-organizing multiobjective particle swarm optimization algorithm (SMPSO-MM), where the self-organizing map network is used to establish the neighborhood structure of individuals in the population. The results show that the algorithm has good performance in dealing with complex problems. Li et al. [32] proposed a multiobjective particle swarm optimization algorithm based on self-organizing species formation (SS-MOPSO), where the proposed method uses the species formation strategy to form the niches to search PS. In addition, a self-organization mechanism is proposed to improve the efficiency of population formation and the performance of the algorithm.
(3) 基于新进化范式的多模态多目标进化算法 [31], [32] 。Liang[31]提出了一种自组织多目标粒子群优化算法（SMPSO-MM），利用自组织图网络建立种群中个体的邻域结构。结果表明，该算法在处理复杂问题时具有良好的性能。Li 等人[32]提出了一种基于自组织物种形成的多目标粒子群优化算法（SS-MOPSO），所提出的方法利用物种形成策略形成壁龛来搜索 PS。此外，还提出了一种自组织机制，以提高种群形成的效率和算法的性能。

(4) Multimodal multiobjective integration algorithm based on different strategies [33], [34]. This kind of algorithm integrates the complementary advantages of different strategies in MMOEA to further improve the performance of the algorithm. The MMO-CLRPSO proposed by Zhang et al. [33] uses a new decision variable clustering method to form multiple subpopulations, where the particle swarm optimization algorithm is used to make the subpopulations evolve separately, and the ring topology is used to enhance the information interaction ability between subpopulations.
(4) 基于不同策略的多模态多目标集成算法[33]，[34]。这种算法综合了 MMOEA 中不同策略的互补优势，进一步提高了算法的性能。Zhang 等[33]提出的 MMO-CLRPSO 采用新的决策变量聚类方法形成多个子群，其中利用粒子群优化算法使子群分别演化，利用环状拓扑结构增强子群间的信息交互能力。

2.3. Constrained multimodal multiobjective optimization
2.3.有约束多模态多目标优化

The research on CMMOPs has just started, and there are few related research. Liang et al. [6] proposed a CMMODE that employs the species formation mechanism to form niches and combines CDP to locate more CPSs. Hence, the ability of algorithms that retain feasible solutions is greatly improved. This method is a preliminary attempt of CMMOPs. Through the direct integration of niche technology and CDP, the multimodal characteristics and complex constraints of the problem are effectively processed. Ming et al. [7] used a coevolutionary framework, where the method involves the simultaneous evolution of two populations. One population focus on the original CMMOPs, and the other population focus on MMOP. Li et al. [35] proposed DSRDE, which divides the population into multiple species and establishes a ring topology between each species to enhance the diversity of species.
关于 CMMOPs 的研究刚刚起步，相关研究较少。Liang 等人[6]提出了一种 CMMODE，利用物种形成机制来形成壁龛，并结合 CDP 来定位更多的 CPS。因此，保留可行解的算法能力大大提高。该方法是 CMMOPs 的初步尝试。通过利基技术与 CDP 的直接结合，有效地处理了问题的多模态特征和复杂约束条件。Ming 等人[7]采用了协同进化框架，该方法涉及两个种群的同时进化。一个种群侧重于原始的 CMMOP，另一个种群侧重于 MMOP。Li 等人[35] 提出的 DSRDE 将种群分为多个物种，并在每个物种之间建立环状拓扑结构，以提高物种的多样性。

In summary, for CMO, the current research on dealing with constraint has obtained satisfactory results, which can locate and retain complete CPF. However, the multimodality features are not considered. For MMO, the existing algorithms primarily focus on identifying equivalent global PS and local PS but do not consider constraint features. In CMMOP, it involves multimodality features and constraint features, and a mere combination of CHT and existing MMOEAs may not effectively deal with CMMOPs. Therefore, it is crucial to design specific strategies for dealing with CMMOPs, where the primary task is to search for multiple CPSs in feasible regions.
总之，对于 CMO 来说，目前关于处理约束条件的研究取得了令人满意的成果，可以定位和保留完整的 CPF。然而，多模态特征并未被考虑在内。对于 MMO，现有算法主要侧重于识别等效的全局 PS 和局部 PS，但没有考虑约束特征。在 CMMOP 中，它涉及多模态特征和约束特征，仅仅结合 CHT 和现有的 MMOEA 可能无法有效处理 CMMOP。因此，设计处理 CMMOP 的特定策略至关重要，因为 CMMOP 的主要任务是在可行区域内搜索多个 CPS。

3. Proposed method 3.建议的方法

3.1. Procedure of SCCMMO 3.1.南昌市委组织部的工作程序

The main framework of SCCMMO is shown in Fig. 2. When the iteration number

g

is less than the maximum iterations number

G_{max}

, the constrained population and unconstrained population form the corresponding constrained subpopulations and unconstrained subpopulations by subspace generation strategy in subspace. After that, the whole evolutionary process is divided into two stages. In the first stage, the constrained and unconstrained subpopulations continuously evolve through the subspace maintenance strategy, which can identify the potential subspaces of multiple Pareto solution sets, maintain the diversity of the population in decision space, and improve the algorithm’s efficiency. In the second stage, the constrained and unconstrained subpopulations generate offspring based on the proposed subspace-type perception strategy and subspace-type-based coevolution strategy. On the one hand, by utilizing the knowledge accumulated in the first stage, these strategies can perceive the potential relationship between constrained and unconstrained Pareto solution sets in the subspaces. On the other hand, by matching different evolutionary strategies to different types of subspaces, the search efficiency of the population can be improved, and the quality of the solution sets can be enhanced.
SCCMMO 的主要框架如图 2 所示。当迭代次数

g

小于最大迭代次数

G_{max}

时，约束种群和非约束种群在子空间中通过子空间生成策略形成相应的约束子种群和非约束子种群。之后，整个进化过程分为两个阶段。在第一阶段，约束子群和无约束子群通过子空间维持策略不断进化，可以识别多个帕累托解集的潜在子空间，保持决策空间中种群的多样性，提高算法效率。在第二阶段，基于所提出的子空间类型感知策略和基于子空间类型的协同进化策略，受约束子种群和无约束子种群产生后代。一方面，通过利用第一阶段积累的知识，这些策略可以感知子空间中受约束和无约束帕累托解集之间的潜在关系。另一方面，通过将不同的进化策略与不同类型的子空间相匹配，可以提高群体的搜索效率，并提高解集的质量。

The procedure of SCCMMO is presented in Algorithm 1. Initially, two populations

P_{1}

(constrained population) and

P_{2}

(unconstrained population) are randomly generated, and each population has

N P

individuals. The constrained population focuses on the original CMMOP, and the unconstrained population focuses on MMOP. Then, the decision space is divided into multiple subspaces by the subspace generation strategy and two populations in each subspace form corresponding subpopulation. Further, the entire evolution process is divided into two stages based on the iteration number. And a control factor

s

is introduced to control the stage switching. When the iteration number is less than or equal to

s \times G_{m a x}

, the proposed method performs the first stage, where the subspace maintenance strategy is adopted to identify and maintain the potential subspace of the CPSs and UPSs. When the iteration number is greater than

s \times G_{m a x}

, the proposed method executes the second stage, where the subspace-type perception and subspace-type-based coevolution strategies are employed.
SCCMMO 的程序如算法 1 所示。最初，随机生成两个种群

