Manifold-assisted coevolutionary algorithm for constrained multi-objective optimization

doi:10.1016/j.swevo.2024.101717

Swarm and Evolutionary Computation
蜂群和进化计算

Volume 91, December 2024, 101717
第 91 卷，2024 年 12 月，第 101717 期

, , , ,

^a: School of Computer Science and technology, Zhengzhou University of Light Industry, Zhengzhou 450000, China
郑州轻工业学院计算机科学与技术学院，中国郑州 450000

^b: Faculty of Electrical Engineering and Computer Science, Ningbo University, Ningbo 315211, China
宁波大学电气工程与计算机科学学院，中国宁波 315211

^c: School of Electrical and Computer Engineering, Oklahoma State University, Stillwater, OK 74078 United States
俄克拉荷马州立大学电气与计算机工程学院，美国俄克拉荷马州斯蒂尔沃特 74078

Received 17 June 2024, Revised 29 July 2024, Accepted 23 August 2024, Available online 30 August 2024, Version of Record 30 August 2024.
2024 年 6 月 17 日收到，2024 年 7 月 29 日修订，2024 年 8 月 23 日接受，2024 年 8 月 30 日在线提供，2024 年 8 月 30 日记录版本。

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Abstract https://linux.do/t/topic/111737

In constrained multi-objective optimization problems (CMOPs), constraints often fragment the Pareto solution space into multiple feasible and infeasible regions. This fragmentation presents a challenge for evolutionary optimization methods as feasible regions can be discrete and isolated by infeasible areas, making exploration difficult and leading to populations getting trapped in local optima. To address these issues, this paper introduces a manifold assisted coevolutionary algorithm for solving CMOPs. Firstly, a guided feasible search strategy is proposed to explore feasible regions, especially those isolated by infeasible barriers. This is achieved by estimating directions to the Constrained Pareto Set (CPS). Secondly, a manifold learning-based exploration strategy is employed to spread the population along the Pareto Set (PS) manifold by estimating the manifold distribution. Moreover, two populations are exploited, where the first population serves as the primary population, considering both constraints and objectives to explore the feasible region and search along the CPS. The second population, on the other hand, does not consider constraints and serves as an auxiliary population to explore the Unconstrained PS. By cooperating, these two populations effectively approach and cover separated CPS segments. The proposed algorithm is evaluated against seven state-of-the-art algorithms on 37 CMOP test functions and 5 CMOPs with fraudulent constraints. The experimental results clearly demonstrate that our algorithm can reliably locate multiple CPSs and is considered state-of-the-art in handling CMOPs.
在受约束的多目标优化问题（CMOPs）中，约束条件通常会将帕累托求解空间分割成多个可行和不可行区域。这种分割给进化优化方法带来了挑战，因为可行区域可能是离散的，并被不可行区域所隔离，从而使探索变得困难，并导致种群陷入局部最优状态。为解决这些问题，本文介绍了一种用于求解 CMOP 的流形辅助协同进化算法。首先，本文提出了一种引导可行搜索策略，以探索可行区域，尤其是那些被不可行障碍隔离的区域。这是通过估计约束帕雷托集（CPS）的方向来实现的。其次，采用基于流形学习的探索策略，通过估计流形分布，将种群沿着帕累托集合（PPS）流形扩散。此外，还利用了两个种群，其中第一个种群作为主种群，同时考虑约束条件和目标，探索可行区域并沿着 CPS 进行搜索。另一方面，第二个群体不考虑约束条件，而是作为辅助群体探索无约束 PS。通过合作，这两个群体可以有效地接近和覆盖分离的 CPS 段。我们在 37 个 CMOP 测试函数和 5 个带有欺诈性约束的 CMOP 上，对所提出的算法与七种最先进的算法进行了评估。实验结果清楚地表明，我们的算法能够可靠地定位多个 CPS，在处理 CMOP 方面被认为是最先进的。

Keywords https://linux.do/t/topic/111737

Manifold

Adaptive sampling

Constrained multi-objective optimization

Multiple population

Coevolution

多目标优化多群体进化

1. Introduction https://linux.do/t/topic/111737

Numerous real-world optimization scenarios involve optimizing multiple conflicting objectives while satisfying a number of constraints. These problems are referred to as constrained multi-objective optimization problems (CMOPs) [1]. For example, in engineering design, CMOPs are used to optimize the structural integrity and performance of components while adhering to safety and material constraints [2,12]. In resource allocation, CMOPs help balance competing objectives such as cost reduction and environmental impact while meeting regulatory constraints [3]. Similarly, in operational planning, CMOPs are employed to optimize scheduling and logistics while considering constraints such as capacity limits and delivery deadlines [4]. Generally, CMOPs can be expressed as follows:(1) $\begin{matrix} Minimize : F (x) = (f_{1} (x), f_{2} (x), \dots, f_{m} (x)) \\ Subject to : g_{i} (x) \leq 0, i = 1, \dots, p \\ h_{j} (x) = 0, j = 1, \dots, q \end{matrix}$ where $F (x)$ stands for the objective vector to be optimized which consists of m objectives; $x \in Ω$ is the decision vector; $g_{i} (x)$ and $h_{j} (x)$ represents the inequality and equality constraints, respectively. In the absence of constraints, the main goal in multi-objective optimization is to search for a set of well-converged and well-spread trade-off solutions, recognized as the Pareto Set (PS) within the decision space, and the corresponding Pareto Front (PF) within the objective space. The PS consists of decision vectors for which no improvement in any objective can be made without deteriorating at least one other objective. The PF, on the other hand, is the set of objective vectors that correspond to the PS and represents the trade-offs between different objectives. The introduction of constraints significantly complicates the derivation of the PF in CMOPs. Constraints can fragment the feasible space into disconnected regions, render a large portion of the search space infeasible, and potentially make the PF of unconstrained multi-objective problems (MOPs) partially or entirely infeasible. To distinguish between the PF/PS in constrained and unconstrained scenarios, the terms Constrained PS/PF (CPS/CPF) and Unconstrained PS/PF (UPS/UPF) are commonly used [[5], [6], [7], [8]]. The CPS/CPF refers to the set of trade-off solutions and their corresponding objective vectors that satisfy all constraints, whereas the UPS/UPF refers to those solutions and vectors without considering constraints.
现实世界中的许多优化方案都涉及在满足一系列约束条件的同时优化多个相互冲突的目标。这些问题被称为约束多目标优化问题（CMOPs）[1]。例如，在工程设计中，CMOPs 用于优化部件的结构完整性和性能，同时遵守安全和材料约束 [2,12]。在资源分配中，CMOPs 可帮助平衡各种相互竞争的目标，如降低成本和环境影响，同时满足法规约束[3]。同样，在运营规划中，CMOPs 可用于优化调度和物流，同时考虑容量限制和交货期限等约束[4]。一般来说，CMOPs 可表示如下：其中， $F (x)$ 代表待优化的目标向量，由 m 个目标组成； $x \in Ω$ 为决策向量； $g_{i} (x)$ 和 $h_{j} (x)$ 分别代表不等式约束和等式约束。在没有约束条件的情况下，多目标优化的主要目标是寻找一组融合度高、分布均匀的权衡解决方案，即决策空间中的帕累托集合（PS）和目标空间中相应的帕累托前沿（PF）。帕累托集合由一些决策向量组成，对于这些决策向量，如果不损害至少一个其他目标，就无法改善任何目标。另一方面，PF 是与 PS 相对应的目标向量集，代表了不同目标之间的权衡。在 CMOP 中，约束条件的引入大大增加了 PF 的推导难度。约束会将可行空间分割成互不相连的区域，使大部分搜索空间变得不可行，并可能使无约束多目标问题（MOP）的 PF 部分或完全不可行。为了区分有约束和无约束情况下的 PF/PS，通常使用有约束 PS/PF（CPS/CPF）和无约束 PS/PF（UPS/UPF）这两个术语[[5], [6], [7], [8]]。CPS/CPF 指的是满足所有约束条件的权衡解及其相应的目标向量集，而 UPS/UPF 指的是不考虑约束条件的解和向量。

To address CMOPs, various constraint handling techniques have been proposed. Noteworthy approaches include the constrained dominance principle (CDP) [9], penalty function methods [10], $ε$ constrained [3,11] and stochastic ranking [13], among others. Additionally, infeasible solutions are exploited to navigate and explore the CPF for enhanced solution distribution. Consequently, several coevolutionary frameworks involving multiple populations, such as CCMO [14] and C-TAEA [15], have been introduced to intensify the interplay between feasible and infeasible solutions, aiding in boundary search. Furthermore, two-stage algorithms, including PPS [16] and CMOEA-MS [17], guide the population toward the UPF in the first stage and disperse it along the CPF in the second stage.
为解决 CMOP 问题，人们提出了各种约束处理技术。值得注意的方法包括约束支配原理（CDP）[9]、惩罚函数法[10]、 $ε$ 约束[3,11]和随机排序[13]等。此外，还利用不可行解来导航和探索 CPF，以增强解的分布。因此，一些涉及多个种群的协同进化框架（如 CCMO [14] 和 C-TAEA [15]）已被引入，以加强可行解与不可行解之间的相互作用，帮助进行边界搜索。此外，包括 PPS [16] 和 CMOEA-MS [17] 在内的两阶段算法在第一阶段将种群引向 UPF，在第二阶段沿 CPF 分散。

Despite the development of various constraint handling techniques, limitations persist in the existing MOEAs. One primary challenge in CMOPs is the expansion of the infeasible region due to constraints, making it challenging to identify a small feasible region [18]. A common design is to gradually approach the feasible region by reducing constraint violations and relaxing constraint conditions [19,20]. Nevertheless, the presence of the infeasible region can act as a barrier on the path to the CPF. This barrier can cause algorithms to become trapped in the outer feasible region, restricting their search to local CPFs and impeding the discovery of the global CPF. As illustrated in Fig. 1, LIR-CMOP7 [57] exhibits multiple local CPS/CPF with significant infeasible gaps between local and global CPS/CPF. The population as shown in black dots faces a challenge during the search process, as it often becomes trapped in local CPS/CPF, compromising its ability to traverse the infeasible domain and explore the global CPS/CPF.
尽管开发了各种约束处理技术，但现有的 MOEAs 仍然存在局限性。CMOP 面临的一个主要挑战是，由于约束条件的存在，不可行区域不断扩大，这使得确定一个小的可行区域变得十分困难 [18]。一种常见的设计方法是通过减少违反约束和放宽约束条件来逐步接近可行区域 [19,20]。然而，不可行区域的存在会成为通向 CPF 的障碍。这种障碍会导致算法被困在外部可行区域，从而将搜索范围限制在局部 CPF 上，阻碍全局 CPF 的发现。如图 1 所示，LIR-CMOP7 [57] 显示了多个局部 CPS/CPF，局部 CPS/CPF 与全局 CPS/CPF 之间存在明显的不可行间隙。黑点所示的群体在搜索过程中面临着挑战，因为他们经常被困在局部 CPS/CPF 中，影响了他们穿越不可行域和探索全局 CPS/CPF 的能力。

While many constrained multi-objective evolutionary algorithms (CMOEAs) [19] attempt to increase exploration ability and avoid trapping in local optima by enhancing diversity, this approach is often inefficient. To explore the feasible region, infeasible solutions could be used to provide guidance. For instance, infeasible solutions are guided into the feasible region to approach the PS in a shift-based penalty method [21]. To avoid trapping into local optima, effective mechanisms and efficient offspring reproduction operators have been proposed. Zhu et al. [22] introduced a detect-and-escape strategy to detect when the search becomes trapped and help the population find global feasible regions. Qu and Suganthan [23] proposed a diversity promotion strategy to prevent the population from falling into local optima. While increasing diversity can enhance the probability of discovering feasible regions, it is evident that not all feasible regions are worth exploring. Exploring in the directions that approaches the CPS is more likely to lead to the discovery of the feasible region where the global/local CPS resides. In this paper, we introduce the guided feasible search strategy to explore feasible regions, especially those near the PS and isolated by infeasible areas.
虽然许多受限多目标进化算法（CMOEAs）[19] 试图通过增强多样性来提高探索能力，避免陷入局部最优，但这种方法往往效率低下。为了探索可行区域，可以利用不可行解来提供指导。例如，在基于移动的惩罚方法中，不可行解被引导到可行区域以接近 PS [21]。为了避免陷入局部最优，人们提出了有效的机制和高效的后代繁殖算子。Zhu 等人[22]提出了一种检测-逃逸策略，用于检测搜索是否陷入困境，并帮助群体找到全局可行区域。Qu 和 Suganthan [23] 提出了一种多样性促进策略，以防止种群陷入局部最优。虽然增加多样性可以提高发现可行区域的概率，但显然并非所有可行区域都值得探索。向接近 CPS 的方向探索更有可能发现全局/局部 CPS 所在的可行区域。在本文中，我们引入了引导式可行搜索策略来探索可行区域，尤其是那些靠近 CPS 且被不可行区域隔离的区域。

Another challenge arises from disconnected feasible regions, leading to the CPF being composed of multiple disjoint segments. Covering all these segments effectively becomes a daunting task. To identify CPF fragments, an indicator-based constrained multi-objective optimization algorithm (ICMA) [25] was proposed, in which an indicator considering both cost value-based distance and constraints is defined. It partitions promising areas into multiple subregions and eliminates crowded individuals based on the proposed indicator. It is worth noticing that both cost value-based distance and region division are processed in the objective space. It is known that the PS of continuous MOPs form a segmented continuous manifold in the decision space under mild conditions [24]. Even though the presence of constraints disrupts the manifold structure of the PS, exploration along the manifold can help discover disjoint CPS segments. Most existing CMOEAs focus on approximating and covering the PF, with limited exploration of the PS manifold and its potential benefits for solving constrained problems. Therefore, a manifold-assisted coevolutionary algorithm, MACA for short, is proposed in the paper. The contribution of the work is as follows:
另一个挑战来自互不相连的可行区域，这导致 CPF 由多个互不相连的部分组成。有效覆盖所有这些片段成为一项艰巨的任务。为了识别 CPF 片段，有人提出了一种基于指标的约束多目标优化算法 (ICMA)[25]，其中定义了一种既考虑基于成本值的距离又考虑约束条件的指标。它将有希望的区域划分为多个子区域，并根据提出的指标消除拥挤的个体。值得注意的是，基于成本值的距离和区域划分都是在目标空间中处理的。众所周知，连续澳门威尼斯人网址的 PS 在温和条件下会在决策空间中形成一个分割的连续流形 [24]。尽管约束条件的存在会破坏 PS 的流形结构，但沿着流形进行探索有助于发现不相连的 CPS 段。现有的 CMOEA 大多侧重于近似和覆盖 PF，而对 PS 流形及其在解决约束问题方面的潜在优势探索有限。因此，本文提出了一种流形辅助协同进化算法，简称 MACA。这项工作的贡献如下：

1)
We introduce a guided feasible search strategy, which estimates the directions to approach the CPS. Through searching along these estimated directions, this approach could effectively explore the feasible regions, especially those isolated by infeasible barriers. Moreover, it is benefit for escaping the local optima.
我们引入了一种引导式可行搜索策略，它能估计出接近 CPS 的方向。通过沿着这些估计方向进行搜索，这种方法可以有效地探索可行区域，尤其是那些被不可行障碍隔离的区域。此外，它还有利于摆脱局部最优状态。
2)
We propose a manifold learning-based exploration strategy. By constructing a model to estimate the manifold distribution of both CPS and UPS, this strategy effectively spreads the population along the CPS manifold and assists in locating separated CPS manifold segments. It enhances the algorithm's ability to explore the complex structure of CPS.
我们提出了一种基于流形学习的探索策略。通过构建一个模型来估计 CPS 和 UPS 的流形分布，该策略能有效地将种群沿着 CPS 流形扩散，并帮助定位分离的 CPS 流形段。它增强了算法探索 CPS 复杂结构的能力。
3)
We employ a dual-population coevolutionary algorithm in which the primary population explores the feasible region and approximates CPS by considering both constraints and objective values, while the auxiliary population explores UPF focusing only on objectives and provides assistance to the primary population. Through extensive experimentation on 37 test CMOPs, the proposed algorithm achieves superior performance compared to seven state-of-the-art designs.
我们采用了一种双种群协同进化算法，其中主种群探索可行区域，并通过同时考虑约束条件和目标值来逼近 CPS，而辅助种群则只针对目标探索 UPF，并为主种群提供帮助。通过对 37 个测试 CMOP 的广泛实验，与七种最先进的设计相比，所提出的算法取得了卓越的性能。

The rest of this article is organized as follows. Section 2 introduces related work and motivation. Section 3 describes the proposed MACA algorithm and its details. Section 4 presents and analyzes the experimental results. Section 5 concludes the article.
本文接下来的内容安排如下。第 2 节介绍相关工作和动机。第 3 节介绍拟议的 MACA 算法及其细节。第 4 节介绍并分析了实验结果。第 5 节是本文的结论。

2. Related work and motivation
2.相关工作和动机

2.1. Existing constraint handling techniques
2.1.现有的约束处理技术

In addressing CMOPs, researchers have developed various evolutionary algorithms, broadly categorized into three main types. The first type is feasibility-first approaches, which prioritize feasible solutions throughout the evolutionary process. A notable method is the CDP [9], which emphasizes the preservation of feasibility while assessing solutions based on Pareto dominance. CDP can be seamlessly integrated with a range of unconstrained optimization algorithms to address CMOPs. However, the search process is more likely to converge to local CPFs when the initially identified feasible region fails to encompass the entire CPS. Additionally, there are methods that relax constraints to some extent, considering some infeasible solutions as feasible [26,27] or penalize infeasible solutions [28,29]. These algorithms often rely on introduced parameters to scale the influence of constraints, making them sensitive to parameter choices. The second type is methods that balance feasibility satisfaction and objective optimization, which aim to find a trade-off between feasibility and optimality [30]. To balance the constraints and the objective functions, some CMOEAs incorporate the ranking of individuals both in constraint and objective space, such as promoting the ranking of some infeasible solutions [31] and utilizing a weighted ranking method [32]. Additionally, certain CMOEAs maintain two separate archives, one geared towards convergence while the other towards diversity. This two-archive based approach aims to steer the optimization process towards feasible solutions while preserving diversity. Moreover, some CMOEAs divide the entire evolutionary process into multiple stages, each employing different constraint handling strategies, such as ToP [33], CMOEA-MS [17], CIC-MOEA/D [34] and CMOES [35]. Additionally, multi-population-based coevolutionary frameworks are employed to tackle CMOPs. In these frameworks such as CCMO, CMOEA-PP [36], one population is dedicated to solving the primary problem, while the other population addresses a derived helper problem [14]. In response to specific challenges encountered in CMOPs, certain algorithms have been designed. Notable among these is the Push and Pull Algorithm (PPS) [16], which effectively explores the boundaries of the feasible region. Additionally, the detect and escape strategy [22] has been designed to overcome the issue of stagnation at local optima. Other algorithms can be classified into the third type. Some algorithms transform constrained optimization problems into multi-objective optimization problems, treating constraints as one or multiple external objectives. Additionally, dynamic allocation [[37], [38], [39], [40]], evolutionary multi-tasking framework (EMCMO) [41] has also been introduced to address CMOPs.
为解决 CMOP 问题，研究人员开发了各种进化算法，大致可分为三大类。第一类是可行性优先方法，即在整个进化过程中优先考虑可行的解决方案。一种值得注意的方法是 CDP [9]，它强调在根据帕累托优势评估解决方案的同时保留可行性。CDP 可与一系列无约束优化算法无缝集成，以解决 CMOP 问题。然而，当最初确定的可行区域未能涵盖整个 CPS 时，搜索过程更有可能收敛到局部 CPF。此外，还有一些方法可以在一定程度上放松约束，将一些不可行解视为可行解 [26,27] 或对不可行解进行惩罚 [28,29]。这些算法通常依赖引入的参数来调节约束条件的影响，因此对参数选择很敏感。第二类是兼顾可行性满足和目标优化的方法，其目的是在可行性和最优性之间寻求权衡[30]。为了平衡约束条件和目标函数，一些 CMOEA 将个体在约束条件和目标空间中的排序纳入其中，例如促进一些不可行解的排序[31]，以及利用加权排序法[32]。此外，某些 CMOEA 还保留了两个独立的档案库，一个用于收敛，另一个用于多样性。这种基于双档案的方法旨在将优化过程引向可行的解决方案，同时保持多样性。此外，一些 CMOEA 将整个进化过程分为多个阶段，每个阶段采用不同的约束处理策略，如 ToP [33]、CMOEA-MS [17]、CIC-MOEA/D [34] 和 CMOES [35]。此外，还采用了基于多群体的协同进化框架来处理 CMOP。在这些框架（如 CCMO、CMOEA-PP [36]）中，一个群体致力于解决主要问题，而另一个群体则解决衍生的辅助问题 [14]。为了应对 CMOPs 中遇到的特定挑战，人们设计了一些算法。其中值得注意的是推拉算法（PPS）[16]，它能有效探索可行区域的边界。此外，还设计了检测和逃避策略 [22] 来克服局部最优停滞问题。其他算法可分为第三类。有些算法将约束优化问题转化为多目标优化问题，将约束视为一个或多个外部目标。此外，动态分配 [[37]、[38]、[39]、[40]]、进化多任务框架（EMCMO）[41] 也被引入解决 CMOP 问题。

2.2. Existing manifold learning techniques
2.2.现有的流形学习技术

Leveraging problem domain knowledge to generate promising and high-quality offspring has seen significant progress in the field of unconstrained continuous optimization. One of the most notable advancements involves utilizing the regularity property, specifically the manifold characteristics of the PS and PF. Early pioneering work in this area includes the RM-MEDA proposed by Zhou [24], which employs local Principal Component Analysis (Local PCA) to approximate the manifold structure of the PS. Subsequently, many research efforts have built upon RM-MEDA, continuously establishing new variants, such as RMEA [42]. Additionally, various manifold learning models have been proposed, including techniques like locally linear embedding (LLE) [43], generative topographic mapping (GTM) [44], generative adversarial networks (GAN) [45], and more. More recently, Zhou et al. [24] have synthesized the concepts from RM-MEDA and its variations to introduce the regularity evolution framework. This framework combines the structure vector and perturbation vector to enhance the exploration of the manifold structure in MOPs.
在无约束连续优化领域，利用问题领域知识生成有前途的高质量子代取得了重大进展。其中最显著的进展之一就是利用了正则特性，特别是 PS 和 PF 的流形特征。该领域的早期开创性工作包括 Zhou [24] 提出的 RM-MEDA，它采用局部主成分分析（Local PCA）来近似 PS 的流形结构。随后，许多研究工作以 RM-MEDA 为基础，不断建立新的变体，如 RMEA [42]。此外，还提出了各种流形学习模型，包括局部线性嵌入（LLE）[43]、生成地形映射（GTM）[44]、生成对抗网络（GAN）[45]等技术。最近，Zhou 等人[24] 综合了 RM-MEDA 及其变体的概念，提出了正则性演化框架。该框架结合了结构向量和扰动向量，以加强对澳门威尼斯人网址流形结构的探索。

2.3. Motivation 2.3.动机

When dealing with constrained multi-objective optimization problems, it is challenging for algorithms to strike a balance between maintaining diversity, convergence, and constraint satisfaction. The designed algorithm should aim to maintain diversity to explore the entire feasible region while rapidly converging to local optima within that region. Fig. 2, Fig. 3 illustrate the experimental results of eight CMOEAs on LIR-CMOP8 and LIR-CMOP10 test problems, respectively. These figures reveal a clear separation between the feasible region containing the global CPF and the region containing local CPF in both LIR-CMOP8 and LIR-CMOP10. In these scenarios, several algorithms, such as A-NSGA-III [46], C-MOEA/D [47], ToP [33], C-TAEA [15], POCEA [48], MOEA/D-DAE [22], CCMO and PPS often become trapped or partially stuck in local optima, hindering their ability to explore the global CPF. The issue arises from multiple factors. On one hand, the algorithms may lack sufficient exploration power. On the other hand, the excessive gravitational pull from strong local attractors makes it challenging for the algorithms to escape their influence domains. Providing a force that guides algorithms towards the global optimum could enable them to continue exploration, ultimately discovering the feasible region where the global optimum resides.
在处理有约束的多目标优化问题时，算法如何在保持多样性、收敛性和约束满足之间取得平衡是一个挑战。所设计的算法既要保持多样性，探索整个可行区域，又要在该区域内快速收敛到局部最优。图 2 和图 3 分别展示了八个 CMOEA 在 LIR-CMOP8 和 LIR-CMOP10 测试问题上的实验结果。这些图显示了在 LIR-CMOP8 和 LIR-CMOP10 中，包含全局 CPF 的可行区域和包含局部 CPF 的区域之间的明显分离。在这些情况下，A-NSGA-III [46]、C-MOEA/D [47]、ToP [33]、C-TAEA [15]、POCEA [48]、MOEA/D-DAE [22]、CCMO 和 PPS 等几种算法经常陷入或部分陷入局部最优状态，阻碍了它们探索全局 CPF 的能力。这一问题由多种因素造成。一方面，算法可能缺乏足够的探索能力。另一方面，强局部吸引子的引力过大，使得算法难以摆脱其影响域。提供一种引导算法走向全局最优的力量，可以让算法继续探索，最终发现全局最优所在的可行区域。

Moreover, even when the population reaches the global/local feasible regions, they tend to concentrate on specific regions, resulting in insufficient coverage of the CPF. As depicted in Fig. 2, algorithms such as C-MOEA/D, POCEA, MOEA/D-DAE, CCMO and PPS are capable of covering the whole CPF. This phenomenon occurs because the CPF often resides on the boundaries between feasible and infeasible regions. The influence of attractors results in individuals experiencing different attraction forces and selection pressures along these boundaries. Consequently, individuals tend to gather in regions where objective values decrease rapidly, leading to insufficient exploration of the entire boundary and preventing complete coverage of the CPF. For this kind of continuous manifold, increasing the search along the manifold can enhance the coverage of the entire CPF manifold surface. Furthermore, due to the impact of constraints, the feasible region is often partitioned into disconnected, scattered regions as shown in the case of LIR-CMOP10 in Fig. 3. Consequently, the originally continuous PS manifold is separated into disconnected segments. Algorithms encounter challenges when exploring these disconnected regions, leading to incomplete coverage of the PS manifold surface. As illustrated in Fig. 3, algorithms such as ToP, C-TAEA, CCMO, and PPS successfully reach the global feasible region. However, they fail to comprehensively cover the entire CPF. For these disjoint CPF manifold segments, estimating the manifold structure and searching along the manifold can effectively bridge the fragmented areas, exploring separated feasible CPF manifold segments.
此外，即使群体达到了全局/局部可行区域，他们也倾向于集中在特定区域，导致 CPF 的覆盖范围不足。如图 2 所示，C-MOEA/D、POCEA、MOEA/D-DAE、CCMO 和 PPS 等算法能够覆盖整个 CPF。出现这种现象的原因是，CPF 通常位于可行区域和不可行区域之间的边界上。吸引子的影响导致个体在这些边界上受到不同的吸引力和选择压力。因此，个体往往会聚集在目标值迅速降低的区域，导致对整个边界的探索不足，无法完全覆盖 CPF。对于这种连续流形，增加沿流形的搜索可以提高整个 CPF 流形面的覆盖率。此外，由于约束条件的影响，可行区域往往被分割成互不相连的分散区域，如图 3 中 LIR-CMOP10 的情况所示。因此，原本连续的 PS 流形被分割成断开的部分。算法在探索这些断开区域时会遇到困难，导致 PS 流形表面覆盖不全。如图 3 所示，ToP、C-TAEA、CCMO 和 PPS 等算法成功地到达了全局可行区域。但是，它们无法全面覆盖整个 CPF。对于这些不连贯的 CPF 流形段，估计流形结构并沿流形搜索可以有效地弥合破碎区域，探索分离的可行 CPF 流形段。

3. Proposed algorithm 3.拟议算法

3.1. The procedure of the proposed algorithm
3.1.拟议算法的程序

As presented in Algorithm 1, the proposed coevolutionary algorithm assisted by manifold structure (MACA) starts with two randomly initialized populations, P₁ and P₂, each containing NP individuals. In each generation, guided feasible search is applied to P₁ to generate offspring population Q₁ and manifold learning-based exploration is conducted on both P₁ and P₂, resulting in the generation of Q₂ and Q₃, respectively. Then, P₁ and P₂ are combined and NP individuals are selected to generate the offspring population Q₄ through GA based reproduction. Subsequently, P₁ is combined with Q₁, Q₂, and Q₄ and further truncated by the CDP method, while P₂ is combined with Q₁, Q₃, and Q₄ and subjected to environmental selection by truncation strategy in SPEA2 [49]. Finally, if the termination condition is met, P₁ is returned as the final output.
如算法 1 所示，拟议的流形结构辅助协同进化算法（MACA）从两个随机初始化种群 P ₁ 和 P ₂ 开始，每个种群包含 NP 个个体。在每一代中，对 P ₁ 进行引导可行搜索，生成子代种群 Q ₁ ；对 P ₁ 和 P ₂ 进行基于流形学习的探索，分别生成 Q ₂ 和 Q ₃ 。然后，合并 P ₁ 和 P ₂ 并通过基于 GA 的繁殖，选择 NP 个体生成子代群体 Q ₄ 。随后，P ₁ 与 Q ₁ 、Q ₂ 和 Q ₄ 结合，并通过 CDP 方法进一步截断，而 P ₂ 与 Q ₁ 、Q ₃ 和 Q ₄ 结合，并通过 SPEA2 的截断策略进行环境选择[49]。最后，如果满足终止条件，则返回 P ₁ 作为最终输出。

Algorithm 1. General Framework of MACA.
算法 1.MACA 的总体框架。

Input: NP (population size)
输入NP（种群数量）
Output: P₁ 输出：P ₁
1: Initializing the population P₁ and P₂ with NP individuals, respectively;
1: 分别用 NP 个体初始化种群 P ₁ 和 P ₂ ；
2: while termination criterion is not satisfied do
2: 如果不满足终止标准，则执行
3: Q₁

\leftarrow

Guided feasible search (P₁);
3: Q ₁

\leftarrow

引导可行搜索（P ₁ ）；
4: Q₂

\leftarrow

Manifold learning-based exploration (P₁);
4: Q ₂

\leftarrow

基于歧面学习的探索（P ₁ ）；
5: Q₃

\leftarrow

Manifold learning-based exploration (P₂);
5: Q ₃

\leftarrow

基于歧面学习的探索（P ₂ ）；
6: Temp

\leftarrow

Select NP individuals from P₁∪P₂;
6: Temp

\leftarrow

从 P ₁ ∪P ₂ 中选择 NP 个体；
7: Q₄

\leftarrow

Perform GA on Temp to generate offspring;
7: Q ₄

\leftarrow

对 Temp 进行 GA 计算，生成后代；
8: P₁

\leftarrow

P₁∪Q₁∪Q₂∪Q₄;
8: p ₁

\leftarrow

p ₁ ∪q ₁ ∪q ₂ ∪q ₄ ；
9: P₂

\leftarrow

P₂∪Q₁∪Q₃∪Q₄;
9: p ₂

\leftarrow

p ₂ ∪q ₁ ∪q ₃ ∪q ₄ ；
10: P₁

\leftarrow

use the CDP method as environmental selection to select NP individuals from P₁;
10：P ₁

\leftarrow

使用 CDP 方法作为环境选择，从 P ₁ 中选择 NP 个体；
11: P₂

\leftarrow

select NP individuals from P₂ using the truncation strategy in SPEA2 as environmental selection;
11：P ₂

\leftarrow

使用 SPEA2 中的截断策略作为环境选择，从 P ₂ 中选择 NP 个体；
12: endwhile 12: 终止

Note that the MACA employs the similar idea of CCMO [14] that P₁ handles the original problem, while P₂ solely considers the objective values without constraints. Different from [14], MACA not only involves weak cooperation by sharing offspring between populations but also incorporates additional offspring generation through the merged populations to promote convergence. Fig. 4 shows the flowchart of the proposed MACA.
需要注意的是，MACA 采用了与 CCMO [14] 类似的思想，即 P ₁ 处理原始问题，而 P ₂ 只考虑目标值而不考虑约束条件。与[14]不同的是，MACA 不仅涉及种群间分享后代的弱合作，还通过合并种群产生额外的后代来促进收敛。图 4 显示了所提出的 MACA 的流程图。