P_{1}

（受约束种群）和

P_{2}

（无约束种群），每个种群有

N P

个个体。受约束种群侧重于原始的 CMMOP，无约束种群侧重于 MMOP。然后，通过子空间生成策略将决策空间划分为多个子空间，每个子空间中的两个种群形成相应的子种群。此外，整个进化过程根据迭代次数分为两个阶段。并引入控制因子

s

来控制阶段的切换。当迭代次数小于或等于

s \times G_{m a x}

时，建议的方法执行第一阶段，采用子空间维护策略来识别和维护 CPS 和 UPS 的潜在子空间。当迭代次数大于

s \times G_{m a x}

时，建议的方法执行第二阶段，采用子空间类型感知和基于子空间类型的协同进化策略。

3.2. Subspace generation and maintenance
3.2.子空间的生成和维护

(1) Subspace generation (1) 子空间生成

To find as many CPSs as possible, this paper proposes a subspace generation strategy. By dividing subspace, it retains the diversity of solutions in decision space and enhances the search efficiency of the algorithm. Further, the proposed method adopts the grid division technology to help realize subspace generation, where the grid division method is similar to GG-MOPSO [36] and GSMPSO-MM [37]. The specific process is as follows:
为了找到尽可能多的 CPS，本文提出了子空间生成策略。通过划分子空间，保留了决策空间解的多样性，提高了算法的搜索效率。此外，本文提出的方法采用网格划分技术来帮助实现子空间生成，其中网格划分方法类似于 GG-MOPSO [36] 和 GSMPSO-MM [37]。具体过程如下：

For the

m

-dimensional decision space, suppose that it is divided into

k

contiguous segment at each dimension, and the

k^{m}

grids are generated. The grid width

w_{d}

in the

d

th dimensional is determined by Eq. (5).
对于

m

维决策空间，假设在每个维度上将其划分为

k

个连续段，并生成

k^{m}

个网格。第

d

维的网格宽度

w_{d}

由公式 (5) 决定。(5)

w_{d} = \frac{U_{d} - L_{d}}{k}, d = 1, 2, \dots, m

where

U_{d}

and

L_{d}

are the upper and lower boundary of the

d

th dimension in decision space respectively. For the individual

x

, the position

U_{d} (x)

d

th dimension is calculated by Eq. (6).
其中

U_{d}

和

L_{d}

分别为决策空间中

d

维度的上边界和下边界。对于个体

x

，其

d

维度的位置

U_{d} (x)

由公式 (6) 计算得出。(6)

C_{d} (x) = ⌊\frac{x_{d} - L_{d}}{w_{d}}⌋ + 1, d = 1, 2, \dots, m

where “

⌊⌋

” represents the floor function. As shown in Fig. 3, the search space is a two-dimensional decision space, and the upper and lower boundary is 1 and 5, where each dimension is divided into four segments, i.e.

k

= 4. Fig. 3(a) shows the divided subspace and the grid width

w_{1}

w_{2}

. It can be calculated by Eq. (5) and the results are both equal to 1. Fig. 3(b)shows an illustration about the distribution of the solutions, where the population size is 30. The position of each two-dimension individual can be obtained by Eq. (6). For example, the position

C_{1} (A)

of the individual

A

in 1th dimension of the grid is 1, and the position

C_{2} (A)

of the individual

A

in 2th dimension of the grid is 4. Then, the density of all the subspaces can be obtained, as shown in Fig. 3(c).
其中，"

⌊⌋

"表示下限函数。如图 3 所示，搜索空间为二维决策空间，上下边界分别为 1 和 5，其中每个维度分为四段，即

k

= 4。图 3(a) 显示了划分后的子空间和网格宽度

w_{1}

w_{2}

。可通过公式 (5) 计算，结果均等于 1。图 3(b) 显示了解的分布情况，其中种群规模为 30。每个二维个体的位置可以由公式 (6) 得出。例如，网格 1 维中个体

A

的位置

C_{1} (A)

为 1，网格 2 维中个体

A

的位置

C_{2} (A)

为 4。

In summary, the initial constrained population

P_{1}

is divided into multiple constrained subpopulations

P_{1, 1}

P_{1, 2}

, …,

P_{1, j}

(

j

=1,2,…,

k^{m}

) while the initial unconstrained population

P_{2}

is divided into multiple unconstrained subpopulations

P_{2, 1}

P_{2, 2}

, …,

P_{2, j}

(

j

=1,2,…,

k^{m}

). The distribution and the number of constrained and unconstrained subpopulations in each subspace can be obtained by subspace generation strategy.
总之，初始约束种群

P_{1}

被划分为多个约束子种群

P_{1, 1}

P_{1, 2}

, ...,

P_{1, j}

(

j

=1,2,...,

k^{m}

) ，而初始非约束种群

P_{2}

被划分为多个非约束子种群

P_{2, 1}

P_{2, 2}

, ...,

P_{2, j}

(

j

=1,2,...,

k^{m}

) 。通过子空间生成策略可以得到每个子空间中受约束子群和无约束子群的分布和数量。

(2) Subspace maintenance (2) 子空间维护

To improve the search capabilities of the algorithm, it is essential to maintain the generated subspaces. On the one hand, it helps to locate the potential multiple CPSs. On the other hand, the two subpopulations evolve near the CPS and UPS in each subspace, which lays the foundation for the subsequent subspace-type perception. The specific steps are as follows:
为了提高算法的搜索能力，必须维护生成的子空间。一方面，它有助于定位潜在的多个 CPS。另一方面，在每个子空间中的 CPS 和 UPS 附近演化出两个子群，这为后续的子空间类型感知奠定了基础。具体步骤如下

When the number of individuals in the subspace is greater than 0, the proposed method regards the subspace as a promising subspace. Otherwise, it is a hopeless subspace. For promising subspace, the subspace maintenance strategy is adopted to identify and maintain the CPS and UPS. In detail, the proposed strategy first assigns

n

individuals for each promising subspace, where the

n

is determined by using Eq. (7):
当子空间中的个体数大于 0 时，建议的方法将该子空间视为有希望的子空间。否则，则为无望子空间。对于有希望的子空间，采用子空间维护策略来识别和维护 CPS 和 UPS。具体来说，建议的策略首先为每个有希望的子空间分配

n

个个体，其中

n

是通过公式（7）确定的：(7)

n = \frac{N P}{k^{m} - q}

where

N P

is the population size,

k^{m}

is the number of subspaces, and

q

is the number of hopeless subspaces. If the number of individuals in the promising subspace is less than

n

, the individuals in the remaining subspaces are randomly selected until the number of individuals in the subpopulation is equal to

n

. Then, the

n

individuals in each promising subspace use DE/current-to-

o p

best/1 to generate

n

offspring. Finally, the offspring (including constrained offspring and unconstrained offspring) are merged into the parents and the environment selection criterion is used to update the population, where NSGA-II with CDP is performed to find feasible and well-distributed solutions for the constrained population and NSGA-II for unconstrained population.
其中，

N P

为种群数量，

k^{m}

为子空间数量，

q

为无望子空间数量。如果有希望子空间中的个体数量少于

n

，则随机选择其余子空间中的个体，直到子群中的个体数量等于

n

。然后，每个有希望子空间中的

n

个体使用 DE/current 到

o p

best/1 的方法产生

n

个后代。最后，将子代（包括受约束子代和无约束子代）合并到父代中，并使用环境选择准则更新种群，其中，对受约束种群使用带 CDP 的 NSGA-II 寻找可行且分布良好的解，对无约束种群使用 NSGA-II 寻找可行且分布良好的解。

3.3. Subspace-type perception
3.3.子空间类型感知

To fully utilize the information about feasible and infeasible solutions in the subspace, this paper adopts the subspace-type perception strategy, where the subspace is divided into seven types based on the distribution relationship between CPS and UPS in subspace. The detailed process is shown in Fig. 4. Firstly, according to the feasibility ratio of unconstrained subpopulation, the subspace is divided into Type I - Type IV. Then, according to whether there is constrained subpopulation in the subspace, it is further divided into type (a) - type (g).
为了充分利用子空间中可行解和不可行解的信息，本文采用了子空间类型感知策略，即根据 CPS 和 UPS 在子空间中的分布关系，将子空间划分为七种类型。具体过程如图 4 所示。首先，根据无约束子群的可行性比率，将子空间划分为类型 I - 类型 IV。然后，根据子空间中是否存在受约束子群，将其进一步划分为类型（a）-类型（g）。

The procedure of subspace-type perception strategy is shown in Algorithm 2. First, if unconstrained subpopulation is located in the infeasible region, the subspace belongs to Type I, and if there is a constrained subpopulation in this subspace, the subspace belongs to type (a), otherwise, it belongs to type (b). Then, if the unconstrained subpopulation lies in the feasible region, then it belongs to Type III, and if there is a constrained subpopulation in this subspace, the subspace belongs to type (e), otherwise, it belongs to type (f). Further, if one part of the unconstrained subpopulation lies in the feasible region while the rest lies in the infeasible region, the subspace belongs to Type II, and if there is constrained subpopulation in this subspace, the subspace belongs to type (c), otherwise, it belongs to type (d). Finally, if there is no unconstrained subpopulation in the subspace, the subspace belongs to Type IV, and if there are only constrained subpopulation in this subspace, the subspace belongs to type (g). Thus, the seven types of subspaces are divided by subspace-type perception strategy.
子空间类型认知策略的流程如算法 2 所示。首先，如果无约束子群位于不可行区域，则该子空间属于类型 I，如果该子空间中有约束子群，则该子空间属于类型（a），否则属于类型（b）。然后，如果无约束子群位于可行区域内，则属于类型 III，如果该子空间中存在约束子群，则该子空间属于类型（e），否则属于类型（f）。此外，如果无约束子群的一部分位于可行区域，而其余部分位于不可行区域，则该子空间属于类型 II，如果该子空间中有约束子群，则该子空间属于类型 (c)，否则属于类型 (d)。最后，如果子空间中没有无约束子群，则该子空间属于类型 IV，如果该子空间中只有约束子群，则该子空间属于类型（g）。因此，七种类型的子空间是按子空间类型感知策略划分的。

Fig. 5 exhibits the diagram of the constraint and unconstrained Pareto solution sets relation type in subspace after subspace-type perception. The shadow part represents the feasible region while the other part represents infeasible region. The red circle and blue circle represent CPS and UPS, respectively.
图 5 展示了子空间类型感知后子空间中约束和无约束帕累托解集关系类型的示意图。阴影部分代表可行区域，另一部分代表不可行区域。红圈和蓝圈分别代表 CPS 和 UPS。

3.4. Subspace-type-based coevolution
3.4.基于子空间类型的协同进化

To find more feasible solutions with higher quality, the subspace-type-based coevolution strategy is adopted in this paper. By matching the best evolutionary strategy for the unconstrained subpopulations and constrained subpopulations, the information interaction is strengthened to enhance the utilization efficiency of solutions.
为了找到更多更高质量的可行解，本文采用了基于子空间类型的协同进化策略。通过匹配无约束子群和受约束子群的最佳进化策略，加强信息交互，提高解的利用效率。

Algorithm 3 gives the procedure of subspace-type-based coevolution strategy. The main task in subspace-type-based coevolution is to assign appropriate strategies for

P_{2, j}

, and

P_{1, j}

is updated in the same manner with the subspace maintenance stage because of the good search ability. Specifically, if the subspace belongs type (a),

P_{2, j}

and

P_{1, j}

implement DE/current-to-

o p

best/1 to generate offspring

O_{2, j}

and

O_{1, j}

. If the subspace belongs type (b),

P_{2, j}

implements GA to generate offspring

O_{2, j}

. Then, if the subspace belongs type (c),

P_{2, j}

implements DE/current-to-

o p

best/1, DE/current_to_rand_1 and GA to generate offspring

O_{2, j}

, and

P_{1, j}

implements DE/current-to-

o p

best/1 to generate offspring

O_{1, j}

. If the subspace belongs type (d) or type (f), the

P_{2, j}

implements DE/current-to-

o p

best/1 to generate offspring

O_{2, j}

. Further, if the subspace belongs type (e),

P_{2, j}

implements GA and DE/current_to_rand_1 to generate offspring

O_{2, j}

, and

P_{1, j}

implement DE/current-to-

o p

best/1 to generate offspring

O_{1, j}

. In addition, If the subspace belongs type (g),

P_{1, j}

implements DE/current-to-

o p

best/1 to generate offspring

O_{1, j}

. After that, all constrained offspring are merged into a combined constrained offspring

O_{1}

and all unconstrained offspring are merged into a combined unconstrained offspring

O_{2}

. Finally, all offspring (including

O_{1}

and

O_{2}

) are merged into the parents and the environment selection criterion is adopted.
算法 3 给出了基于子空间类型的协同演化策略的流程。基于子空间类型的协同演化的主要任务是为

P_{2, j}

分配合适的策略，而

P_{1, j}

由于具有良好的搜索能力，其更新方式与子空间维护阶段相同。具体来说，如果子空间属于（a）型，则

P_{2, j}

和

P_{1, j}

执行 DE/current-to-

o p

best/1 生成子代

O_{2, j}

和

O_{1, j}

。如果子空间属于（b）型，则

P_{2, j}

执行 GA 生成子代

O_{2, j}

。然后，如果子空间属于类型（c），则

P_{2, j}

实现 DE/current-to-

o p

best/1、DE/current_to_rand_1 和 GA 来生成后代

O_{2, j}

，而

P_{1, j}

实现 DE/current-to-

o p

best/1 来生成后代

O_{1, j}

。如果子空间属于类型（d）或类型（f），则

P_{2, j}

实现 DE/current-to-

o p

best/1 来生成后代

O_{2, j}

。此外，如果子空间属于类型（e），

P_{2, j}

实现 GA 和 DE/current_too_rand_1 生成子代

O_{2, j}

，

P_{1, j}

实现 DE/current-to-

o p

best/1 生成子代

O_{1, j}

。此外，如果子空间属于类型（g），

P_{1, j}

实现 DE/current-to-

o p

best/1 生成子代

O_{1, j}

。之后，所有受约束子代合并成一个组合受约束子代

O_{1}

，所有非受约束子代合并成一个组合非受约束子代

O_{2}

。最后，所有子代（包括

O_{1}

和

O_{2}

）合并为父代，并采用环境选择标准。

According to the different evolutionary strategies that unconstrained subpopulation adopts in subspace, we divided them into four groups (

S_{1}

–

S_{4}

), as shown in Fig. 6.
根据无约束子群在子空间中采取的不同进化策略，我们将它们分为四组（

S_{1}

S_{4}

），如图 6 所示。

First group (

S_{1}

): This group contains type (a), type (d), type (f), and type (g), and the used strategy is DE/current-to-

o p

best/1 [38]. For type (a), the unconstrained subpopulation and the constrained subpopulation are far apart, and the unconstrained subpopulation are infeasible solution. Thus, the diversity of the solutions should be improved to help the unconstrained subpopulation cross the infeasible area. For type (d), the subspace only contains an unconstrained subpopulation. Moreover, the solutions of the unconstrained population are partially feasible, which indicates that the constrained population may converge to other feasible regions and lose the feasible solutions in the subspace. Therefore, the population needs some new search directions. For type (f), the subspace only contains an unconstrained subpopulation. Moreover, the solutions of the unconstrained population are completely feasible, which indicates that the constrained population may converge to other feasible regions and lose the feasible solutions in the subspace. In DE/current-to-

o p

best/1, the learning exemplar is selected from the surrounding constrained subpopulation, which can provide some new search directions, thereby enhancing the diversity of solutions. For type (g), the subspace only contains a constrained subpopulation. In DE/current-to-

o p

best/1, the learning exemplar is selected from itself, which can guide the population to the optimal. Thus, DE/current-to-