3.2. Guided feasible search strategy
3.2.指导下的可行搜索策略

In the presence of constraints, the feasible region often contracts to a small space, posing a significant challenge for exploration within these constrained boundaries. Furthermore, infeasible regions may act as barriers, blocking the path toward searching for the CPF and hindering algorithms from discovering the feasible regions and potentially causing them to get trapped in local optima. Relying on conventional exploration methods in such scenarios is impractical. Through problem reformulation [[50], [51], [52], [53]], the multi-objective optimization problem in high-dimensional space can be transformed into multiple low-dimensional subproblems, using specific weight variables to track specific points on the PS. This transformation helps estimate promising search directions in the decision space. Inspired by this concept, directed sampling techniques [54] have been introduced, generating guiding solutions to accelerate convergence through search direction identification and targeted solution sampling. Building upon this inspiration, this paper introduces the Guided feasible search method, aiming to estimate search directions toward the CPS, especially in the process of exploring potential feasible regions.
在存在约束条件的情况下，可行区域往往会收缩到一个很小的空间，这给在这些约束边界内进行探索带来了巨大挑战。此外，不可行区域可能会成为障碍，阻挡搜索 CPF 的路径，阻碍算法发现可行区域，并可能导致算法陷入局部最优状态。在这种情况下，依靠传统的探索方法是不切实际的。通过问题重构[[50]、[51]、[52]、[53]]，可以将高维空间中的多目标优化问题转化为多个低维子问题，使用特定的权重变量来跟踪 PS 上的特定点。这种转换有助于估计决策空间中的搜索方向。受这一概念的启发，人们引入了定向采样技术 [54]，通过搜索方向识别和有针对性的解决方案采样，生成指导性解决方案以加速收敛。受此启发，本文介绍了引导式可行搜索方法，旨在估算面向 CPS 的搜索方向，尤其是在探索潜在可行区域的过程中。

Algorithm 2 shows the pseudocode of the proposed guided feasible search. In guided feasible search, a set of reference vectors, denoted as $W = {w_{1}, w_{2}, . . ., w_{N w}}$ , is utilized at first. The number of reference vectors, denoted as $N w$ , is predetermined. Initially, all individuals in population P₁ are assigned to the nearest reference vector which is determined by the minimum angle with each individual. If any reference vector $w_{i}$ remains unassigned, the nearest individual will be assigned to it. Furthermore, all reference vectors that have only one individual assigned to them are identified, and these individuals are gathered into set S, which helps generate the guided direction. For each individual in S, the guided directions are generated as follows.(2) $v^{l, i} = X_{j} - L$ (3) $v^{u, i} = X_{j} - U$ where $X_{j}$ represents the solution in the population that has the smallest distance to the ideal point along $w_{i}$ . L and U denote the lower and upper bounds, respectively, with $v^{l, i}$ and $v^{u, i}$ representing the search directions extended from the intersection points L and U to $X_{j}$ , respectively, resulting in a total of $2 \times N w$ search directions. Subsequently, samples are generated along these directions. Assuming that, $N_{s}$ samples are generated along each search direction, and the total number of generated solutions is $2 \times N w \times N_{s}$ .
算法 2 显示了拟议的引导式可行搜索的伪代码。在有向导的可行搜索中，首先利用一组参考向量（表示为 $W = {w_{1}, w_{2}, . . ., w_{N w}}$ ）。参考向量的数量（用 $N w$ 表示）是预先确定的。最初，群体 P ₁ 中的所有个体都被分配到最近的参考向量上，该向量由与每个个体的最小角度决定。如果任何参考向量 $w_{i}$ 仍未分配，则最近的个体将被分配到该向量上。此外，所有只分配给一个个体的参考矢量都会被识别出来，这些个体会被集合到集合 S 中，从而帮助生成引导方向。对于 S 中的每个个体，引导方向的生成过程如下。 (2) $v^{l, i} = X_{j} - L$ (3) $v^{u, i} = X_{j} - U$ 其中 $X_{j}$ 代表群体中沿 $w_{i}$ 与理想点距离最小的解。L 和 U 分别表示下界和上界， $v^{l, i}$ 和 $v^{u, i}$ 分别表示从交点 L 和 U 延伸到 $X_{j}$ 的搜索方向，因此总共有 $2 \times N w$ 个搜索方向。随后，沿着这些方向生成样本。假设沿每个搜索方向生成了 $N_{s}$ 个样本，则生成的解决方案总数为 $2 \times N w \times N_{s}$ 个。

Algorithm 2. Pseudocode of Guided feasible search.
算法 2.引导可行搜索的伪代码。

Input: P₁ 输入：P ₁
Output: Q₁ 输出：Q ₁
1: Initialize the reference vector $W = {w_{1}, w_{2}, . . ., w_{N w}}$ ; 1: 初始化参考向量 $W = {w_{1}, w_{2}, . . ., w_{N w}}$ ；
2: Assign each individual in P₁ to its closest reference vector; 2: 将 P ₁ 中的每个个体分配给最接近的参考向量；
3: S= Ø; 3: S= Ø；
4: for i=1:Nw do
5: if the reference vector $w_{i}$ has not been assigned any individual then 5：如果参考向量 $w_{i}$ 没有指定任何个体，那么
6: Assign the nearest individual to the reference vector; 6: 将最近的个体分配给参考向量；
7: end if 7: 如果结束
8: if the reference vector $w_{i}$ has been assigned one individual then 8：如果参考向量 $w_{i}$ 已分配给一个个体，那么
9: Put the individual into S; 9: 将个人放入 S；
10: end if 10: 如果结束
11: end for 11: 结束 for
12: for each individual in S do 12: 对于 S 中的每个个体
13: Q₁ $\leftarrow$ Sampling along the guided directions according to Eq. (2) and (3); 13:Q ₁ $\leftarrow$ 根据公式 (2) 和 (3) 沿引导方向取样；
14: end for 14: 结束 for

When dealing with constraints, the primary objective is to explore areas that have not been adequately explored or remain unexplored. Hence, the sampling strategy is applied exclusively to sparse regions, where the reference vectors are only assigned with one or no individual. Only these individuals are selected to construct guided search directions. If a reference vector is assigned more than one individual, those individuals will not be used to generate samples, which differentiates this approach from [54]. On the other side, since the sampling methods are costly, only sampling in the spare region will save the computational resource.
在处理制约因素时，主要目标是探索尚未充分探索或仍未探索的区域。因此，采样策略只适用于稀疏区域，在这些区域中，参考向量只分配给一个或没有分配给任何个体。只有这些个体被选中来构建引导搜索方向。如果一个参考向量被分配了多个个体，这些个体将不会被用于生成样本，这也是这种方法与 [54] 方法的不同之处。另一方面，由于采样方法成本较高，只在空闲区域采样将节省计算资源。

Fig. 5 illustrates a scenario depicting the sampled solutions of the guided feasible search. In this scenario, there are two local CPS and one global CPS, and the feasible regions are separated. The individuals x₁, x₂, x₃, x₄, x₅ belong to population P₁. Assuming x₁ is selected to produce the guided direction, the estimated directions v^l^,1 from the Lower (L) to x₁ and v^u^,1 from the Upper (U) to x₁ are shown. The samples on these directions, indicated by pentagram symbols, have a high likelihood of exploring previously undiscovered feasible regions when approaching the CPS.
图 5 展示了一个场景，描述了引导式可行搜索的采样解。在这个场景中，有两个局部 CPS 和一个全局 CPS，可行区域是分开的。个体 x ₁ , x ₂ , x ₃ , x ₄ , x ₅ 属于种群 P ₁ 。假设 x ₁ 被选中以产生引导方向，则从下部（L）到 x ₁ 的估计方向 v ^,1 和从上部（U）到 x ₁ 的估计方向 v ^u ^,1 如图所示。这些方向上的样本（用五角星符号表示）在接近 CPS 时很有可能探索到之前未发现的可行区域。

3.3. The manifold learning based exploration strategy
3.3.基于流形学习的探索策略

Due to the influence of constraints, the feasible region can become fragmented, potentially disrupting the regularity property of the PS and PF. However, traces of regularity may still exist. The CPS may appear as fragmented segments within the UPS or on the boundary between feasible and infeasible regions. By modeling the manifold of the PS and then sampling on this captured manifold, it becomes evident that the generated offspring possess the capability to explore the fragmented feasible regions and the separated CPS. Fig. 6 shows a scenario that CPS is fragmented segments within the UPS. As indicated by [5], the CPS may partially overlap or coincide with the UPS, be segmented, or entirely distinct. Regardless of the specific configuration, learning the regularity of the UPS or CPS is advantageous not only for exploring fragmented feasible regions but also for achieving a more accurate approximation of the PS.
由于约束条件的影响，可行区域可能变得支离破碎，从而可能破坏 PS 和 PF 的正则性。然而，规则性的痕迹可能仍然存在。CPS 可能会在 UPS 内或可行区域与不可行区域的边界上显示为支离破碎的片段。通过对 PS 流形建模，然后在捕捉到的流形上采样，可以发现生成的子代具有探索破碎的可行区域和分离的 CPS 的能力。图 6 显示的是 CPS 在 UPS 中被分割成片段的情况。正如文献[5]所指出的，CPS 可能与 UPS 部分重叠或重合，也可能被分割或完全分离。无论具体配置如何，学习 UPS 或 CPS 的规律性不仅有利于探索破碎的可行区域，还能更准确地近似 PS。

Note that in the proposed algorithm, manifold learning-based exploration is implemented for both population P₁ and population P₂. Population P₁ primarily comprises feasible solutions, so the manifold learning-based exploration on this population leverages the discovered manifold structure of the CPS. In contrast, P₂ disregards constraint influences and focuses on exploring the manifold structure of the UPS. Through this structured exploration, the algorithm can uncover hidden, small, and fragmented CPS.
请注意，在提议的算法中，基于流形学习的探索是针对群体 P ₁ 和群体 P ₂ 实施的。种群 P ₁ 主要包括可行解，因此对该种群的基于流形学习的探索利用了已发现的 CPS 流形结构。与此相反，P ₂ 不考虑约束影响，而专注于探索 UPS 的流形结构。通过这种结构化探索，该算法可以发现隐藏的、小的和零散的 CPS。

Due to the challenge of forming a well-defined manifold structure in the early stages of algorithm evolution, instead of modeling the entire PS manifold, modeling local PS structures through clustering is adopted. There are various ways to establish probabilistic models, such as Local PCA, GTM, and GAN, among others. For the sake of simplicity, we have opted for the classic approach used in RM-MEDA [24], which is local PCA as the probability modal. The reason is that the purpose of manifold learning here is not to precisely capture the structure of the manifold. Instead, it aims to explore the extended regions of the manifold to uncover hidden feasible regions. Hence, a less precise modeling approach is recommended.
由于在算法演化的早期阶段难以形成定义明确的流形结构，因此采用了通过聚类对局部 PS 结构建模，而不是对整个 PS 流形建模。建立概率模型的方法有多种，如局部 PCA、GTM 和 GAN 等。为了简单起见，我们选择了 RM-MEDA [24] 中使用的经典方法，即局部 PCA 作为概率模型。原因是流形学习的目的不是精确捕捉流形的结构。相反，它的目的是探索流形的扩展区域，发现隐藏的可行区域。因此，建议采用不那么精确的建模方法。

Algorithm 3 details the application of the manifold learning process. The Population is separated into K disjoint clusters $S^{1}$ , $S^{2}$ , ··· $S^{K}$ at first. The smallest hyper-rectangle containing the projections of all the points of $S^{i}$ in the affine $(m - 1)$ -dimensional principal subspace of $S^{i}$ is(4) $ϕ^{i} = {x \in R^{D} | x = {\bar{x}}^{i} + \sum_{i = 1}^{m - 1} α_{k} U_{k}^{i}, a_{k}^{i} \leq α_{k} \leq b_{k}^{i}, k = 1, \dots, m - 1}$ where $a_{k}^{i} = min_{x \in S^{i}} {(x - {\bar{x}}^{i})}^{T} U_{k}^{i}$ , $b_{k}^{i} = max_{x \in S^{i}} {(x - {\bar{x}}^{i})}^{T} U_{k}^{i}$ , $α_{k}$ is a random number between $a_{k}^{i}$ and $b_{k}^{i}$ , and $D$ is the dimension of the decision space.
算法 3 详细介绍了流形学习过程的应用。首先，将 "人口 "分成 K 个互不相交的簇 $S^{1}$ 、 $S^{2}$ 、 --- $S^{K}$ 。在 $S^{i}$ 的仿射 $(m - 1)$ 维主子空间中，包含 $S^{i}$ 所有点的投影的最小超矩形为 (4) $ϕ^{i} = {x \in R^{D} | x = {\bar{x}}^{i} + \sum_{i = 1}^{m - 1} α_{k} U_{k}^{i}, a_{k}^{i} \leq α_{k} \leq b_{k}^{i}, k = 1, \dots, m - 1}$ ，其中 $a_{k}^{i} = min_{x \in S^{i}} {(x - {\bar{x}}^{i})}^{T} U_{k}^{i}$ , $b_{k}^{i} = max_{x \in S^{i}} {(x - {\bar{x}}^{i})}^{T} U_{k}^{i}$ , $α_{k}$ 为 $a_{k}^{i}$ 和 $b_{k}^{i}$ 之间的随机数， $D$ 为决策空间的维数。

Algorithm 3. Pseudocode of Manifold Learning based exploration.
算法 3.基于 Manifold Learning 的探索伪代码。

Input: Population, the number of clusters K 输入人口、集群数 K
Output: offspring 输出：后代
1: Divide Population into K disjoint clusters $S^{1}$ , $S^{2}$ , ··· $S^{K}$ ; 1: 将人口划分为 K 个不相邻的群 $S^{1}$ , $S^{2}$ , --- $S^{K}$ ；
2: for each cluster do 2: 对每个群组做
3: Let ${\bar{x}}^{i}$ be the mean of cluster $S^{i}$ and $U_{k}^{i}$ be the $k t h$ principal component, build probability model for $S^{i}$ by (4); 3: 设 ${\bar{x}}^{i}$ 为 $S^{i}$ 聚类的平均值， $U_{k}^{i}$ 为 $k t h$ 主成分，根据（4）建立 $S^{i}$ 的概率模型；
4.: Extend $ϕ^{i}$ by 50% along each direction $U_{k}^{i}$ to get $φ^{i}$ by (5) ; 4.:将 $ϕ^{i}$ 沿 $U_{k}^{i}$ 的每个方向延长 50%，得到 $φ^{i}$ （5）；
5: Randomly sample a point $x^{'}$ from $φ^{i}$ uniformly, $i = 1, 2, \dots, K$ and generate new solutions; 5: 从 $φ^{i}$ 中随机均匀地抽取 $x^{'}$ 点， $i = 1, 2, \dots, K$ 并生成新的解；
6: Offspring ← Reproduction by generate new solutions as (6); 6：子代 ← 通过产生新的解决方案进行繁殖，如 (6)；
7: end for 7: 结束 for

And, set(5) $ψ^{i} = {x \in R^{D} | x = {\bar{x}}^{i} + \sum_{k = 1}^{m - 1} α_{k} U_{k}^{i}, a_{k}^{i} - 0.25 (b_{k}^{i} - a_{k}^{i}) \leq α_{k}^{i} \leq b_{k}^{i} + 0.25 (b_{k}^{i} - a_{k}^{i}), i = 1, \dots, m - 1}$ 并设置 (5) $ψ^{i} = {x \in R^{D} | x = {\bar{x}}^{i} + \sum_{k = 1}^{m - 1} α_{k} U_{k}^{i}, a_{k}^{i} - 0.25 (b_{k}^{i} - a_{k}^{i}) \leq α_{k}^{i} \leq b_{k}^{i} + 0.25 (b_{k}^{i} - a_{k}^{i}), i = 1, \dots, m - 1}$

Each individual can be considered as an independent observation of the probabilistic model:(6) $x = x^{'} + ε^{'}$ where $ε^{'} \in N (0, σ^{i} I)$ is a noise vector, and $σ^{i} = \frac{1}{D - m + 1} \sum_{i = m}^{D} λ_{j}^{i}$ , $λ_{j}^{i}$ is the kth largest eigenvalue of the covariance matrix of the points in $S^{i}$ , I is the $D \times D$ identity matrix.
每个个体都可视为概率模型的一个独立观测值： (6) $x = x^{'} + ε^{'}$ 其中 $ε^{'} \in N (0, σ^{i} I)$ 是噪声向量， $σ^{i} = \frac{1}{D - m + 1} \sum_{i = m}^{D} λ_{j}^{i}$ , $λ_{j}^{i}$ 是 $S^{i}$ 中各点协方差矩阵的第 k 个最大特征值，I 是 $D \times D$ 的身份矩阵。

3.4. Analysis of MACA 3.4.分析 MACA

To analyze the effectiveness and understand the mechanisms of the proposed MACA, we have conducted experiments on several constrained multi-objective test functions using PlatEMO [55].
为了分析拟议澳门威尼斯人官网具的有效性并了解其机制，我们使用 PlatEMO [55]对多个受限多目标测试函数进行了实验。

Fig. 7 provides a detailed visualization of the population distribution, illustrating population P₁ and its offspring Q₁ generated through the guided feasible search strategy on the LIR-CMOP10 test function. As indicated in Fig. 7, during the early generations, the population contains few feasible solutions, clustered within easily accessible regions. Consequently, population P₁, denoted by the yellow pentagon, converges within specific local CPF regions. Through the guided feasible search, offspring Q₁ exhibits significant dispersion in regions close to the CPF. This dispersion inadvertently enhances the possibility of exploring other feasible domains, increasing the chance to escape local optima. Even in cases where some feasible domains remain undiscovered, Q₁ is distributed in areas proximate to the CPF. By incorporating Q₁ into population P₂ in the subsequent step, the convergence toward the UPF is accelerated, while a high degree of diversity is maintained.
图 7 提供了种群分布的详细可视化图示，展示了在 LIR-CMOP10 测试函数上通过引导可行搜索策略生成的种群 P ₁ 及其后代 Q ₁ 。如图 7 所示，在早期的几代中，种群包含的可行解很少，都集中在容易获得的区域内。因此，由黄色五边形表示的种群 P ₁ 收敛在特定的局部 CPF 区域内。通过引导可行搜索，后代 Q ₁ 在接近 CPF 的区域表现出显著的分散性。这种分散无意中提高了探索其他可行域的可能性，增加了摆脱局部最优的机会。即使在一些可行域仍未被发现的情况下，Q ₁ 也会分布在靠近 CPF 的区域。通过在后续步骤中将 Q ₁ 纳入群体 P ₂ ，加速了向 UPF 的收敛，同时保持了高度的多样性。

Fig. 8 further illustrates the population distributions of population P₁ and population P₂, as well as their offspring Q₂ and Q₃, generated through manifold learning-based exploration, at different stages for the two-objective LIR-CMOP11 test problem. LIR-CMOP11 exhibits substantial infeasible regions, with local CPF resembling a comb-like pattern. Moreover, its global CPF poses a greater challenge, as it is scattered across tiny, discontinuous isolated fragments. In Fig. 8(a), it is evident that P₁ converges in specific regions of the local CPF in the early stage, leading to the relatively dispersed Q₂. Meanwhile, P₂ rapidly converges toward the UPF, and its offspring Q₃ are predominantly distributed near the manifold regions of the UPF. Through interactions between P₁ and P₂, P₁ escapes the confines of the local CPF and converges toward the global CPF, as illustrated in Fig. 8(b). Notably, P₁ begins to explore portions of the global CPF. Offspring Q₂ generated during this phase predominantly follow the manifold, enabling exploration of previously undiscovered and isolated areas. Through continuous interactions, in the later stages, as illustrated in Fig. 8(c), the populations effectively cover the entire global CPF, achieving comprehensive coverage.
图 8 进一步说明了在双目标 LIR-CMOP11 测试问题的不同阶段，通过基于流形学习的探索生成的种群 P ₁ 和种群 P ₂ 及其后代 Q ₂ 和 Q ₃ 的种群分布情况。LIR-CMOP11 显示出大量不可行区域，局部 CPF 类似于梳状图案。此外，它的全局 CPF 散布在微小、不连续的孤立片段中，这对我们提出了更大的挑战。从图 8（a）中可以明显看出，P ₁ 在早期阶段收敛于局部 CPF 的特定区域，导致 Q ₂ 相对分散。同时，P ₂ 迅速向 UPF 收敛，其后代 Q ₃ 主要分布在 UPF 的流形区域附近。如图 8（b）所示，通过 P ₁ 和 P ₂ 之间的相互作用，P ₁ 摆脱了局部 CPF 的束缚，向全局 CPF 靠拢。值得注意的是，P ₁ 开始探索全局 CPF 的部分区域。在这一阶段产生的后代 Q ₂ 主要遵循流形，从而能够探索以前未被发现的孤立区域。在后期阶段，如图 8（c）所示，通过持续的相互作用，种群有效地覆盖了整个全局 CPF，实现了全面覆盖。

In MACA, the interaction between populations P₁ and P₂ occurs through their merger to create offspring Q₄, which is subsequently integrated separately into P₁ and P₂. Fig. 9 illustrates the individual distribution of Q₄ at different stages while addressing the MW6 benchmark problem. It is evident that MW6 presents narrow and discrete feasible regions. During the early stages, the population explores limited feasible solutions, causing Q₄ to predominantly disperse within the infeasible domain. However, as evolution progresses, guided by the combined efforts of guided feasible search and manifold learning-based exploration from both P₁ and P₂, more feasible solutions are discovered in P₁, while well-distributed nondominated solutions emerge in P₂. The selection of superior solutions from these sets further refines Q₄, progressively guiding it toward the CPF. Guided feasible search and manifold learning-based exploration play pivotal roles in introducing diversity and augmenting exploration capabilities throughout the search process. The generation of Q₄ through the amalgamation of P₁ and P₂, facilitated by genetic algorithms, significantly enhances the algorithm's convergence, enabling precise and swift convergence to the CPF.
在 MACA 中，种群 P ₁ 和 P ₂ 通过合并产生后代 Q ₄ ，随后分别整合为 P ₁ 和 P ₂ 。图 9 展示了处理 MW6 基准问题时 Q ₄ 在不同阶段的个体分布情况。很明显，MW6 呈现出狭窄且离散的可行区域。在早期阶段，群体探索的可行方案有限，导致 Q ₄ 主要分散在不可行区域内。然而，随着演化的进展，在 P ₁ 和 P ₂ 的引导式可行搜索和基于流形学习的探索的共同努力下，P ₁ 中发现了更多的可行解，而 P ₂ 中则出现了分布良好的非优势解。从这些集合中选择出的优秀解决方案进一步完善了 Q ₄ ，逐步引导其向 CPF 方向发展。在整个搜索过程中，引导可行搜索和基于流形学习的探索在引入多样性和增强探索能力方面发挥了关键作用。在遗传算法的帮助下，通过 P ₁ 和 P ₂ 的合并生成 Q ₄ ，大大提高了算法的收敛性，使其能够精确、快速地收敛到 CPF。

4. Experimental studies 4.实验研究

4.1. Experimental setting
4.1.实验环境

4.1.1. Test problems and performance metrics
4.1.1.测试问题和性能指标

We utilize four benchmark sets of CMOPs: MW [56], LIR-CMOP [57], DAS-CMOP [58] and FCP [25]. Four benchmark sets of CMOPs are adopted: MW [56], LIR-CMOP [57], DAS-CMOP [58], and FCP [25]. Each suite offers distinct features that thoroughly test the algorithm's robustness. The LIR-CMOP suite includes 14 complex problems characterized by narrow feasible regions and intricated variable relationships, testing the algorithm's ability to locate and explore feasible regions. The MW suite consists of 14 problems featuring diverse constraints and isolated Pareto optimal solutions, providing multi-scale challenges. The DAS-CMOP suite describes constraints using three main types of difficulty, which can be combined to form various constraint variants, evaluating the algorithm's flexibility and adaptability. The FCP suite contains CMOPs with fraudulent constraints, adding another layer of complexity. Together, these test functions ensured a comprehensive evaluation of the proposed algorithm across diverse and complex scenarios, demonstrating its robustness and effectiveness in solving CMOPs. The dimensions of the decision space (D) were configured as 15, 30, 30 and 30 for MW, LIR-CMOP, DAS-CMOP and FCP, respectively. The number of objectives was kept consistent with the original configuration. According to the classification provided in [5], CMOPs can be categorized into four types: Type-I (CPF equals UPF), Type-II (CPF is part of UPF), Type-III (partial overlap between CPF and UPF), and Type-IV (no overlap between CPF and UPF). MW and LIR-CMOP test suites clearly assign functions to these categories, forming the basis for the experimental results presented in the following subsections. All experiments were conducted on the evolutionary multi-objective optimization platform, PlatEMO [55].
我们使用了四个 CMOP 基准集：MW [56]、LIR-CMOP [57]、DAS-CMOP [58] 和 FCP [25]。我们采用了四个 CMOP 基准集：MW [56]、LIR-CMOP [57]、DAS-CMOP [58] 和 FCP [25]。每套基准都具有不同的特点，可全面测试算法的鲁棒性。LIR-CMOP 套件包括 14 个复杂问题，这些问题的特点是可行区域狭窄，变量关系错综复杂，因此需要测试算法定位和探索可行区域的能力。MW 套件由 14 个问题组成，这些问题具有不同的约束条件和孤立的帕累托最优解，提供了多尺度挑战。DAS-CMOP 套件使用三种主要难度类型来描述约束条件，可将其组合成各种约束条件变体，从而评估算法的灵活性和适应性。FCP 套件包含具有欺诈性约束的 CMOP，增加了另一层复杂性。这些测试功能共同确保了对所提出的算法在各种复杂情况下的全面评估，证明了该算法在解决 CMOP 时的稳健性和有效性。对于 MW、LIR-CMOP、DAS-CMOP 和 FCP，决策空间（D）的维数分别为 15、30、30 和 30。目标数量与原始配置保持一致。根据文献[5]的分类，CMOP 可分为四种类型：I 型（CPF 等于 UPF）、II 型（CPF 是 UPF 的一部分）、III 型（CPF 和 UPF 部分重叠）和 IV 型（CPF 和 UPF 没有重叠）。MW 和 LIR-CMOP 测试套件将功能明确归入这些类别，为下面各小节介绍的实验结果奠定了基础。所有实验都是在进化多目标优化平台 PlatEMO 上进行的[55]。

To evaluate solution quality, this study utilizes two primary indicators: Inverted Generational Distance (IGD) and Hypervolume (HV).
为了评估解决方案的质量，本研究采用了两个主要指标：倒代距离 (IGD) 和超体积 (HV)。

IGD [59] measures the mean distance from the points in the true PF $P^{*}$ to their closet to solution in the obtained population $P$ :(7) $IGD (P^{*}, P) = \frac{\sum_{x \in P^{*}} d i s t (x, P)}{| P^{*} |}$ where $d i s t (x, P)$ defines the Euclidean distance between $x$ and its nearest neighbor in $P$ . A small IGD indicates that the obtained population is close to the true PF as well as has a good distribution.
IGD [59]测量的是从真实 PF $P^{*}$ 中的点到它们在所获得的群体 $P$ ：其中 $d i s t (x, P)$ 定义了 $x$ 与其最近邻 $P$ 之间的欧氏距离。IGD 越小，说明得到的群体越接近真实 PF，且分布良好。

HV [60] measures the volume of the objective space enclosed by the obtained population $P$ with respect to a reference point $z^{r} = (z_{1}^{r}, . . . ., z_{M}^{r})$ :(8) $H V (P) = (L (\cup_{x \in P} [f (x_{1}), z_{1}^{r}] \times . . . \times [f (x_{M}), z_{M}^{r}]))$ where $L (\cdot)$ denotes the Lebesgue measure. A large HV value represents a good dominance relation. In our experiments, each objective is first normalized into [0,1] according to the ideal point and nadir point of the constrained PF. Then, the reference point $z^{r}$ is set to ${(1.1, . . ., 1.1)}^{T}$ .The obtained solutions that do not dominate reference point $z^{r}$ are discarded for the HV calculation.
HV[60]测量的是相对于参考点 $z^{r} = (z_{1}^{r}, . . . ., z_{M}^{r})$ ： (8) $H V (P) = (L (\cup_{x \in P} [f (x_{1}), z_{1}^{r}] \times . . . \times [f (x_{M}), z_{M}^{r}]))$ 其中 $L (\cdot)$ 表示 Lebesgue 度量。HV 值越大，表示支配关系越好。在我们的实验中，首先根据受约束 PF 的理想点和天底点，将每个目标归一化为 [0,1]。然后，将参考点 $z^{r}$ 设为 ${(1.1, . . ., 1.1)}^{T}$ .在计算 HV 时，将舍弃不支配参考点 $z^{r}$ 的解。

IGD quantifies the average distance between generated solutions and the true PF, while HV assesses the volume encompassed by the obtained solutions. A low IGD suggests the proximity of the obtained population to the true PF, indicating good distribution. Conversely, a high HV value signifies strong dominance relations. These metrics collectively measure both convergence and diversity. The Wilcoxon test (at a significance level of α = 0.05) determines whether the results are statistically better (+), worse (−), or similar (=) to the MACA algorithm.
IGD 用于量化生成的解决方案与真实 PF 之间的平均距离，而 HV 则用于评估获得的解决方案所包含的体积。IGD 值越低，说明生成的群体与真实 PF 越接近，表明分布良好。相反，HV 值高则表示支配关系强。这些指标共同衡量了收敛性和多样性。Wilcoxon 检验（显著性水平为 α = 0.05）可确定结果在统计上与 MACA 算法相比是更好（+）、更差（-）还是相似（=）。

4.1.2. Competing algorithms
4.1.2.竞争算法

The CMOEAs used for comparison in this study were C-TAEA [15], CCMO [14], CMOES [61], MFO-MOEA/D [62], MOEA/D-2WA [63], ToP [33], and PPS [16]. Among them, C-TAEA and CCMO are representative two-population-based optimization methods, ToP and PPS are well-regarded two-stage optimization approaches, MOEA/D-2WA represents decomposition-based multi-objective optimization algorithm, CMOES represents an optimization method based on search strategy expansion, while MFO-MOEA/D enbodies an algorithm that optimizes based on different tasks. Note that we follow the original parameter settings of all compared algorithms.
本研究中用于比较的 CMOEA 包括 C-TAEA [15]、CCMO [14]、CMOES [61]、MFO-MOEA/D [62]、MOEA/D-2WA [63]、ToP [33] 和 PPS [16]。其中，C-TAEA 和 CCMO 是具有代表性的基于双群体的优化方法，ToP 和 PPS 是广受好评的两阶段优化方法，MOEA/D-2WA 代表了基于分解的多目标优化算法，CMOES 代表了基于搜索策略扩展的优化方法，而 MFO-MOEA/D 则体现了基于不同任务的优化算法。请注意，我们沿用了所有比较算法的原始参数设置。

4.1.3. Parameter settings
4.1.3.参数设置

Population Size is set to 100, and the crossover probability and distribution index of SBX are set to 1 and 20, respectively, while the mutation probability and distribution index of PM are set to 1/D and 20, respectively. K in manifold learning based exploration is set to 10. The number of function evaluations are set to 100,000 for MW problems and DAS-CMOP problems, and 300,000 for LIR-CMOP problems and FCP problems. We run each algorithm 30 times independently. Note that the metrics only consider feasible solutions, if an algorithm consistently fails over 30 runs, results are marked as ‘‘NaN.’’ Note that in subsequent studies, the cluster quantity is set to M+10 which is consistent with the recommendation in the original reference, and the number of samples along each search direction is set to 30.
种群规模设为 100，SBX 的交叉概率和分布指数分别设为 1 和 20，PM 的突变概率和分布指数分别设为 1/D 和 20。基于流形学习的探索中的 K 设置为 10。MW 问题和 DAS-CMOP 问题的函数评估次数设为 100,000 次，LIR-CMOP 问题和 FCP 问题的函数评估次数设为 300,000 次。我们将每种算法独立运行 30 次。请注意，这些指标只考虑可行的解决方案，如果算法在 30 次运行中持续失败，结果将标记为 "NaN"。请注意，在后续研究中，集群数量设置为 M+10，这与原始参考文献中的建议一致，每个搜索方向的样本数量设置为 30。

4.2. Comparison between MACA and state-of-the-art CMOEAs
4.2.MACA 与最先进 CMOEA 的比较

Table 1 provides an overview of the IGD results, indicating that MACA has outperformed other CMOEAs in average 20 out of 37 test instances. This includes 7 out of 14 instances in the MW test suite, all 6 instances in the DAS-CMOP test suite, and 7 out of 14 instances in the LIR-CMOP test suite. The consistent excellence of MACA across diverse problem sets highlights its robustness and efficiency. To statistically validate the superiority of MACA, we conduct the Friedman test on the experimental results of the compared algorithms. Lower ranking scores indicate better algorithm performance. The summarized results are displayed in Fig. 10. It is observed that MACA has consistently obtained lower ranking scores, affirming its status as a leading CMOEA. Next, we provide an in-depth analysis of the results obtained from each of the test suites.
表 1 提供了 IGD 结果概览，表明 MACA 在 37 个测试实例中平均有 20 个实例的表现优于其他 CMOEA。这包括 MW 测试套件中 14 个实例中的 7 个，DAS-CMOP 测试套件中所有 6 个实例，以及 LIR-CMOP 测试套件中 14 个实例中的 7 个。MACA 在不同的问题集中始终表现出色，这突出表明了它的稳健性和高效性。为了从统计学角度验证 MACA 的优越性，我们对比较算法的实验结果进行了 Friedman 测试。排名分数越低，说明算法性能越好。汇总结果如图 10 所示。从图中可以看出，MACA 一直获得较低的排名分数，这肯定了其作为领先 CMOEA 的地位。接下来，我们将对每个测试套件的结果进行深入分析。

Table 1. IGD values of MACA and the compared algorithms on benchmarks.
表 1.MACA 和比较算法在基准上的 IGD 值。