o p

best/1 is more suitable to deal with

S_{1}

.
第一组（

S_{1}

）：这一组包含类型（a）、类型（d）、类型（f）和类型（g），采用的策略是 DE/current-to-

o p

best/1 [38]。对于（a）型，无约束子群和有约束子群相距甚远，无约束子群是不可行解。因此，应提高解的多样性，帮助无约束子群跨越不可行区域。对于类型（d），子空间只包含一个无约束子群。此外，无约束群体的解是部分可行的，这表明受约束群体可能会收敛到其他可行区域，从而失去子空间中的可行解。因此，群体需要一些新的搜索方向。对于类型（f），子空间只包含一个无约束子群。此外，无约束群体的解是完全可行的，这表明受约束群体可能会收敛到其他可行区域，并失去子空间中的可行解。在 DE/current-to-

o p

best/1 中，学习范例是从周围的受约束子群中选择的，这可以提供一些新的搜索方向，从而增强解的多样性。对于类型（g），子空间只包含一个受限子群。在 DE/current-to-

o p

best/1 中，学习范例是从自身中选择的，可以引导群体达到最优。因此，DE/current-to

o p

best/1 更适合处理

S_{1}

。

Second group (

S_{2}

): This group contains type (b), and the used strategy is GA. For type (b), the subspace only contains unconstrained subpopulation and all of them are infeasible solutions, so it should consider assigning fewer computing resources to the unconstrained subpopulation to avoid wasting computing resources. GA has a strong convergence ability and consumes fewer computational resources during evolution. Thus, GA is more suitable for solving

S_{2}

.
第二组（

S_{2}

）：这一组包含类型（b），使用的策略是 GA。对于类型（b），子空间只包含无约束子群，且所有子群都是不可行解，因此应考虑为无约束子群分配较少的计算资源，以避免浪费计算资源。GA 具有较强的收敛能力，在进化过程中消耗的计算资源较少。因此，GA 更适合求解

S_{2}

。

Third group (

S_{3}

): This group contains type (c), and the used strategies are GA, DE/current_to_rand_1 and DE/current-to-

o p

best/1. For type (c), which can be considered as the combination of type (a) and type (e). Some solutions could be used to improve the performance directly, and the remaining partial solutions need some new search directions. Thus, for

S_{3}

, GA, DE/current_to_rand_1 and DE/current-to-

o p

best/1 are more suitable.
第三组（

S_{3}

）：这一组包含类型（c），使用的策略有 GA、DE/current_to_rand_1 和 DE/current-to

o p

best/1。对于类型（c），可以认为是类型（a）和类型（e）的组合。有些方案可以直接用来提高性能，剩下的部分方案则需要一些新的搜索方向。因此，对于

S_{3}

，GA、DE/current_to_rand_1 和 DE/current-to-

o p

best/1 更为合适。

Fourth group (

S_{4}

): This group contains type (e), and the used strategies are DE/current_to_rand_1 and GA. For type (e), the location and distribution of the potential UPS and CPS are close to each other. The unconstrained subpopulation can be directly used to improve the diversity and distribution of the corresponding constrained subpopulation. DE randomly selects individuals from unconstrained population, which can improve the diversity. GA has a stronger convergence ability. GA has a stronger convergence ability. Thus, DE/current_to_rand_1 and GA are adopted to guide the evolution of unconstrained subpopulation to find more feasible solutions.
第四组 (

S_{4}

)：该组包含类型（e），使用的策略是 DE/current_too_rand_1 和 GA。对于类型（e），潜在 UPS 和 CPS 的位置和分布相互接近。无约束子群可直接用于改善相应约束子群的多样性和分布。DE 从无约束子群中随机选择个体，可以提高多样性。GA 具有更强的收敛能力。GA 具有更强的收敛能力。因此，采用 DE/current_to_rand_1 和 GA 来引导无约束子群的演化，以找到更多的可行解。

3.5. Complexity of the SCCMMO framework
3.5.SCCMMO 框架的复杂性

The complexity of SCCMMO is composed of subspace generation, environmental selection and subspace-type perception. Specifically, the complexity of subspace generation is

O

(

N_{s} m N^{2}

), where

m

is the number of decision variables,

N_{s}

is the number of subpopulation, and

N

is the number of individuals. The worst complexity of environmental selection is

O

(

m N^{2}

). For the subspace-type perception, the worst complexity is

O

(

m N^{2}

). Therefore, for SCCMMO, the total computational complexity is

O

(

N_{s} m N^{2}

).
SCCMMO 的复杂度由子空间生成、环境选择和子空间类型感知三部分组成。具体来说，子空间生成的复杂度为

O

(

N_{s} m N^{2}

) ，其中

m

为决策变量数，

N_{s}

为子群体数，

N

为个体数。环境选择的最差复杂度为

O

(

m N^{2}

) 。对于子空间型感知，最差复杂度为

O

(

m N^{2}

)。因此，对于 SCCMMO，总计算复杂度为

O

(

N_{s} m N^{2}

) 。

4. Experiment setup 4.实验装置

Abundant experiments are conducted in this section to demonstrate the performance of the proposed SCCMMO. Firstly, the benchmark problems, the performance indicators, the compared algorithms and the parameter settings are introduced. Secondly, the experimental results are analyzed. Then, the ablation experiment and parameter sensitivity analysis of SCCMMO are studied. Finally, SCCMMO is validated for the location-selection problem. All experiments in this paper are conducted on PlatEMO [39].
本节将进行大量实验来证明所提出的 SCCMMO 的性能。首先，介绍基准问题、性能指标、比较算法和参数设置。其次，分析实验结果。然后，研究 SCCMMO 的烧蚀实验和参数敏感性分析。最后，针对位置选择问题对 SCCMMO 进行了验证。本文所有实验均在 PlatEMO [39] 上进行。

4.1. Experimental settings
4.1.实验设置

In this paper, CMMF benchmarks are used to evaluate the performance of SCCMMO in dealing with CMMOPs, which consist of CMMF1-CMMF17. As illustrated in Table 1, these benchmarks are divided into four categories based on the different characteristics. Category-I contains CMMF1, CMMF4, CMMF6, CMMF10, CMMF15 and CMMF17 with the same feasible regions and the same CPSs. Category-II contains CMMF7 and CMMF11 with the same feasible regions and the different CPSs. Category-III contains CMMF5 and CMMF12 with the different feasible regions and the same CPSs. Category-IV contains CMMF2, CMMF3, CMMF8, CMMF9, CMMF13, CMMF14 and CMMF16 with the different feasible regions and the different CPSs. Three performance metrics are employed to evaluate the quality of the results obtained by compared algorithms, including IGDX [40], IGD [41] and CPSP [6]. Specifically, IGDX is used to assess the quality of solutions in decision space, IGD measures the diversity and convergence of solutions in the objective space, and CPSP evaluates the feasibility, convergence and diversity of the solutions.
本文使用 CMMF 基准来评估 SCCMMO 处理 CMMOP 的性能，这些基准包括 CMMF1-CMMF17。如表 1 所示，这些基准根据不同特性分为四类。第一类包含 CMMF1、CMMF4、CMMF6、CMMF10、CMMF15 和 CMMF17，它们具有相同的可行区域和相同的 CPS。第二类包含 CMMF7 和 CMMF11，它们具有相同的可行区域和不同的 CPS。第三类包含 CMMF5 和 CMMF12，可行区域不同，CPS 相同。第四类包含 CMMF2、CMMF3、CMMF8、CMMF9、CMMF13、CMMF14 和 CMMF16，它们具有不同的可行区域和不同的 CPS。我们采用了三种性能指标来评估比较算法所获得结果的质量，包括 IGDX [40]、IGD [41] 和 CPSP [6]。具体来说，IGDX 用于评估决策空间中解决方案的质量，IGD 衡量目标空间中解决方案的多样性和收敛性，CPSP 评估解决方案的可行性、收敛性和多样性。

To assess the effectiveness of the proposed algorithm, ten algorithms are selected as comparison algorithms in this paper, including C-TAEA [42], PPS [19], NSGA-II

+

ARSBX [43], CMOEA-MS [44], DNNSGA-II [27], MO_Ring_PSO_SCD [28], MMEA-WI [45], TriMOEA-TA&R [46], CMMOCEA [7] and CMMODE [6]. Among them, C-TAEA, PPS, NSGA-II

+

ARSBX and DCNSGA-III are designed for solving CMOPs. CMMOCEA and CMMODE are designed for solving CMMOPs. DNNSGA-II, MO_Ring_PSO_SCD, MMEA-WI and TriMOEA-TA&R are designed for solving MMOPs. To ensure a fair comparison and enable DNNSGA-II, MO_Ring_PSO_SCD, MMEA-WI and TriMOEA-TA&R to handle CMMOPs, we combine CDP with MO_Ring_PSO_SCD, DNNSGA-II, MMEA-WI and TriMOEA-TA&R, namely C-MO_Ring_PSO_SCD, C-DNNSGA-II, C-MMEA-WI and C-TriMOEA-TA&R.
为了评估所提算法的有效性，本文选取了十种算法作为比较算法，包括 C-TAEA [42]、PPS [19]、NSGA-II

+

ARSBX [43]、CMOEA-MS [44]、DNNSGA-II [27]、MO_Ring_PSO_SCD [28]、MMEA-WI [45]、TriMOEA-TA&R [46]、CMMOCEA [7] 和 CMMODE [6]。其中，C-TAEA、PPS、NSGA-II

+

ARSBX 和 DCNSGA-III 是为求解 CMOP 而设计的。CMMOCEA 和 CMMODE 是为求解 CMMOP 而设计的。DNNSGA-II、MO_Ring_PSO_SCD、MMEA-WI 和 TriMOEA-TA&R 是为求解 MMOP 而设计的。为了确保公平比较，并使 DNNSGA-II、MO_Ring_PSO_SCD、MMEA-WI 和 TriMOEA-TA&R 能够处理 CMMOP，我们将 CDP 与 MO_Ring_PSO_SCD、DNNSGA-II、MMEA-WI 和 TriMOEA-TA&R 结合起来，即 C-MO_Ring_PSO_SCD、C-DNNSGA-II、C-MMEA-WI 和 C-TriMOEA-TA&R。

Table 1. Categories of the employed test functions.
表 1.采用的测试功能类别。

Category 类别	Test functions 测试功能	Characteristics 特点
Category-I 第一类	CMMF1 CMMF4 CMMF6	Same feasible regions 相同的可行区域
Category-I 第一类	CMMF10 CMMF15 CMMF17	Same CPSs 相同的 CPS
Category-II 第二类	CMMF7	Same feasible regions 相同的可行区域
Category-II 第二类	CMMF11	Different CPSs 不同的 CPS
Category-III 第三类	CMMF5	Different feasible regions 不同的可行区域
Category-III 第三类	CMMF12	Same CPSs 相同的 CPS
Category-IV 第四类	CMMF2 CMMF13 CMMF8 CMMF9	Different CPSs 不同的 CPS
Category-IV 第四类	CMMF13 CMMF14 CMMF16	Different feasible regions 不同的可行区域

The parameter settings of all the compared algorithms adhere to the recommendations in the original literature. For SCCMMO, the parameter settings are as follows. The population size is 100, and the maximum number of evaluations of the algorithm is 20000. The values of crossover probability

C R

and mutation operator

F

in DE/current_to_ rand_1 and DE/current-to-

o p

best/1 are the same as those of CMMODE . All algorithms are run 31 times independently on the benchmark to obtain the average (AVG) and standard deviation (STD). The Wilconxon rank sum test [47] with

α

= 0.05 is used to evaluate the difference between SCCMMO and other algorithms. The symbol ‘＋ /

-

/=’ represents that the proposed algorithm is better, worse, and similar to the compared algorithm.
所有比较算法的参数设置均遵循原始文献中的建议。SCCMMO 的参数设置如下。种群规模为 100，算法的最大评估次数为 20000。DE/current_too_ rand_1 和 DE/current-to-

o p

best/1 中的交叉概率

C R

和变异算子

F

的值与 CMMODE 相同。所有算法都在基准上独立运行 31 次，以获得平均值 (AVG) 和标准偏差 (STD)。使用

α

= 0.05 的 Wilconxon 秩和检验[47]来评估 SCCMMO 与其他算法之间的差异。符号"＋ /

-

/="代表所提算法与比较算法的优劣和相似程度。

4.2. Experimental results
4.2.实验结果

According to Table 2, SCCMMO obtains six best IGD values on CMMF1-CMMF17, and C-TAEA obtains the best IGD values in CMMF1, CMMF6,CMMF8. The reason is that C-TAEA employs a dual-archive to maintain the convergence of the solutions and it devotes more attention to finding CPF in objective space. Meanwhile, PPS obtains two optimal IGD values, since PPS considers all constraints as a whole in the second stage and can find narrow part of CPF. Therefore, PPS can achieve better results in the face of narrow feasible Pareto front problems such as CMMF10, CMMF13 and CMMF14.
表 2 显示，SCCMMO 在 CMMF1-CMMF17 上获得了六个最佳 IGD 值，而 C-TAEA 在 CMMF1、CMMF6、CMMF8 上获得了最佳 IGD 值。这是因为 C-TAEA 采用了双存档来保持解的收敛性，而且它更注重在目标空间中寻找 CPF。同时，由于 PPS 在第二阶段将所有约束条件作为一个整体来考虑，并能找到 CPF 的狭义部分，因此能得到两个最优 IGD 值。因此，面对狭窄可行帕累托前沿问题（如 CMMF10、CMMF13 和 CMMF14），PPS 可以取得更好的结果。

Table 2. The IGD statistical results (AVE $\pm$ STD) obtained by SCCMMO and competing algorithms.
表 2.SCCMMO 和其他算法得出的 IGD 统计结果（AVE $\pm$ STD）。

Table 3. The IGDX statistical results (AVE $\pm$ STD) obtained by SCCMMO and competing algorithms.
表 3.SCCMMO 和其他算法得出的 IGDX 统计结果（AVE $\pm$ STD）。

Table 4. The CPSP statistical results (AVE $\pm$ STD) obtained by SCCMMO and competing algorithms.
表 4.SCCMMO 和其他算法得出的 CPSP 统计结果（AVE $\pm$ STD）。

Further, Table 3 demonstrates that SCCMMO obtains nine best IGDX values on CMMF1-CMMF17, indicating the proposed method has good performance in solving multimodal problems. The IGDX values of C-TAEA,PPS,NSGA-II

+

ARSBX and CMOEA-MS on most test functions are poor since these algorithms are difficult to maintain the diversity in decision space during the evolutionary process. CMMODE and CMMOCEA are superior to other comparison algorithms, but the Wilcoxon rank sum test results are still worse than SCCMMO.
此外，表 3 显示，SCCMMO 在 CMMF1-CMMF17 上获得了 9 个最佳 IGDX 值，表明所提出的方法在解决多模态问题方面具有良好的性能。C-TAEA、PPS、NSGA-II

+

ARSBX 和 CMOEA-MS 在大多数测试函数上的 IGDX 值都很差，因为这些算法在进化过程中很难保持决策空间的多样性。CMMODE 和 CMMOCEA 优于其他比较算法，但 Wilcoxon 秩和检验结果仍不如 SCCMMO。