Type 类型	Problem https://linux.do/t/topic/111737	C-TAEA	CCMO	CMOES	MFO-MOEA/D	MOEA/D-2WA	PPS	ToP	MACA
I	MW2	1.6469e-2 (5.92e-3) -	1.9057e-2 (5.60e-3) -	1.2160e-2 (7.36e-3) =	4.0963e-2 (2.59e-2) -	2.2620e-2 (1.07e-2) -	1.1736e-1 (8.37e-2) -	1.6363e-1 (1.72e-1) -	9.9427e-3 (5.14e-3)
	MW4	4.6625e-2 (4.38e-4) -	6.1667e-2 (5.82e-3) -	4.1783e-2 (4.89e-4) +	4.1426e-2 (8.50e-4) +	4.1314e-2 (1.21e-4) +	5.7412e-2 (2.38e-3) -	NaN (NaN) NaN (无)	4.4745e-2 (7.47e-4)
	MW14	1.1218e-1 (4.22e-3) -	5.4275e-1 (1.55e-1) -	8.1685e-2 (2.74e-2) +	2.7525e+0 (1.07e+0) -	2.1164e-1 (2.56e-3) -	1.3924e-1 (1.18e-2) -	3.2885e-1 (3.62e-1) -	1.0308e-1 (2.41e-3)
II	MW1	2.3649e-3 (1.36e-3) -	3.1848e-2 (5.67e-2) -	1.6784e-3 (6.23e-5) =	3.6439e-2 (7.64e-2) -	3.7930e-3 (2.54e-3) -	3.1277e-3 (2.97e-4) -	6.8462e-1 (0.00e+0) =	1.6792e-3 (1.21e-5)
	MW5	1.2818e-2 (2.88e-3) -	2.9887e-2 (2.73e-2) -	5.6483e-4 (4.33e-4) +	1.5152e-1 (3.07e-1) =	1.4091e-2 (5.48e-3) -	3.3053e-1 (3.64e-1) -	NaN (NaN) NaN (无)	7.5140e-4 (2.09e-4)
	MW6	8.6475e-3 (4.55e-3) -	3.7992e-2 (8.83e-2) -	1.0347e-2 (8.87e-3) -	2.5343e-1 (3.34e-1) -	1.8663e-2 (1.05e-2) -	5.2884e-1 (3.56e-1) -	6.9223e-1 (3.98e-1) -	3.8763e-3 (2.19e-3)
	MW8	5.5253e-2 (4.00e-3) -	4.9396e-2 (4.74e-3) -	5.1499e-2 (1.51e-2) -	7.7649e-2 (4.12e-2) -	5.3600e-2 (1.53e-2) -	1.8978e-1 (1.12e-1) -	7.6745e-1 (3.65e-1) -	4.2973e-2 (9.89e-4)
III	MW3	5.0537e-3 (3.11e-4) =	9.9505e-3 (3.11e-3) -	4.8319e-3 (1.66e-4) +	5.5934e-3 (6.01e-4) -	5.9562e-3 (5.57e-4) -	6.1820e-3 (3.89e-4) -	7.0157e-1 (3.33e-1) -	4.9639e-3 (2.24e-4)
	MW7	6.6809e-3 (4.40e-4) -	1.3065e-2 (3.18e-3) -	4.2279e-3 (4.13e-4) +	5.0138e-3 (2.59e-4) =	5.2574e-3 (2.64e-4) -	5.5028e-3 (4.37e-4) -	3.4577e-2 (2.51e-2) -	4.8892e-3 (3.96e-4)
	MW10	1.3453e-2 (9.22e-3) -	6.1148e-2 (1.22e-1) -	1.7674e-2 (2.05e-2) -	1.7604e-1 (1.69e-1) -	3.4620e-2 (2.81e-2) -	3.2430e-1 (2.16e-1) -	1.3229e-1 (0.00e+0) =	3.5266e-3 (1.03e-4)
	MW13	3.3831e-2 (1.90e-2) -	8.3267e-2 (3.38e-2) -	4.2008e-2 (3.38e-2) -	1.4296e-1 (8.71e-2) -	7.8658e-2 (3.64e-2) -	4.7087e-1 (3.33e-1) -	6.9738e-1 (5.33e-1) -	1.5114e-2 (8.40e-3)
IV	MW9	8.5060e-3 (7.08e-4) -	1.1483e-1 (2.54e-1) -	6.1381e-2 (8.20e-2) -	2.0704e-1 (3.15e-1) -	1.1521e-2 (2.10e-3) -	1.2919e-1 (2.58e-1) -	8.4248e-1 (1.32e-1) -	4.9778e-3 (2.83e-4)
	MW11	1.4201e-2 (1.08e-3) -	6.1610e-2 (8.49e-2) -	5.6442e-3 (9.43e-4) +	2.4045e-1 (3.71e-1) -	4.4515e-2 (1.36e-1) -	7.4371e-3 (3.99e-4) -	6.3018e-1 (2.19e-1) -	6.3883e-3 (2.48e-4)
	MW12	7.6989e-3 (6.80e-4) -	1.2473e-2 (8.38e-3) -	3.6325e-2 (4.54e-2) =	2.5431e-1 (3.24e-1) -	5.9761e-3 (3.21e-4) -	1.0378e-1 (2.24e-1) -	9.6114e-1 (2.28e-1) -	5.0309e-3 (1.27e-4)
I	LIR-CMOP5	9.7427e-1 (4.09e-1) -	2.6681e-1 (4.93e-2) -	2.0835e-1 (9.28e-2) -	1.2568e+0 (1.68e-2) -	3.5351e-1 (3.69e-2) -	6.9047e-3 (7.53e-4) +	1.1703e+0 (2.14e-2) -	8.7645e-3 (9.71e-4)
	LIR-CMOP6	1.3459e+0 (3.05e-4) -	2.7928e-1 (7.57e-2) -	2.2647e-1 (9.12e-2) -	1.3501e+0 (3.13e-3) -	3.9827e-1 (5.79e-2) -	5.1984e-2 (2.44e-1) -	1.2613e+0 (2.61e-1) -	8.0290e-3 (3.26e-4)
	LIR-CMOP13	1.0844e-1 (1.97e-3) -	1.0804e-1 (4.44e-2) -	8.5703e-2 (1.57e-2) +	1.3167e+0 (2.23e-3) -	1.3027e+0 (2.16e-4) -	1.2792e-1 (3.64e-3) -	1.3112e+0 (1.01e-1) -	1.0308e-1 (2.10e-3)
	LIR-CMOP14	1.1131e-1 (1.01e-3) -	9.6024e-2 (7.31e-4) +	8.5334e-2 (1.68e-2) +	1.2731e+0 (2.05e-3) -	1.2589e+0 (3.75e-4) -	1.5656e-1 (2.11e-1) -	1.2597e+0 (1.09e-1) -	9.8133e-2 (1.01e-3)
II	LIR-CMOP9	4.4355e-1 (9.02e-2) -	3.6072e-1 (1.63e-1) -	2.4624e-1 (1.02e-1) -	1.3804e+0 (3.06e-1) -	4.8002e-1 (1.88e-1) -	3.3294e-1 (1.12e-1) -	4.5509e-1 (1.12e-1) -	1.6701e-1 (1.32e-1)
II	LIR-CMOP10	2.3149e-1 (9.96e-2) -	7.8520e-2 (4.57e-2) -	6.6936e-2 (3.62e-2) -	1.0753e+0 (1.01e-1) -	1.6523e-1 (1.15e-1) -	2.0469e-2 (5.27e-2) =	3.7824e-1 (7.42e-2) -	6.3391e-3 (3.86e-4)
III	LIR-CMOP11	1.8481e-1 (3.08e-2) -	4.5681e-2 (3.22e-2) -	2.7330e-2 (2.82e-2) -	1.1860e+0 (1.50e-1) -	4.9128e-1 (2.82e-1) -	2.2393e-1 (1.10e-1) -	3.6704e-1 (1.02e-1) -	3.4664e-3 (5.63e-4)
III	LIR-CMOP12	1.8666e-1 (1.14e-1) -	1.8028e-1 (9.04e-2) -	7.7308e-2 (4.19e-2) -	1.1011e+0 (1.76e-1) -	3.5923e-1 (1.20e-1) -	1.2944e-1 (6.58e-2) -	2.4883e-1 (8.97e-2) -	5.5884e-2 (6.03e-2)
IV	LIR-CMOP1	1.6785e-1 (7.40e-2) -	1.8069e-1 (4.59e-2) -	1.1578e-1 (4.22e-2) =	3.3608e-1 (3.83e-2) -	1.9968e-1 (1.47e-2) -	1.2084e-2 (1.02e-2) +	3.0065e-1 (2.27e-2) -	1.0657e-1 (4.06e-2)
	LIR-CMOP2	1.5288e-1 (4.49e-2) +	1.7080e-1 (3.58e-2) +	9.9256e-2 (1.99e-2) +	2.9725e-1 (2.94e-2) +	1.7273e-1 (1.93e-2) +	6.3101e-3 (5.53e-4) +	2.6075e-1 (1.72e-2) +	6.7321e-1 (6.57e-3)
	LIR-CMOP3	2.2603e-1 (8.31e-2) -	2.0983e-1 (5.49e-2) -	9.4233e-2 (2.61e-2) =	3.2251e-1 (4.32e-2) -	2.3961e-1 (4.49e-2) -	1.6641e-2 (2.90e-2) +	3.3946e-1 (1.11e-2) -	9.2958e-2 (4.18e-2)
	LIR-CMOP4	2.4471e-1 (9.19e-2) -	2.1871e-1 (5.85e-2) -	1.1549e-1 (3.39e-2) -	3.0690e-1 (2.92e-2) -	2.3745e-1 (2.57e-2) -	4.3869e-2 (6.16e-2) +	3.1362e-1 (1.65e-2) -	5.7908e-2 (1.51e-2)
	LIR-CMOP7	2.0546e-1 (3.10e-1) -	1.0887e-1 (3.30e-2) -	8.9314e-2 (4.18e-2) -	3.4861e-1 (2.63e-1) -	1.5784e-1 (2.33e-2) -	1.0218e-1 (4.34e-2) -	6.3416e-1 (7.57e-1) -	8.4092e-3 (4.09e-4)
	LIR-CMOP8	5.6839e-1 (5.88e-1) -	1.6122e-1 (4.86e-2) -	1.5132e-1 (6.62e-2) -	6.0721e-1 (4.39e-1) -	2.5228e-1 (2.84e-2) -	1.0915e-1 (6.42e-2) -	1.4090e+0 (5.65e-1) -	8.1411e-3 (3.61e-4)
	DAS-CMOP1	1.9093e-1 (1.76e-2) -	7.4823e-1 (2.89e-2) -	4.9551e-1 (3.03e-1) -	7.6533e-1 (6.47e-2) -	8.0276e-1 (7.95e-2) -	7.3000e-1 (1.10e-1) -	7.9754e-1 (4.37e-2) -	6.9686e-3 (8.96e-4)
	DAS-CMOP2	1.0590e-1 (3.34e-2) -	2.6666e-1 (2.77e-2) -	1.7532e-1 (9.24e-2) -	5.9246e-1 (2.32e-1) -	2.9679e-1 (4.74e-2) -	2.4753e-1 (2.15e-2) -	7.2302e-1 (1.07e-1) -	6.8491e-3 (5.65e-4)
	DAS-CMOP3	1.7410e-1 (5.72e-2) -	3.4173e-1 (2.15e-2) -	2.7597e-1 (3.22e-2) -	6.8822e-1 (1.12e-1) -	6.7628e-1 (2.14e-1) -	3.0296e-1 (9.36e-2) -	7.5082e-1 (5.58e-2) -	2.6070e-2 (2.28e-2)
	DAS-CMOP4	1.1725e-2 (2.68e-3) -	8.7122e-1 (2.79e-1) -	6.6918e-3 (7.84e-3) -	1.0109e+0 (5.25e-2) -	9.5558e-1 (1.05e-1) -	7.6306e-1 (2.64e-1) -	NaN (NaN) NaN (无)	5.5807e-3 (2.29e-3)
	DAS-CMOP5	7.6657e-3 (6.83e-4) -	7.1743e-1 (1.06e-1) -	5.0112e-3 (3.10e-3) +	9.0246e-1 (7.05e-2) -	9.0601e-1 (8.77e-2) -	3.8203e-1 (1.72e-1) -	NaN (NaN) NaN (无)	6.1977e-3 (1.20e-3)
	DAS-CMOP6	2.7149e-2 (5.20e-3) -	8.9456e-1 (8.38e-2) -	5.1909e-2 (3.45e-2) =	9.6609e-1 (1.49e-1) -	1.0212e+0 (1.95e-1) -	5.8057e-1 (2.72e-1) -	NaN (NaN) NaN (无)	2.1482e-2 (4.55e-3)
	DAS-CMOP7	3.8574e-2 (5.05e-3) =	8.1882e-1 (1.72e-1) -	3.3767e-2 (3.47e-3) +	1.0397e+0 (2.48e-1) -	1.0643e+0 (1.38e-1) -	4.9219e-1 (3.84e-1) -	NaN (NaN) NaN (无)	3.7606e-2 (1.99e-3)
	DAS-CMOP8	5.7053e-2 (8.60e-3) -	9.7387e-1 (3.12e-1) -	3.9581e-2 (7.86e-4) +	1.2254e+0 (1.73e-1) -	1.2088e+0 (1.29e-1) -	7.6992e-1 (3.75e-1) -	NaN (NaN) NaN (无)	5.0230e-2 (7.81e-3)
	DAS-CMOP9	2.7459e-1 (6.80e-2) -	4.3575e-1 (7.25e-2) -	2.9589e-1 (8.82e-2) -	7.3217e-1 (1.69e-1) -	8.1670e-1 (9.67e-2) -	6.0675e-1 (8.26e-2) -	6.8294e-1 (1.51e-1) -	4.7412e-2 (2.18e-3)
	+ / - / =	1/34/2	2/35/0	12/19/6	2/33/2	2/35/0	5/31/1	1/27/2

Fig. 11 presents the results of the Friedman test for all compared algorithms on the three test suites. It is evident that MACA outperforms all other peer algorithms in all three test suites. In the MW test suite, CMOES secures the second position, followed by C-TAEA, PPS, CCMO, MOEA/D-2WA, MFO-MOEA/D and ToP. In the LIR-CMOP test suite, CMOES ranks second, followed by PPS in the third position, while MFO-MOEA/D performs the least favorably. In the DAS-CMOP test suite, the results align with those in the MW test suite, except for PPS, which exhibits subpar performance.
图 11 显示了所有比较算法在三个测试套件上的弗里德曼测试结果。很明显，在所有三个测试套件中，MACA 都优于所有其他同行算法。在 MW 测试套件中，CMOES 稳居第二，紧随其后的是 C-TAEA、PPS、CCMO、MOEA/D-2WA、MFO-MOEA/D 和 ToP。在 LIR-CMOP 测试套件中，CMOES 排名第二，PPS 紧随其后排名第三，而 MFO-MOEA/D 的表现最差。在 DAS-CMOP 测试套件中，结果与 MW 测试套件中的结果一致，但 PPS 除外，表现不佳。

Table 2 shows the HV values of the MACA and seven compared algorithms. It is observed that the MACA achieves the best HV values on 20 out of 37 test instances. This includes 7 out of 14 instances in the MW test suite, 5 out of 9 instances in the DAS-CMOP test suite, and 8 out of 14 instances in the LIR-CMOP test suite. The consistent excellence of MACA across diverse problem sets highlights its robustness and efficiency. Additional observations are drawn in the following.
表 2 显示了 MACA 和七种比较算法的 HV 值。可以看出，在 37 个测试实例中，MACA 在 20 个实例上达到了最佳 HV 值。这包括 MW 测试套件中 14 个实例中的 7 个，DAS-CMOP 测试套件中 9 个实例中的 5 个，以及 LIR-CMOP 测试套件中 14 个实例中的 8 个。MACA 在不同的问题集中始终表现出色，这突出表明了它的稳健性和高效性。以下是其他观察结果。