Table 4 shows the CPSP value of SCCMMO and compared algorithms on test functions. The results indicate that SCCMMO outperforms the other algorithms by achieving the best CPSP value on eleven out of seventeen test functions. In addition, CMMOCEA demonstrates superior performance on CMMF1, CMMF5, CMMF10, and CMMF11. The reason is that the CPSs on CMMF1, CMMF10 and CMMF11 are discrete and the feasible regions are narrow, while the mating pool of CMMOCEA is selected based on the diversity performance in decision space, which can help find these regions. As for CMMF5, CMMOCEA obtains the best CPSP value, but there was no significant difference with SCCMMO. However, from the Wilcoxon rank sum test results shown on Table 4, SCCMMO outperforms CMMOCEA and other compared algorithms on CMMF test functions.
表 4 显示了 SCCMMO 和比较算法在测试功能上的 CPSP 值。结果表明，在 17 个测试函数中，SCCMMO 在 11 个函数上取得了最佳 CPSP 值，优于其他算法。此外，CMMOCEA 在 CMMF1、CMMF5、CMMF10 和 CMMF11 上表现出更优越的性能。这是因为 CMMF1、CMMF10 和 CMMF11 上的 CPS 是离散的，可行区域较窄，而 CMMOCEA 的交配池是根据决策空间的多样性性能选择的，这有助于找到这些区域。对于 CMMF5，CMMOCEA 获得了最好的 CPSP 值，但与 SCCMMO 没有显著差异。不过，从表 4 所示的 Wilcoxon 秩和检验结果来看，SCCMMO 在 CMMF 检验函数上的表现优于 CMMOCEA 和其他比较算法。

Fig. 7 illustrates the CPFs and CPSs obtained by eleven algorithms on CMMF15, where the red points represent true CPF/CPS and the blue circles represent obtained CPF/CPS. It can be concluded that SCCMMO demonstrates superior performance in decision space and finds all complete CPSs. Additionally, the solutions generated by SCCMMO in the objective space exhibit a commendable distribution.
图 7 展示了 11 种算法在 CMMF15 上获得的 CPF 和 CPS，其中红点代表真实 CPF/CPS，蓝圈代表获得的 CPF/CPS。由此可以得出结论，SCCMMO 在决策空间中表现优异，能找到所有完整的 CPS。此外，SCCMMO 在目标空间生成的解呈现出值得称道的分布。

4.3. Ablation experiments
4.3.烧蚀实验

To evaluate the effectiveness of the proposed subspace generation and maintenance strategy and subspace-type-based coevolution strategy, this subsection compares the performance of the three method variants, including SCCMMO without subspace generation strategy (SCCMMO-1), SCCMMO without subspace-type-based coevolution strategy (SCCMMO-2), and SCCMMO without the two strategies simultaneously (SCCMMO-3).
为了评估所提出的子空间生成和维护策略以及基于子空间类型的协同演化策略的有效性，本小节比较了三种方法变体的性能，包括不使用子空间生成策略的 SCCMMO（SCCMMO-1）、不使用基于子空间类型的协同演化策略的 SCCMMO（SCCMMO-2）和不同时使用这两种策略的 SCCMMO（SCCMMO-3）。

Table 5, Table 6 respectively exhibit the IGD and IGDX results of SCCMMO and its variant. It can be observed that SCCMMO obtains fourteen best IGD values and thirteen best IGDX values out of seventeen, indicating that SCCMMO outperforms other variants. Specifically, the IGDX results of SCCMMO are better than SCCMMO-1 in ten test functions, indicating that the subspace generation strategy can effectively decompose the decision space and find multiple CPSs. Further, the IGDX of SCCMMO-1 on CMMF3 achieves the best value but there was no significant difference with SCCMMO. In addition, the IGDX results of SCCMMO on ten test functions and its IGD results on fourteen test functions are better than SCCMMO-2, which shows the efficacy of the subspace-type-based coevolution strategy. Through the judgment of the relationship types between two subpopulations, the evolution of the population is accurately guided, the diversity, convergence and feasibility of the solutions in different subspaces are balanced. Although SCCMMO-2 achieved the best IGDX on CMMF14, CMMF15, and CMMF16, there was no significant difference with SCCMMO. The IGDX and IGD values of SCCMMO-3 are the worst. The reason is that it does not consider the diversity of solutions in the decision space and the convergence of solutions in the objective space.
表 5 和表 6 分别列出了 SCCMMO 及其变体的 IGD 和 IGDX 结果。可以看出，在 17 个 IGD 值中，SCCMMO 获得了 14 个最佳 IGD 值和 13 个最佳 IGDX 值，表明 SCCMMO 优于其他变体。具体来说，在十个测试函数中，SCCMMO 的 IGDX 结果优于 SCCMMO-1，这表明子空间生成策略能有效分解决策空间并找到多个 CPS。此外，SCCMMO-1 在 CMMF3 上的 IGDX 达到了最佳值，但与 SCCMMO 没有显著差异。此外，SCCMMO 在十个测试函数上的 IGDX 结果和在十四个测试函数上的 IGD 结果均优于 SCCMMO-2，这表明了基于子空间类型的协同演化策略的有效性。通过对两个子群之间关系类型的判断，准确地引导了子群的进化，平衡了不同子空间解的多样性、收敛性和可行性。虽然 SCCMMO-2 在 CMMF14、CMMF15 和 CMMF16 上取得了最好的 IGDX，但与 SCCMMO 没有显著差异。SCCMMO-3 的 IGDX 和 IGD 值最差。原因是它没有考虑决策空间中解决方案的多样性和目标空间中解决方案的收敛性。

Table 5. Statistical results of IGD(AVG $\pm$ STD) obtained by different variants.
表 5.不同变量得出的 IGD（AVG $\pm$ STD）统计结果。

Table 6. Statistical results of IGDX(AVG $\pm$ STD) obtained by different variants.
表 6.不同变量得出的 IGDX（AVG $\pm$ STD）统计结果。

4.4. Parameter sensitivity analysis
4.4.参数敏感性分析

(1) Number of segments

k

(1) 段数

k

In SCCMMO, the parameter

k

represents the number of segments divided in each dimension, and its value significantly impacts the algorithm’s search capabilities. A larger

k

value represents more subspaces will be divided in decision space, which enhances the subpopulation’s ability to exploit relatively smaller subspaces. Therefore, the local search capability of the algorithm is improved and the global search ability is diminished. Conversely, a smaller

k

value enhances the global search ability but decreases the local search capability. Therefore, finding a suitable

k

value is crucial for enhancing the algorithm’s performance. In this experiment, the results of the SCCMMO with different parameters

k

on CMMF1 - CMMF17 are compared. In detail,

k

is set to six different values, including 1, 2, 3, 4, 5, 6.
在 SCCMMO 中，参数

k

代表在每个维度上划分的分段数，其值对算法的搜索能力有很大影响。

k

值越大，表示在决策空间中划分的子空间越多，从而增强了子群利用相对较小的子空间的能力。因此，算法的局部搜索能力会得到提高，而全局搜索能力则会减弱。相反，

k

值越小，全局搜索能力越强，但局部搜索能力越弱。因此，找到一个合适的

k

值对提高算法性能至关重要。在本实验中，比较了在 CMMF1 - CMMF17 上使用不同参数

k

的 SCCMMO 的结果。具体而言，

k

设置为六个不同的值，包括 1、2、3、4、5、6。

As shown in Fig. 8, the four subgraphs correspond to Category I - IV shown in Table 1, respectively. The blue line represents the value of IGD while the green line represents the value of IGDX. It can be observed that when

k

is equal to 4, IGD and IGDX get the best value at the same time, and the performance of SCCMMO is optimal.
如图 8 所示，四个子图分别对应表 1 中的类别 I - IV。蓝线代表 IGD 的值，绿线代表 IGDX 的值。可以看出，当