1) MW Test Suites: This test suite encompasses all four types of CPF, featuring both continuous and discrete CPFs. From Table 1, it is observed that MACA consistently outperforms other algorithms in Type II, III, and IV scenarios, delivering superior results. While there is a minor performance gap observed in MW4, MACA clearly demonstrates exceptional competence in handling instances within this suite. Specifically, MACA obtains the minimum average IGD values on seven instances (MW2, MW6, MW8, MW9, MW10, MW12, and MW13) and the maximum average HV values on seven instances (MW2, MW6, MW7, MW8, MW9, MW10, and MW13). Fig. 12 provides the obtained CPF of all compared algorithms on MW2. From Fig. 12, it is clear that only MACA can satisfactorily cover the entire CPF with well-distributed solutions. In contrast, other algorithms such as ToP and PPS struggle due to insufficient population diversity, hindering their smooth traversal to the CPF. MOEA/D-2WA, MFO-MOEA/D, C-TAEA, and CCMO, although able to converge to the CPF, face challenges in covering the entire PF due to the fragmented nature of feasible regions.
1) MW 测试套件：该测试套件包括所有四种类型的 CPF，既有连续 CPF，也有离散 CPF。从表 1 中可以看出，MACA 在类型 II、III 和 IV 场景中的性能始终优于其他算法，并提供了出色的结果。虽然在 MW4 中观察到的性能差距较小，但 MACA 在处理该套件中的实例时显然表现出了非凡的能力。具体来说，MACA 在七个实例（MW2、MW6、MW8、MW9、MW10、MW12 和 MW13）中获得了最小平均 IGD 值，在七个实例（MW2、MW6、MW7、MW8、MW9、MW10 和 MW13）中获得了最大平均 HV 值。图 12 提供了所有比较算法在 MW2 上获得的 CPF。从图 12 中可以明显看出，只有 MACA 能以良好的分布解覆盖整个 CPF。相比之下，ToP 和 PPS 等其他算法则由于种群多样性不足而难以顺利穿越 CPF。MOEA/D-2WA、MFO-MOEA/D、C-TAEA 和 CCMO 虽然能够收敛到 CPF，但由于可行区域的分散性，在覆盖整个 PF 方面面临挑战。
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Fig. 12. Populations with the median IGD obtained by MW2 for algorithms (a) C-TAEA (b) CCMO (c) CMOES (d) MFO-MOEA/D (e) MOEA/D-2WA (f) PPS (g) ToP, and (h) MACA.
图 12.MW2 算法 (a) C-TAEA (b) CCMO (c) CMOES (d) MFO-MOEA/D (e) MOEA/D-2WA (f) PPS (g) ToP 和 (h) MACA 得出的 IGD 中位数种群。

Table 2. HV values of MACA and the compared algorithms on benchmarks.
表 2.MACA 和比较算法在基准上的 HV 值。

Type 类型	Problem https://linux.do/t/topic/111737	C-TAEA	CCMO	CMOES	MFO-MOEA/D	MOEA/D-2WA	PPS	ToP	MACA
I	MW2	5.6032e-1 (1.05e-2) -	5.5591e-1 (9.42e-3) -	5.6573e-1 (1.24e-2) =	5.2329e-1 (3.48e-2) -	5.5024e-1 (1.70e-2) -	4.2940e-1 (9.43e-2) -	3.9230e-1 (1.33e-1) -	5.7170e-1 (9.26e-3)
	MW4	8.3817e-1 (2.94e-4) +	8.1023e-1 (7.35e-3) -	8.4051e-1 (6.25e-4) +	8.4131e-1 (8.67e-4) +	8.4149e-1 (1.23e-4) +	8.1246e-1 (5.37e-3) -	NaN (NaN) NaN (无)	8.3358e-1 (1.18e-3)
	MW14	4.6723e-1 (2.24e-3) +	2.8120e-1 (7.19e-2) -	4.7091e-1 (2.20e-3) +	5.5870e-2 (1.09e-1) -	4.4388e-1 (2.16e-3) -	4.4720e-1 (6.44e-3) -	3.7316e-1 (1.41e-1) -	4.6596e-1 (1.70e-3)
II	MW1	4.8812e-1 (2.78e-3) -	4.5010e-1 (4.83e-2) -	4.9004e-1 (5.06e-5) +	4.5848e-1 (6.43e-2) -	4.8791e-1 (4.56e-3) -	4.8715e-1 (9.36e-4) -	0.0000e+0 (0.00e+0) =	4.8983e-1 (4.09e-5)
	MW5	3.1673e-1 (2.07e-3) -	2.9864e-1 (1.64e-2) -	3.2435e-1 (2.56e-4) +	2.7761e-1 (9.50e-2) =	3.1698e-1 (6.51e-3) -	2.1703e-1 (1.12e-1) -	NaN (NaN) NaN (无)	3.2422e-1 (1.39e-4)
	MW6	3.1592e-1 (7.42e-3) -	2.9340e-1 (3.68e-2) -	3.1205e-1 (1.15e-2) -	2.1371e-1 (9.56e-2) -	3.0450e-1 (1.46e-2) -	1.0446e-1 (9.38e-2) -	8.3180e-2 (9.28e-2) -	3.2646e-1 (3.91e-3)
	MW8	5.1900e-1 (1.52e-2) -	5.2270e-1 (1.36e-2) -	5.3925e-1 (7.28e-3) =	4.6836e-1 (7.72e-2) -	5.2834e-1 (3.57e-2) =	2.8375e-1 (9.80e-2) -	1.0473e-1 (1.37e-1) -	5.4287e-1 (5.20e-3)
III	MW3	5.4467e-1 (6.26e-4) +	5.3730e-1 (2.01e-3) -	5.4428e-1 (3.73e-4) +	5.4350e-1 (9.89e-4) -	5.4373e-1 (7.13e-4) =	5.4326e-1 (5.89e-4) -	1.0949e-1 (1.72e-1) -	5.4400e-1 (1.88e-4)
	MW7	4.0922e-1 (6.39e-4) -	4.0614e-1 (1.97e-3) -	4.1229e-1 (4.79e-4) =	4.1108e-1 (4.02e-4) -	4.1070e-1 (3.43e-4) -	4.1175e-1 (3.49e-4) -	3.7765e-1 (1.95e-2) -	4.1229e-1 (1.37e-4)
	MW10	4.4014e-1 (1.11e-2) -	4.0596e-1 (6.34e-2) -	4.3127e-1 (2.44e-2) -	3.3727e-1 (8.54e-2) -	4.2117e-1 (2.31e-2) -	2.5988e-1 (1.01e-1) -	3.5341e-1 (0.00e+0) =	4.5429e-1 (2.35e-4)
	MW13	4.6291e-1 (9.12e-3) -	4.3965e-1 (1.92e-2) -	4.5597e-1 (1.73e-2) -	4.0225e-1 (5.38e-2) -	4.4662e-1 (1.39e-2) -	2.5586e-1 (1.11e-1) -	2.5782e-1 (1.17e-1) -	4.7332e-1 (5.59e-3)
IV	MW9	3.9264e-1 (2.18e-3) -	3.2650e-1 (1.30e-1) -	3.8607e-1 (7.29e-2) -	2.7923e-1 (1.77e-1) -	3.8409e-1 (4.65e-3) -	3.1447e-1 (1.44e-1) -	0.0000e+0 (0.00e+0) -	3.9705e-1 (2.22e-3)
	MW11	4.4273e-1 (9.23e-4) -	4.2016e-1 (3.48e-2) -	4.4768e-1 (2.14e-4) +	3.9401e-1 (8.31e-2) -	4.3499e-1 (3.63e-2) -	4.4741e-1 (1.54e-4) +	2.8592e-1 (4.71e-2) -	4.4660e-1 (4.08e-4)
	MW12	6.0084e-1 (8.46e-4) -	5.9556e-1 (7.25e-3) -	6.0488e-1 (2.73e-4) +	3.9885e-1 (2.63e-1) -	6.0268e-1 (6.36e-4) -	5.1407e-1 (1.91e-1) -	0.0000e+0 (0.00e+0) -	6.0384e-1 (3.18e-4)
I	LIR-CMOP5	4.1241e-2 (7.00e-2) -	1.6631e-1 (1.90e-2) -	1.6631e-1 (2.05e-2) -	0.0000e+0 (0.00e+0) -	1.3860e-1 (1.59e-2) -	2.9158e-1 (2.66e-4) +	0.0000e+0 (0.00e+0) -	2.8968e-1 (8.13e-4)
	LIR-CMOP6	0.0000e+0 (0.00e+0) -	1.2319e-1 (1.62e-2) -	1.2547e-1 (1.84e-2) -	0.0000e+0 (0.00e+0) -	9.5626e-2 (2.49e-3) -	1.9055e-1 (3.60e-2) -	5.7171e-3 (1.74e-2) -	1.9573e-1 (1.41e-4)
	LIR-CMOP13	5.4690e-1 (1.59e-3) +	5.3715e-1 (5.34e-2) +	5.5039e-1 (1.49e-3) +	2.9659e-4 (1.32e-4) -	4.4485e-4 (1.02e-5) -	5.1908e-1 (5.50e-3) -	7.0785e-3 (2.26e-2) -	5.3268e-1 (2.93e-3)
	LIR-CMOP14	5.4629e-1 (8.14e-4) -	5.5334e-1 (1.56e-3) +	5.5226e-1 (1.39e-3) +	8.0691e-4 (2.05e-4) -	9.8295e-4 (3.84e-5) -	5.1606e-1 (9.73e-2) -	9.6018e-3 (2.50e-2) -	5.4742e-1 (1.74e-3)
II	LIR-CMOP9	3.8878e-1 (5.77e-2) -	4.3578e-1 (7.59e-2) -	4.7501e-1 (3.47e-2) -	1.0256e-1 (3.39e-2) -	3.8510e-1 (5.34e-2) -	4.6494e-1 (4.04e-2) =	3.8721e-1 (7.91e-2) -	5.0201e-1 (5.43e-2)
II	LIR-CMOP10	5.9064e-1 (4.47e-2) -	6.7183e-1 (1.85e-2) -	6.7141e-1 (1.23e-2) -	6.2875e-2 (4.87e-2) -	5.9267e-1 (7.93e-2) -	7.0042e-1 (2.44e-2) -	5.1024e-1 (4.37e-2) -	7.0573e-1 (3.33e-4)
III	LIR-CMOP11	6.2127e-1 (1.56e-2) -	6.7122e-1 (1.56e-2) -	6.7842e-1 (1.44e-2) -	1.2437e-1 (4.94e-2) -	5.3531e-1 (1.12e-1) -	5.5433e-1 (7.00e-2) -	4.5486e-1 (7.95e-2) -	6.9346e-1 (2.68e-4)
III	LIR-CMOP12	5.2638e-1 (4.97e-2) -	5.3345e-1 (4.96e-2) -	5.7509e-1 (1.99e-2) -	3.0078e-1 (6.56e-2) -	5.3797e-1 (3.37e-2) -	5.6384e-1 (2.95e-2) -	4.9669e-1 (4.51e-2) -	5.9323e-1 (3.17e-2)
IV	LIR-CMOP1	1.5591e-1 (2.22e-2) -	1.4976e-1 (1.59e-2) -	1.7036e-1 (1.58e-2) -	9.9837e-2 (1.27e-2) -	1.4123e-1 (7.72e-3) -	2.3554e-1 (3.04e-3) +	1.1449e-1 (8.90e-3) -	1.8476e-1 (1.92e-2)
	LIR-CMOP2	2.9226e-1 (2.46e-2) -	2.7355e-1 (2.06e-2) -	3.0708e-1 (9.92e-3) -	2.1576e-1 (1.37e-2) -	2.8472e-1 (8.55e-3) -	3.5956e-1 (2.52e-4) -	2.2674e-1 (1.17e-2) -	8.7400e-1 (2.03e-4)
	LIR-CMOP3	1.2089e-1 (2.22e-2) -	1.3057e-1 (1.80e-2) -	1.6815e-1 (1.03e-2) =	9.8236e-2 (1.12e-2) -	1.1805e-1 (1.47e-2) -	2.0057e-1 (1.33e-2) +	9.2425e-2 (5.94e-3) -	1.6712e-1 (1.62e-2)
	LIR-CMOP4	2.2017e-1 (3.60e-2) -	2.2564e-1 (2.64e-2) -	2.6331e-1 (1.48e-2) -	1.8550e-1 (1.32e-2) -	2.1860e-1 (1.18e-2) -	2.9850e-1 (2.75e-2) +	1.8656e-1 (1.28e-2) -	2.9035e-1 (8.05e-3)
	LIR-CMOP7	2.3493e-1 (5.00e-2) -	2.5148e-1 (1.04e-2) -	2.5094e-1 (8.77e-3) -	1.9418e-1 (3.96e-2) -	2.3894e-1 (7.03e-3) -	2.5431e-1 (1.67e-2) -	1.7308e-1 (1.26e-1) -	2.9375e-1 (2.42e-4)
	LIR-CMOP8	1.6966e-1 (9.19e-2) -	2.4076e-1 (1.10e-2) -	2.4202e-1 (1.12e-2) -	1.5493e-1 (6.35e-2) -	2.2005e-1 (1.88e-3) -	2.5521e-1 (1.73e-2) -	4.5412e-2 (9.31e-2) -	2.9400e-1 (1.64e-4)
	DAS-CMOP1	1.6488e-1 (7.47e-3) -	5.1238e-3 (4.04e-3) -	2.2177e-2 (2.15e-2) -	5.0761e-3 (9.72e-3) -	4.0866e-3 (4.91e-3) -	9.5959e-3 (2.93e-2) -	1.5819e-3 (4.31e-3) -	2.0862e-1 (1.51e-3)
	DAS-CMOP2	3.0087e-1 (9.75e-3) -	2.5016e-1 (5.66e-3) -	2.6261e-1 (3.17e-3) -	9.9259e-2 (1.00e-1) -	2.3356e-1 (2.13e-2) -	2.5401e-1 (7.59e-3) -	3.9675e-2 (3.78e-2) -	3.5315e-1 (4.92e-4)
	DAS-CMOP3	2.4441e-1 (1.27e-2) -	2.0826e-1 (3.81e-3) -	2.1921e-1 (1.11e-2) -	3.9686e-2 (4.85e-2) -	6.4380e-2 (7.97e-2) -	2.1809e-1 (2.73e-2) -	1.7517e-2 (1.22e-2) -	3.0765e-1 (8.55e-3)
	DAS-CMOP4	1.9544e-1 (5.21e-3) -	5.1642e-3 (1.30e-2) -	2.0393e-1 (4.36e-4) +	0.0000e+0 (0.00e+0) -	0.0000e+0 (0.00e+0) -	2.7950e-2 (4.39e-2) -	NaN (NaN) NaN (无)	1.9768e-1 (4.26e-3)
	DAS-CMOP5	3.4783e-1 (4.86e-4) -	2.6199e-2 (1.85e-2) -	3.5137e-1 (1.95e-4) +	6.5569e-3 (4.90e-3) -	5.4188e-3 (5.81e-3) -	1.5598e-1 (7.75e-2) -	NaN (NaN) NaN (无)	3.4889e-1 (7.67e-4)
	DAS-CMOP6	3.0681e-1 (2.09e-3) -	4.2913e-3 (7.17e-3) -	3.0476e-1 (1.10e-2) =	5.0954e-3 (1.02e-2) -	1.4518e-3 (3.84e-3) -	1.0340e-1 (8.88e-2) -	NaN (NaN) NaN (无)	3.0873e-1 (3.98e-3)
	DAS-CMOP7	2.8721e-1 (1.41e-3) +	2.4109e-2 (2.55e-2) -	2.8766e-1 (3.97e-4) +	1.2732e-2 (3.22e-2) -	3.2932e-3 (5.69e-3) -	1.2169e-1 (1.03e-1) -	NaN (NaN) NaN (无)	2.8325e-1 (1.10e-3)
	DAS-CMOP8	2.0336e-1 (1.91e-3) +	3.4922e-3 (1.34e-2) -	2.0661e-1 (5.24e-4) +	0.0000e+0 (0.00e+0) -	0.0000e+0 (0.00e+0) -	3.3174e-2 (3.25e-2) -	NaN (NaN) NaN (无)	2.0164e-1 (3.07e-3)
	DAS-CMOP9	1.3935e-1 (1.23e-2) -	1.0481e-1 (1.42e-2) -	1.3673e-1 (1.41e-2) -	5.6039e-2 (1.56e-2) -	4.2418e-2 (6.92e-3) -	5.4278e-2 (1.82e-2) -	6.3888e-2 (2.20e-2) -	1.9936e-1 (8.88e-4)
	+ / - / =	6/31/0	2/35/0	13/19/5	1/35/1	1/34/2	5/31/1	0/28/2

Fig. 13 displays the results obtained by MACA on MW test problems. Among these test problems, MW4, MW8 and MW14 are three-objective problem, while the remaining ones are two-objective problems. The CPF of some test problems are fragmented (MW1, MW5-MW11, MW13, MW14), and some of them have local CPFs (MW7, MW9, MW12 and MW13). Despite the challenges posed by the small and fragmented feasible regions typical of MW problems, MACA demonstrates an outstanding capability to approach and cover the entire CPF.
图 13 显示了 MACA 在 MW 测试问题上得到的结果。在这些测试问题中，MW4、MW8 和 MW14 属于三目标问题，其余问题属于二目标问题。一些测试问题的 CPF 是零散的（MW1、MW5-MW11、MW13 和 MW14），还有一些测试问题的 CPF 是局部的（MW7、MW9、MW12 和 MW13）。尽管 MW 问题的典型可行区域较小且分散，这对 MACA 提出了挑战，但 MACA 展示了接近和覆盖整个 CPF 的出色能力。