k

等于 4 时，IGD 和 IGDX 同时得到最佳值，SCCMMO 的性能达到最佳。

(2) Contral factor

s

(2) 对立因子

s

In SCCMMO, the control factor

s

determines when the algorithm stops the first stage and enters the second stage. When

s

is too large, the algorithm cannot make full use of the infeasible solution in the second stage, which may lead to the decrease of the effectiveness of information interaction between constrained and unconstrained population. When

s

is too small, the algorithm accumulates little knowledge during the evolution of the first stage, affecting the perception accuracy of the second stage. Thus, it is very important to find a suitable

s

.
在 SCCMMO 中，控制因子

s

决定算法何时停止第一阶段并进入第二阶段。当

s

过大时，算法在第二阶段无法充分利用不可行解，这可能会导致受约束群体与非受约束群体之间的信息交互效果下降。当

s

太小时，算法在第一阶段的演化过程中积累的知识很少，影响第二阶段的感知精度。因此，找到一个合适的

s

非常重要。

In this experiment,

s

is set to four different values, including 0.25, 0.5, 0.75 and 1.0, to verify their performance. And the corresponding variants are denoted as SCCMMO-0.25, SCCMMO-0.5, SCCMMO, and SCCMMO-1.0, respectively. The results of IGD and IGDX on CMMF are shown in Table 7, Table 8. Specifically, when

s

is equal to 0.75, the algorithm obtains fourteen best results out of seventeen for IGD and eight best results out of seventeen for IGDX. In addition, from the sumup results of the Wilcoxon rank sum test, when

s

is less than 0.75, the performance of the algorithm deteriorates with the decrease of the parameter value. From the sumup results shown on the bottom row of the two tables, when

s

is equal to 0.75, the algorithm can obtain the best performance.
本实验将

s

设为四个不同的值，包括 0.25、0.5、0.75 和 1.0，以验证它们的性能。相应的变体分别称为 SCCMMO-0.25、SCCMMO-0.5、SCCMMO 和 SCCMMO-1.0。IGD 和 IGDX 在 CMMF 上的结果如表 7 和表 8 所示。具体来说，当

s

等于 0.75 时，IGD 算法获得了 17 个最佳结果中的 14 个，IGDX 算法获得了 17 个最佳结果中的 8 个。此外，从 Wilcoxon 秩和检验的汇总结果来看，当

s

小于 0.75 时，算法的性能随着参数值的降低而降低。从两个表格最下面一行的求和结果来看，当

s

等于 0.75 时，算法可以获得最佳性能。

Table 7. Statistical results of IGD(AVG $\pm$ STD) obtained by SCCMMO with different $s$ .
表 7.SCCMMO 在不同 $s$ 条件下得到的 IGD（AVG $\pm$ STD）的统计结果。

Table 8. Statistical results of IGDX(AVG $\pm$ STD) obtained by SCCMMO with different $s$ .
表 8.SCCMMO 在不同 $s$ 条件下得到的 IGDX（AVG $\pm$ STD）的统计结果。

4.5. Application on location-selection problem
4.5.地点选择问题的应用

In this subsection, to verify the effectiveness of the algorithm in dealing with practical problems, this paper adopts the location-selection problem to compare with other ten algorithms which is a minimum distance problem. It is necessary to find the locations that meet the three objectives and several constraint conditions, and the detailed descriptions can be found in CMMODE.
在本小节中，为了验证算法在处理实际问题时的有效性，本文采用了位置选择问题来与其他十种算法进行比较，这是一个最小距离问题。需要找到满足三个目标和几个约束条件的地点，详细说明可参见 CMMODE。

Fig. 9 shows the obtained solutions by different algorithms on constrained multimodal multiobjective location-selection problem. The blue regions are the Pareto regions and the shaded regions indicate the infeasible regions. The blue, green and red points represent schools, hospitals and convenience stores, respectively. It can be observed that C-TAEA, PPS, NSGA-II

+

ARSBX and CMOEA-MS cannot find all the optimal regions. C-DNNSGA-II, C-MO-Ring-PSO-SCD, C-MMEA-WI, C-TriMOEA-TA&R and CMMODE can find all the optimal regions, but the distribution in the region is poor. Both SCCMMO and CMMOCEA can locate all the optimal regions and present good diversity and distribution. However, it can be observed the feasibility and convergence of SCCMMO are slightly better than CMMOCEA. In short, SCCMMO is superior to other competing algorithms in solving the real-word problems.
图 9 显示了不同算法在受限多模态多目标位置选择问题上得到的解决方案。蓝色区域为帕累托区域，阴影区域表示不可行区域。蓝色、绿色和红色点分别代表学校、医院和便利店。可以看出，C-TAEA、PPS、NSGA-II

+

ARSBX 和 CMOEA-MS 无法找到所有最优区域。C-DNNSGA-II、C-MO-Ring-PSO-SCD、C-MMEA-WI、C-TriMOEA-TA&R 和 CMMODE 可以找到所有最优区域，但区域内的分布较差。SCCMMO 和 CMMOCEA 都能找到所有最优区域，并呈现出良好的多样性和分布性。但可以看出，SCCMMO 的可行性和收敛性略好于 CMMOCEA。总之，在解决实词问题方面，SCCMMO 优于其他竞争算法。

5. Conclusion 5.结论

To balance the convergence, feasibility and diversity performance in both decision and objective spaces, this paper designs a subspace strategy based coevolutionary framework for CMMOPs, named SCCMMO.
为了平衡决策空间和目标空间的收敛性、可行性和多样性，本文为 CMMOPs 设计了一种基于子空间策略的协同演化框架，命名为 SCCMMO。

In SCCMMO, we first proposed a subspace generation and maintenance strategy which divides the decision space into several subspaces and maintains the diversity of solutions in subspace. After that, according to the relationship between CPS and UPS in the subspace, we divide the subspace into seven categories and match them with appropriate evolutionary strategies. In this way, SCCMMO can obtain more feasible solutions with good diversity and convergence in decision and objective space. Finally, SCCMMO is compared with ten competitive algorithms and the results show the superiority of the algorithm in solving CMMOPs.
在 SCCMMO 中，我们首先提出了子空间生成和维护策略，将决策空间划分为多个子空间，并保持子空间中解决方案的多样性。之后，根据子空间中 CPS 与 UPS 的关系，我们将子空间划分为七类，并为其匹配适当的进化策略。这样，SCCMMO 就能在决策空间和目标空间中获得更多具有良好多样性和收敛性的可行解。最后，SCCMMO 与十种竞争算法进行了比较，结果表明该算法在求解 CMMOPs 方面具有优越性。

Although the proposed method has made great progress in solving CMMOPs, there are still some shortcomings. Firstly, according to the experimental results of Section 4, we can observe that the performance of proposed method in the objective space is slightly worse than constrained multiobjective algorithms. The reason is that the proposed SCCMMO takes more effort to search for the multiple equivalent CPSs in the decision space, thus resulting in a slightly worse IGD value in objective space. Secondly, the benchmarks applied in this paper are two-dimensional, and performance of proposed method may decline in the face of high-dimensional CMMOPs. Thus, our future work will focus on balancing the objective space and decision space while enhancing the effectiveness of the proposed method on high-dimensional CMMOPs.
尽管所提出的方法在求解 CMMOP 方面取得了很大进展，但仍存在一些不足。首先，根据第 4 节的实验结果，我们可以观察到所提方法在目标空间的性能略差于约束多目标算法。其原因在于，建议的 SCCMMO 在决策空间中搜索多个等效 CPS 时需要花费更多精力，从而导致目标空间中的 IGD 值稍差。其次，本文应用的基准是二维的，面对高维的 CMMOP，建议方法的性能可能会下降。因此，我们今后的工作将侧重于平衡目标空间和决策空间，同时提高所提方法在高维 CMMOP 上的有效性。