2) LIR-CMOP Test Suites: This test suite encompasses all four types of CPF. In some cases, feasible regions are hindered by extensive infeasible areas, and specific PFs consist of only a few fragmented line segments or sparse points, further obstructed by these sizable infeasible regions.
2) LIR-CMOP 测试套件：该测试套件包括所有四种类型的 CPF。在某些情况下，可行区域会受到大面积不可行区域的阻碍，而特定的 PF 只由一些零散的线段或稀疏的点组成，这些大面积的不可行区域又进一步阻碍了 PF。

Table 1, Table 2 show that MACA achieves the best performance in terms of IGD and HV, followed by PPS and CMOES. Specifically, MACA, PPS, and CMOES obtain the smallest average IGD values in 7/14, 5/14, and 2/14 instances, respectively, while MACA, PPS, CCMO, and CMOES produce the largest average HV results in 8/14, 4/14, 1/14 and 1/14 instances, respectively. For other algorithms, they do not provide any best average IGD or HV values. The results of MACA are slightly inferior to PPS in these test problems (LIR-CMOP 1–5). This can be attributed to the fact that these test problems have extremely small feasible regions, and the UPS is relatively distant from the CPS. The strategy proposed in this paper may not be well-suited for such scenarios. Fig. 14 displays the results obtained by MACA on LIR-CMOP test problems. It is worth noting MACA performs well in dealing with feasible regions obstructed by large infeasible regions (e.g. LIR-CMOP5-LIR-CMOP12) and disconnected regions (e.g. LIR-CMOP9-LIR-CMOP12).
表 1 和表 2 显示，MACA 在 IGD 和 HV 方面表现最佳，其次是 PPS 和 CMOES。具体来说，MACA、PPS 和 CMOES 分别在 7/14、5/14 和 2/14 个实例中获得了最小的平均 IGD 值，而 MACA、PPS、CCMO 和 CMOES 分别在 8/14、4/14、1/14 和 1/14 个实例中产生了最大的平均 HV 值。至于其他算法，它们没有提供任何最佳平均 IGD 或 HV 值。在这些测试问题（LIR-CMOP 1-5）中，MACA 的结果略逊于 PPS。这可能是由于这些测试问题的可行区域极小，而且 UPS 与 CPS 的距离相对较远。本文提出的策略可能不太适合这种情况。图 14 显示了 MACA 在 LIR-CMOP 测试问题上获得的结果。值得注意的是，MACA 在处理被大型不可行区域阻挡的可行区域（如 LIR-CMOP5-LIR-CMOP12）和断开区域（如 LIR-CMOP9-LIR-CMOP12）时表现良好。

3) DAS-CMOP Test Suites: In DAS-CMOP1 to DAS-CMOP6, the CPF is located on the boundaries of one or several feasible regions, and the CPF is composed of disconnected line segments separated by large infeasible regions. Additionally, some feasible regions of DAS-CMOP problems are far from the UPF. Fig. 15 displays the results obtained by MACA on DAS-CMOP test problems. The proposed MACA algorithm struggles to perfectly cover all CPF regions in some particularly fragmented and narrow problems (e.g., DAS-CMOP3, DAS-CMOP6). However, it performs well in all other cases on the DAS-CMOP test suite.
3) DAS-CMOP 测试套件：在 DAS-CMOP1 至 DAS-CMOP6 中，CPF 位于一个或多个可行区域的边界上，CPF 由断开的线段组成，并被较大的不可行区域分隔开来。此外，DAS-CMOP 问题的一些可行区域与 UPF 相距甚远。图 15 显示了 MACA 在 DAS-CMOP 测试问题上获得的结果。在一些特别破碎和狭窄的问题（如 DAS-CMOP3、DAS-CMOP6）中，所提出的 MACA 算法很难完美地覆盖所有 CPF 区域。但是，在 DAS-CMOP 测试套件的所有其他情况下，该算法都表现出色。
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Fig. 15. Experimental results on DAS-CMOP1–9 by MACA with 100 individuals under 100,000 evaluations.
图 15.使用 MACA 对 DAS-CMOP1-9 进行 100 个个体 100,000 次评估的实验结果。

4) Summary: Across 37 test problems encompassing various degrees of challenge in three different test suites, particularly those featuring separated and extensively obstructed feasible regions that are difficult to explore, MACA consistently delivers impressive results. In conclusion, we confirm that MACA is a suitable tool for tackling CMOPs.
4) 总结：在三个不同测试套件中的 37 个测试问题中，特别是在那些分离的、广泛受阻的、难以探索的可行区域中，MACA 始终取得了令人印象深刻的结果。总之，我们证实 MACA 是处理 CMOP 的合适工具。

4.3. Analysis of the effects of the individual components
4.3.各组成部分的效果分析

4.3.1. Effectiveness of the guided feasible search strategy
4.3.1.有指导的可行搜索策略的有效性

To assess the effectiveness of the guided feasible search strategy, we have conducted two experiments. One involved integrating the guided feasible search strategy with NSGA-II-CDP to test its ability to help NSGA-II-CDP explore feasible regions across large infeasible areas. The other experiment compared the performance of MACA with and without the guided feasible search strategy.
为了评估引导式可行搜索策略的有效性，我们进行了两项实验。一个实验是将引导式可行搜索策略与 NSGA-II-CDP 集成，以测试其帮助 NSGA-II-CDP 跨大型不可行区域探索可行区域的能力。另一项实验则比较了 MACA 在使用和不使用引导式可行搜索策略时的性能。

Fig. 16 presents a comparison between NSGA-II-CDP-GF, which incorporates the guided feasible search strategy, and the original NSGA-II-CDP on four representative test functions. It clearly illustrates how the proposed guided feasible search strategy facilitates the exploration of more feasible regions, enabling the population to traverse large infeasible areas and approach the CPF. The comparative experiments demonstrate that the proposed guided feasible search strategy exhibits significant effectiveness in exploring feasible regions, especially in cases involving separated feasible regions obstructed by infeasible areas. Table 3 present the statistical results obtained by NSGA-II-CDP and NSGA-II-CDP-GF on three test suites. Examination of Table 3 reveals that, across all 37 problems, NSGA-II-CDP-GF consistently achieves superior or comparable results in comparison to NSGA-II-CDP, which primarily emphasizes feasibility during convergence. NSGA-II-CDP-GF accomplishes this by guiding the population to explore feasible regions and escape local optima, thereby expediting the convergence process.
图 16 展示了包含引导式可行搜索策略的 NSGA-II-CDP-GF 与原始 NSGA-II-CDP 在四个代表性测试函数上的比较。它清楚地说明了所提出的有向导的可行搜索策略是如何促进探索更多可行区域，使群体穿越大面积的不可行区域并接近 CPF 的。对比实验表明，所提出的有向导的可行搜索策略在探索可行区域方面表现出显著的有效性，尤其是在分离的可行区域被不可行区域阻挡的情况下。表 3 列出了 NSGA-II-CDP 和 NSGA-II-CDP-GF 在三个测试套件上获得的统计结果。表 3 显示，在所有 37 个问题中，NSGA-II-CDP-GF 始终比 NSGA-II-CDP 取得更优或相当的结果，后者在收敛过程中主要强调可行性。NSGA-II-CDP-GF 通过引导群体探索可行区域并摆脱局部最优状态，从而加快了收敛过程。

Table 3. The IGD and HV results of NSGA-II-CDP and NSGA-II-CDP-GF.
表 3.NSGA-II-CDP 和 NSGA-II-CDP-GF 的 IGD 和 HV 结果。

Type 类型	Problem https://linux.do/t/topic/111737	NSGA-II-CDP	NSGA-II-CDP-GF	NSGA-II-CDP	NSGA-II-CDP-GF
Empty Cell	Empty Cell	IGD		HV
I	MW2	2.6369e-2 (1.54e-2) +	3.6562e-2 (3.25e-2)	5.4427e-1 (1.39e-2) =	5.3085e-1 (4.14e-2)
	MW4	5.5589e-2 (2.87e-3) =	5.6737e-2 (2.87e-3)	8.2438e-1 (3.15e-3) =	8.2389e-1 (2.76e-3)
	MW14	1.2392e-1 (6.32e-3) =	1.3279e-1 (1.85e-2)	4.5311e-1 (3.23e-3) +	4.4747e-1 (7.76e-3)
II	MW1	3.6170e-2 (1.05e-1) -	2.6764e-3 (1.88e-3)	4.7361e-1 (3.75e-2) =	4.8839e-1 (3.80e-3)
	MW5	3.0494e-1 (3.45e-1) =	6.7970e-2 (7.18e-2)	1.9011e-1 (9.93e-2) -	2.8210e-1 (4.42e-2)
	MW6	9.1862e-2 (1.52e-1) =	7.9114e-2 (1.39e-1)	2.6961e-1 (5.54e-2) =	2.8293e-1 (3.59e-2)
	MW8	6.2003e-2 (1.02e-2) =	6.1320e-2 (1.65e-2)	5.0313e-1 (1.84e-2) =	4.9918e-1 (3.43e-2)
III	MW3	3.6426e-2 (1.67e-1) =	5.8913e-3 (2.81e-4)	5.4300e-1 (6.17e-4) =	5.4308e-1 (4.93e-4)
	MW7	6.4585e-2 (1.54e-1) =	6.5943e-3 (8.43e-3)	3.9950e-1 (4.23e-2) =	4.1131e-1 (1.58e-3)
	MW10	1.3257e-1 (1.52e-1) +	2.9405e-1 (2.58e-1)	3.4395e-1 (8.57e-2) +	2.8113e-1 (1.25e-1)
	MW13	1.5824e-1 (1.51e-1) +	2.8249e-1 (4.02e-1)	4.1821e-1 (3.78e-2) +	3.8131e-1 (7.19e-2)
IV	MW9	3.7658e-2 (1.28e-1) =	1.5435e-2 (2.02e-2)	3.5998e-1 (9.82e-2) =	3.8223e-1 (1.49e-2)
	MW11	2.9579e-1 (3.39e-1) -	6.7200e-3 (2.23e-4)	3.7138e-1 (8.76e-2) -	4.4765e-1 (1.67e-4)
	MW12	5.5935e-3 (2.02e-4) =	5.5939e-3 (2.31e-4)	5.5573e-1 (1.52e-1) =	6.0404e-1 (1.91e-4)
I	LIR-CMOP5	1.2153e+0 (7.52e-3) -	2.6879e-1 (1.75e-2)	0.0000e+0 (0.00e+0) -	2.1545e-1 (4.70e-3)
	LIR-CMOP6	1.3147e+0 (1.70e-1) -	4.7757e-1 (2.95e-1)	0.0000e+0 (0.00e+0) -	6.1051e-2 (2.12e-2)
	LIR-CMOP13	1.3246e+0 (2.06e-3) =	1.3247e+0 (1.99e-3)	9.0793e-5 (1.51e-4) =	1.3304e-4 (1.46e-4)
	LIR-CMOP14	1.2811e+0 (2.51e-3) =	1.2815e+0 (2.68e-3)	2.9158e-4 (3.02e-4) -	4.7892e-4 (3.45e-4)
II	LIR-CMOP9	8.1604e-1 (7.85e-2) -	5.1786e-1 (5.96e-2)	1.2926e-1 (4.01e-2) -	3.5266e-1 (3.48e-2)
II	LIR-CMOP10	7.5902e-1 (1.60e-1) -	4.1180e-1 (1.17e-3)	1.0737e-1 (7.18e-2) -	4.8969e-1 (1.21e-3)
III	LIR-CMOP11	6.8509e-1 (1.30e-1) -	5.2953e-1 (1.66e-1)	2.3807e-1 (5.90e-2) -	3.7429e-1 (1.02e-1)
III	LIR-CMOP12	6.1530e-1 (1.86e-1) -	3.4320e-1 (9.83e-2)	2.4783e-1 (8.35e-2) -	4.4585e-1 (5.61e-2)
IV	LIR-CMOP1	2.8258e-1 (2.13e-2) =	2.8760e-1 (2.83e-2)	1.1501e-1 (8.72e-3) =	1.1593e-1 (9.42e-3)
	LIR-CMOP2	6.7116e-1 (4.95e-2) -	6.6776e-1 (4.41e-3)	7.0596e-1 (2.80e-2) -	8.7115e-1 (9.42e-4)
	LIR-CMOP3	3.1211e-1 (3.25e-2) =	2.9186e-1 (7.96e-2)	1.0253e-1 (8.50e-3) =	1.1090e-1 (1.77e-2)
	LIR-CMOP4	2.8803e-1 (2.85e-2) +	3.2324e-1 (4.42e-2)	1.9606e-1 (1.11e-2) =	1.8309e-1 (2.57e-2)
	LIR-CMOP7	4.1254e-1 (5.89e-1) -	1.8667e-1 (4.09e-1)	1.6941e-1 (1.13e-1) -	2.5593e-1 (7.05e-2)
	LIR-CMOP8	9.9531e-1 (7.47e-1) =	4.9442e-1 (3.67e-1)	8.8172e-2 (1.10e-1) -	2.0517e-1 (6.33e-2)
	DAS-CMOP1	7.4906e-1 (4.41e-2) -	2.0250e-1 (1.15e-1)	5.7889e-3 (6.17e-3) -	1.4901e-1 (4.14e-2)
	DAS-CMOP2	2.6735e-1 (3.19e-2) -	2.1797e-1 (9.53e-2)	2.4574e-1 (4.07e-3) -	2.5769e-1 (3.00e-2)
	DAS-CMOP3	3.5216e-1 (3.55e-2) -	2.6525e-1 (9.50e-2)	2.0855e-1 (1.52e-3) =	2.1229e-1 (4.88e-2)
	DAS-CMOP4	9.7229e-2 (1.34e-1) =	1.2434e-1 (1.26e-1)	1.4972e-1 (5.05e-2) =	1.7184e-1 (3.27e-2)
	DAS-CMOP5	3.6645e-1 (1.88e-1) -	1.3324e-2 (2.06e-2)	1.4904e-1 (1.01e-1) -	3.4532e-1 (1.11e-2)
	DAS-CMOP6	5.3841e-1 (1.13e-1) -	1.6479e-1 (6.60e-2)	6.7779e-2 (2.66e-2) -	2.7029e-1 (2.30e-2)
	DAS-CMOP7	4.9582e-2 (2.74e-3) =	6.0053e-2 (1.93e-2)	2.8285e-1 (1.47e-3) +	2.7238e-1 (1.41e-2)
	DAS-CMOP8	6.2279e-2 (3.28e-3) +	6.9828e-2 (1.27e-2)	1.9983e-1 (1.57e-3) +	1.9147e-1 (8.94e-3)
	DAS-CMOP9	3.9353e-1 (8.68e-2) -	2.4612e-1 (8.70e-2)	1.1831e-1 (1.57e-2) -	1.4211e-1 (1.84e-2)
Empty Cell	+ / - / =	5/16/16		5/17/15

Furthermore, we have conducted a comparison between MACA with and without the guided feasible search strategy, referred to as MACA-without GF. Table 4 present the statistical results obtained by MACA and MACA-without GF on three test suites. Analysis of Table 4 reveals that, in contrast to MACA-without GF, which lacks the ability to escape local optima in certain problems, our proposed method consistently outperforms or achieves comparable results across all 37 problems. These findings highlight the effectiveness of the approach implemented in MACA, enhancing the population's convergence capability.
此外，我们还比较了采用和不采用引导可行搜索策略的 MACA，简称为不采用 GF 的 MACA。表 4 列出了 MACA 和不带 GF 的 MACA 在三个测试套件上获得的统计结果。对表 4 的分析表明，不带 GF 的 MACA 在某些问题上缺乏摆脱局部最优的能力，相比之下，我们提出的方法在所有 37 个问题上的表现始终优于或达到了相当的结果。这些发现凸显了在 MACA 中实施的方法的有效性，增强了群体的收敛能力。

Table 4. The IGD and HV results of MACA and MACA-without GF.
表 4.MACA 和无 GF MACA 的 IGD 和 HV 结果。