CRediT authorship contribution statement
CRediT 作者贡献声明

Li Yan: Project administration. Shunge Guo: Writing – original draft, Software, Methodology. Jing Liang: Funding acquisition. Boyang Qu: Funding acquisition. Chao Li: Writing – review & editing. Kunjie Yu: Data curation.
Li Yan：项目管理。Shunge Guo：写作 - 原稿、软件、方法论。Jing Liang：资金获取。Boyang Qu：资金获取。李超：写作--审阅和编辑。余坤杰：数据整理数据整理

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 by the National Natural Science Foundation of China (Grant Nos. 62103456, 62373389), the Science and Technology Innovation Team of Colleges and Universities in Henan Province (Grant No. 22IRTSTHN015), the Science and Technology Innovation talents of Colleges and Universities in Henan Province (Grant No. 24HASTIT037), the Key Science and Technology Projects of Henan Province (Grant No. 242102211037), the Research Award Fund for Outstanding Young Teachers in Henan Provincial Institutions of Higher Education (Grant Nos. 2021GGJS111), the Key Research and development projects of Henan Province (Grant No. 241111210100), the Graduate Education Reform Project of Henan Province (Grant Nos. 2023SJGLX188Y), the Leading Talents of Science and Technology in the Central Plain of China (Grant Nos. 254000510055).
本研究得到国家自然科学基金项目（62103456，62373389）、河南省高等学校科技创新团队项目（22IRTSTHN015）、河南省高等学校科技创新人才项目（24HASTIT037）、河南省重点科技项目（242102211037）、河南省高等学校优秀青年教师科研奖励基金（2021GGJS111）、河南省高等学校重点研发计划项目（2021GGJS111）、河南省高等学校优秀青年教师科研奖励基金（2021GGJS111）、河南省高等学校重点研发计划项目（242102211037）、河南省高等学校优秀青年教师科研奖励基金（2021GGJS111）的资助。242102211037）、河南省高等学校优秀青年教师科研奖励基金（批准号：2021GGJS111）、河南省重点研发计划（批准号：241111210100）、河南省研究生教育改革工程（批准号：2023SJGLX188Y）、中原科技领军人才（批准号：254000510055）。

Data availability 数据可用性

The authors are unable or have chosen not to specify which data has been used.
作者无法或选择不说明使用了哪些数据。

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Cited by (0)

View Abstract

[1] [1]
K. Deb, H. Jain
An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraints
IEEE Trans. Evol. Comput., 18 (4) (2013), pp. 577-601
Google Scholar

[2] [2]
C. He, R. Cheng, Y. Tian, X. Zhang, K.C. Tan, Y. Jin
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IEEE Trans. Evol. Comput., 25 (3) (2020), pp. 448-462
Google Scholar

[3] [3]
M. Preuss, C. Kausch, C. Bouvy, F. Henrich
Decision space diversity can be essential for solving multiobjective real-world problems
Multiple Criteria Decision Making for Sustainable Energy and Transportation Systems: Proceedings of the 19th International Conference on Multiple Criteria Decision Making, Auckland, New Zealand, 7th-12th January 2008, Springer (2010), pp. 367-377
View at publisher Crossref View in Scopus Google Scholar

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IEEE/CAA J. Autom. Sin., 8 (2) (2021), pp. 303-318
View at publisher Crossref View in Scopus Google Scholar

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Google Scholar

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J. Liang, H. Lin, C. Yue, K. Yu, Y. Guo, K. Qiao
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IEEE Trans. Evol. Comput., 27 (4) (2022), pp. 1115-1129
View in Scopus Google Scholar

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F. Ming, W. Gong, Y. Yang, Z. Liao
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Outline 概要

Figures (12) 数字 (12)

Tables (8) 桌子 (8)

Swarm and Evolutionary Computation
蜂群和进化计算

Abstract 摘要

Keywords 关键词

1. Introduction 1.导言

2. Related work 2.相关工作

2.1. Constrained multiobjective optimization
2.1.约束多目标优化

2.2. Multimodal multiobjective optimization
2.2.多模式多目标优化

2.3. Constrained multimodal multiobjective optimization
2.3.有约束多模态多目标优化

3. Proposed method 3.建议的方法

3.1. Procedure of SCCMMO 3.1.南昌市委组织部的工作程序

3.2. Subspace generation and maintenance
3.2.子空间的生成和维护

3.3. Subspace-type perception
3.3.子空间类型感知

3.4. Subspace-type-based coevolution
3.4.基于子空间类型的协同进化

3.5. Complexity of the SCCMMO framework
3.5.SCCMMO 框架的复杂性

4. Experiment setup 4.实验装置

4.1. Experimental settings
4.1.实验设置

4.2. Experimental results
4.2.实验结果

4.3. Ablation experiments
4.3.烧蚀实验

4.4. Parameter sensitivity analysis
4.4.参数敏感性分析

4.5. Application on location-selection problem
4.5.地点选择问题的应用

5. Conclusion 5.结论

CRediT authorship contribution statement
CRediT 作者贡献声明

Declaration of competing interest
利益冲突声明

Acknowledgments 致谢

Data availability 数据可用性

References

Cited by (0)

Swarm and Evolutionary Computation蜂群和进化计算

A subspace strategy based coevolutionary framework for constrained multimodal multiobjective optimization problems基于子空间策略的受约束多模态多目标优化问题协同进化框架

Abstract 摘要

Keywords 关键词

1. Introduction 1.导言

2. Related work 2.相关工作

2.1. Constrained multiobjective optimization2.1.约束多目标优化

2.2. Multimodal multiobjective optimization2.2.多模式多目标优化

2.3. Constrained multimodal multiobjective optimization2.3.有约束多模态多目标优化

3. Proposed method 3.建议的方法

3.1. Procedure of SCCMMO 3.1.南昌市委组织部的工作程序

3.2. Subspace generation and maintenance3.2.子空间的生成和维护

3.3. Subspace-type perception3.3.子空间类型感知

3.4. Subspace-type-based coevolution3.4.基于子空间类型的协同进化

3.5. Complexity of the SCCMMO framework3.5.SCCMMO 框架的复杂性

4. Experiment setup 4.实验装置

4.1. Experimental settings4.1.实验设置

4.2. Experimental results4.2.实验结果

4.3. Ablation experiments4.3.烧蚀实验

4.4. Parameter sensitivity analysis4.4.参数敏感性分析

4.5. Application on location-selection problem4.5.地点选择问题的应用

5. Conclusion 5.结论

CRediT authorship contribution statementCRediT 作者贡献声明

Declaration of competing interest利益冲突声明

Acknowledgments 致谢

Data availability 数据可用性

References

Swarm and Evolutionary Computation
蜂群和进化计算

A subspace strategy based coevolutionary framework for constrained multimodal multiobjective optimization problems
基于子空间策略的受约束多模态多目标优化问题协同进化框架

2.1. Constrained multiobjective optimization
2.1.约束多目标优化

2.2. Multimodal multiobjective optimization
2.2.多模式多目标优化

2.3. Constrained multimodal multiobjective optimization
2.3.有约束多模态多目标优化

3.2. Subspace generation and maintenance
3.2.子空间的生成和维护

3.3. Subspace-type perception
3.3.子空间类型感知

3.4. Subspace-type-based coevolution
3.4.基于子空间类型的协同进化

3.5. Complexity of the SCCMMO framework
3.5.SCCMMO 框架的复杂性

4.1. Experimental settings
4.1.实验设置

4.2. Experimental results
4.2.实验结果

4.3. Ablation experiments
4.3.烧蚀实验

4.4. Parameter sensitivity analysis
4.4.参数敏感性分析

4.5. Application on location-selection problem
4.5.地点选择问题的应用

CRediT authorship contribution statement
CRediT 作者贡献声明

Declaration of competing interest
利益冲突声明