Type 类型	Problem https://linux.do/t/topic/111737	MACA-without GF MACA - 无 GF	MACA	MACA-without GF MACA - 无 GF	MACA
Empty Cell	Empty Cell	IGD		HV
I	MW2	4.3199e-3 (1.29e-4) =	4.2988e-3 (1.18e-4)	5.8162e-1 (1.66e-4) =	5.8165e-1 (1.54e-4)
	MW4	4.6532e-2 (8.89e-4) =	4.6969e-2 (5.60e-4)	8.2979e-1 (1.06e-3) =	8.2941e-1 (1.02e-3)
	MW14	1.0288e-1 (1.88e-3) =	1.0256e-1 (2.08e-3)	4.6557e-1 (2.22e-3) =	4.6310e-1 (2.83e-3)
II	MW1	1.6855e-3 (1.49e-5) =	1.6978e-3 (1.38e-5)	4.8972e-1 (5.84e-5) =	4.8969e-1 (2.52e-5)
	MW5	1.0343e-3 (2.91e-4) +	1.4621e-3 (3.95e-4)	3.2401e-1 (1.72e-4) +	3.2372e-1 (2.42e-4)
	MW6	2.7721e-3 (2.17e-5) =	2.7802e-3 (2.24e-5)	3.2839e-1 (5.11e-5) =	3.2842e-1 (2.75e-5)
	MW8	4.3061e-2 (1.53e-3) =	4.3553e-2 (1.59e-3)	5.4458e-1 (4.83e-3) =	5.4169e-1 (7.03e-3)
III	MW3	4.8289e-3 (8.01e-5) =	4.8136e-3 (1.36e-4)	5.4431e-1 (1.80e-4) =	5.4432e-1 (1.78e-4)
	MW7	4.8914e-3 (4.43e-4) =	4.9111e-3 (3.12e-4)	4.1231e-1 (1.57e-4) =	4.1235e-1 (1.17e-4)
	MW10	3.4687e-3 (4.00e-5) =	3.4974e-3 (4.59e-5)	4.5431e-1 (1.66e-4) =	4.5422e-1 (1.92e-4)
	MW13	1.1396e-2 (2.67e-4) +	1.1763e-2 (3.50e-4)	4.7608e-1 (2.09e-4) =	4.7587e-1 (2.23e-4)
IV	MW9	5.1598e-3 (2.65e-4) =	5.1316e-3 (1.78e-4)	3.9554e-1 (1.37e-3) =	3.9589e-1 (1.93e-3)
	MW11	6.4213e-3 (2.51e-4) =	6.2865e-3 (2.36e-4)	4.4675e-1 (3.72e-4) =	4.4689e-1 (1.95e-4)
	MW12	5.1176e-3 (2.28e-4) =	5.0738e-3 (1.11e-4)	6.0368e-1 (6.05e-4) =	6.0376e-1 (3.88e-4)
I	LIR-CMOP5	8.9453e-3 (6.65e-4) -	7.6642e-3 (3.82e-4)	2.8920e-1 (9.80e-4) -	2.9060e-1 (2.53e-4)
	LIR-CMOP6	9.0462e-3 (1.08e-3) -	7.3227e-3 (1.78e-4)	1.9503e-1 (9.26e-4) -	1.9613e-1 (4.83e-5)
	LIR-CMOP13	1.0394e-1 (1.75e-3) =	1.0450e-1 (2.46e-3)	5.3087e-1 (2.45e-3) =	5.2993e-1 (2.47e-3)
	LIR-CMOP14	9.8106e-2 (1.11e-3) =	9.8236e-2 (9.30e-4)	5.4542e-1 (1.88e-3) =	5.4675e-1 (1.40e-3)
II	LIR-CMOP9	1.3126e-2 (8.58e-3) =	5.4303e-2 (1.13e-1)	5.6158e-1 (5.49e-3) =	5.4779e-1 (3.94e-2)
II	LIR-CMOP10	5.8414e-3 (2.48e-4) =	5.8874e-3 (3.16e-4)	7.0626e-1 (2.69e-4) =	7.0630e-1 (2.29e-4)
III	LIR-CMOP11	3.1389e-3 (3.40e-4) =	3.0086e-3 (5.78e-4)	6.9353e-1 (1.56e-4) -	6.9368e-1 (1.43e-4)
III	LIR-CMOP12	4.9580e-3 (2.34e-3) +	7.0618e-2 (5.59e-2)	6.1948e-1 (9.49e-4) +	5.8588e-1 (2.84e-2)
IV	LIR-CMOP1	2.3836e-1 (4.47e-2) =	2.1356e-1 (2.21e-2)	1.3430e-1 (1.84e-2) =	1.3762e-1 (8.38e-3)
	LIR-CMOP2	6.6774e-1 (9.77e-3) =	6.6563e-1 (9.81e-3)	8.7390e-1 (2.50e-4) =	8.7401e-1 (2.56e-4)
	LIR-CMOP3	1.9810e-1 (3.34e-2) =	1.5679e-1 (5.52e-2)	1.3661e-1 (1.40e-2) =	1.4660e-1 (1.74e-2)
	LIR-CMOP4	1.9056e-1 (4.58e-2) =	1.4503e-1 (4.54e-2)	2.3235e-1 (2.06e-2) =	2.5315e-1 (1.86e-2)
	LIR-CMOP7	8.0011e-3 (3.66e-4) -	7.5630e-3 (2.78e-4)	2.9435e-1 (2.08e-4) =	2.9445e-1 (1.42e-4)
	LIR-CMOP8	7.9560e-3 (4.64e-4) -	7.5117e-3 (2.87e-4)	2.9415e-1 (1.46e-4) -	2.9460e-1 (1.49e-4)
	DAS-CMOP1	4.9685e-3 (6.93e-4) +	6.2524e-3 (6.58e-4)	2.0928e-1 (9.59e-4) =	2.0913e-1 (5.99e-4)
	DAS-CMOP2	1.1485e-2 (4.04e-3) -	5.7026e-3 (3.64e-4)	3.4932e-1 (2.47e-3) -	3.5407e-1 (3.78e-4)
	DAS-CMOP3	1.4179e-1 (5.50e-3) -	2.2298e-2 (1.28e-3)	2.6374e-1 (1.68e-3) -	3.0946e-1 (1.80e-3)
	DAS-CMOP4	1.8916e-1 (1.92e-1) -	9.8662e-3 (3.73e-3)	1.1803e-1 (6.75e-2) -	1.9331e-1 (5.61e-3)
	DAS-CMOP5	1.0437e-1 (1.23e-1) -	1.0333e-2 (2.06e-3)	2.8394e-1 (7.59e-2) -	3.4621e-1 (8.07e-4)
	DAS-CMOP6	1.8059e-1 (1.78e-1) -	2.3838e-2 (2.92e-3)	2.2872e-1 (8.71e-2) -	3.0538e-1 (4.57e-3)
	DAS-CMOP7	1.1298e-1 (1.03e-1) -	4.4674e-2 (2.86e-3)	2.4814e-1 (4.93e-2) -	2.8067e-1 (1.05e-3)
	DAS-CMOP8	1.4446e-1 (1.09e-1) -	5.5823e-2 (3.96e-3)	1.5503e-1 (6.55e-2) =	1.9834e-1 (2.38e-3)
	DAS-CMOP9	4.7226e-2 (2.36e-3) +	5.3803e-2 (5.28e-3)	2.0129e-1 (8.57e-4) +	2.0005e-1 (1.00e-3)
Empty Cell	+ / - / =	5/11/21		3/10/24

4.3.2. Effectiveness of the manifold learning based exploration strategy
4.3.2.基于流形学习的探索策略的有效性

Similarly, to confirm the importance of the manifold learning-based feasibility search strategy, we have conducted two experiments. One involves integrating and comparing the manifold learning-based exploration strategy with CCMO, which is also based on two populations. The other experiment focuses on conducting ablation experiments on the proposed algorithm with respect to the manifold learning-based exploration strategy. Fig. 17 presents the experimental results of CCMO and CCMO with manifold learning-based exploration strategy (referred to as CCMO-ML) on four representative test problems. It is evident from the graph that regardless of whether the CPF is continuous or fragmented, CCMO struggles to extend well on the CPF. However, when combined with the manifold learning-based exploration strategy, the population can effectively cover the entire CPF. Experimental results of the ablation experiments also confirm the same effect. The experimental outcomes demonstrate that the manifold learning-based exploration strategy is indeed effective in enabling the algorithm to explore along the manifold surface, aiding in searching and covering the entire CPF.
同样，为了证实基于流形学习的可行性搜索策略的重要性，我们进行了两项实验。一个是将基于流形学习的探索策略与同样基于两个种群的 CCMO 进行整合和比较。另一项实验主要是针对基于流形学习的探索策略，对所提出的算法进行消融实验。图 17 展示了 CCMO 和基于流形学习探索策略的 CCMO（简称 CCMO-ML）在四个代表性测试问题上的实验结果。从图中可以看出，无论 CPF 是连续的还是片段的，CCMO 都很难在 CPF 上很好地扩展。然而，当与基于流形学习的探索策略相结合时，群体可以有效地覆盖整个 CPF。消融实验的结果也证实了同样的效果。实验结果表明，基于流形学习的探索策略确实能有效地使算法沿着流形表面进行探索，帮助搜索和覆盖整个 CPF。

The statistical results obtained by CCMO and CCMO-ML on three test suites are listed in Table 5. CCMO-ML employs the manifold learning-based feasibility search strategy for both populations. It generates two offspring populations using this strategy, then selects superior solutions from each population using their respective environmental selection strategies to proceed to the next iteration. From Table 5, it can be observed that CCMO-ML outperforms CCMO on 24 out of 37 problems.
表 5 列出了 CCMO 和 CCMO-ML 在三个测试套件上获得的统计结果。CCMO-ML 对两个种群都采用了基于流形学习的可行性搜索策略。它利用这一策略生成两个后代种群，然后利用各自的环境选择策略从每个种群中选出优解，进入下一次迭代。从表 5 可以看出，在 37 个问题中，CCMO-ML 在 24 个问题上的表现优于 CCMO。

Table 5. The IGD and HV results of CCMO and CCMO-ML.
表 5.CCMO 和 CCMO-ML 的 IGD 和 HV 结果。

Type 类型	Problem https://linux.do/t/topic/111737	CCMO	CCMO-ML	CCMO	CCMO-ML
Empty Cell	Empty Cell	IGD		HV
I	MW2	1.9331e-2 (7.02e-3) -	9.4182e-3 (4.16e-3)	5.4871e-1 (2.27e-2) -	5.7274e-1 (7.46e-3)
	MW4	4.0859e-2 (4.36e-4) +	4.4562e-2 (8.17e-4)	8.4167e-1 (4.95e-4) +	8.3378e-1 (1.19e-3)
	MW14	9.7771e-2 (1.68e-3) +	1.0276e-1 (2.39e-3)	4.7351e-1 (1.59e-3) +	4.6647e-1 (2.34e-3)
II	MW1	2.8580e-3 (6.70e-3) -	1.6679e-3 (1.94e-5)	4.9005e-1 (5.10e-5) +	4.8988e-1 (9.40e-5)
	MW5	1.5103e-3 (1.77e-3) -	5.1505e-4 (1.58e-4)	3.2402e-1 (7.20e-4) =	3.2437e-1 (1.19e-4)
	MW6	1.9894e-2 (1.30e-2) -	3.7495e-3 (2.17e-3)	3.0295e-1 (1.76e-2) -	3.2694e-1 (3.41e-3)
	MW8	4.6502e-2 (3.51e-3) -	4.3727e-2 (1.42e-3)	5.3162e-1 (1.38e-2) -	5.4077e-1 (6.88e-3)
III	MW3	4.8788e-3 (2.87e-4) +	4.9597e-3 (1.51e-4)	5.4429e-1 (6.03e-4) +	5.4404e-1 (2.70e-4)
	MW7	4.6751e-3 (4.17e-4) =	4.7545e-3 (3.84e-4)	4.1224e-1 (1.23e-3) =	4.1241e-1 (1.54e-4)
	MW10	4.6971e-2 (3.81e-2) -	3.5023e-3 (9.23e-5)	4.1235e-1 (2.80e-2) -	4.5459e-1 (2.01e-4)
	MW13	6.3657e-2 (3.45e-2) -	1.4462e-2 (5.14e-3)	4.4856e-1 (1.50e-2) -	4.7330e-1 (4.97e-3)
IV	MW9	2.8219e-2 (1.29e-1) -	1.8467e-2 (6.72e-2)	3.8502e-1 (7.28e-2) +	3.8324e-1 (6.14e-2)
	MW11	6.1046e-3 (1.92e-4) +	6.2952e-3 (2.07e-4)	4.4742e-1 (2.98e-4) +	4.4705e-1 (2.19e-4)
	MW12	4.6941e-2 (1.61e-1) +	5.3149e-2 (1.84e-1)	5.6727e-1 (1.43e-1) +	5.6412e-1 (1.52e-1)
I	LIR-CMOP5	2.5275e-1 (5.49e-2) -	9.1079e-3 (3.84e-4)	1.6631e-1 (1.90e-2) -	2.8923e-1 (4.08e-4)
	LIR-CMOP6	2.8297e-1 (7.82e-2) -	9.3347e-3 (5.38e-4)	1.2319e-1 (1.62e-2) -	1.9471e-1 (4.44e-4)
	LIR-CMOP13	9.3263e-2 (1.10e-3) +	1.0292e-1 (2.21e-3)	5.5496e-1 (1.35e-3) +	5.3256e-1 (3.16e-3)
	LIR-CMOP14	9.5930e-2 (1.07e-3) +	9.8085e-2 (1.33e-3)	5.5003e-1 (1.87e-2) +	5.4670e-1 (1.78e-3)
II	LIR-CMOP9	3.6775e-1 (1.58e-1) -	1.5552e-2 (5.50e-3)	4.3578e-1 (7.59e-2) -	5.6274e-1 (2.03e-3)
II	LIR-CMOP10	8.3888e-2 (3.77e-2) -	6.0847e-3 (4.04e-4)	6.7183e-1 (1.85e-2) -	7.0598e-1 (3.11e-4)
III	LIR-CMOP11	3.8094e-2 (3.00e-2) -	2.5584e-3 (1.24e-4)	6.7122e-1 (1.56e-2) -	6.9384e-1 (6.77e-5)
III	LIR-CMOP12	1.6570e-1 (9.18e-2) -	6.4115e-3 (3.39e-3)	5.3345e-1 (4.96e-2) -	6.1847e-1 (1.82e-3)
IV	LIR-CMOP1	1.9153e-1 (5.08e-2) +	2.4559e-1 (3.93e-2)	1.4976e-1 (1.59e-2) +	1.2931e-1 (1.50e-2)
	LIR-CMOP2	1.6733e-1 (4.89e-2) +	6.7015e-1 (1.03e-2)	2.7355e-1 (2.06e-2) -	8.7371e-1 (2.21e-4)
	LIR-CMOP3	1.9456e-1 (6.37e-2) =	1.8809e-1 (6.51e-2)	1.3057e-1 (1.80e-2) =	1.3812e-1 (2.29e-2)
	LIR-CMOP4	2.2873e-1 (5.88e-2) =	2.0551e-1 (5.02e-2)	2.2564e-1 (2.64e-2) =	2.2692e-1 (2.26e-2)
	LIR-CMOP7	1.1931e-1 (3.01e-2) -	8.3761e-3 (5.83e-4)	2.5148e-1 (1.04e-2) -	2.9338e-1 (4.34e-4)
	LIR-CMOP8	1.6255e-1 (5.86e-2) -	8.0154e-3 (4.32e-4)	2.4076e-1 (1.10e-2) -	2.9341e-1 (4.95e-4)
	DAS-CMOP1	7.2524e-1 (3.31e-2) -	9.2026e-3 (5.79e-3)	1.4254e-2 (1.74e-2) -	2.0643e-1 (3.71e-3)
	DAS-CMOP2	2.4606e-1 (2.18e-2) -	2.1584e-2 (6.12e-3)	2.5818e-1 (3.91e-3) -	3.4436e-1 (1.96e-3)
	DAS-CMOP3	3.2487e-1 (5.06e-2) -	1.4245e-1 (2.27e-3)	2.1436e-1 (1.32e-2) -	2.6495e-1 (1.01e-3)
	DAS-CMOP4	1.7265e-3 (1.08e-3) +	1.8410e-2 (5.60e-2)	2.0098e-1 (4.08e-3) +	1.9250e-1 (2.55e-2)
	DAS-CMOP5	2.7900e-3 (1.13e-4) +	9.3624e-3 (2.14e-2)	3.5139e-1 (1.36e-4) +	3.4615e-1 (1.38e-2)
	DAS-CMOP6	3.5936e-2 (3.73e-2) =	2.0973e-2 (7.81e-3)	2.9642e-1 (1.72e-2) =	3.0998e-1 (5.19e-3)
	DAS-CMOP7	3.0762e-2 (7.72e-4) +	4.0971e-2 (2.99e-2)	2.8838e-1 (4.57e-4) +	2.8273e-1 (8.38e-3)
	DAS-CMOP8	3.9548e-2 (7.69e-4) +	5.3114e-2 (1.46e-2)	2.0726e-1 (4.38e-4) +	2.0287e-1 (6.05e-3)
	DAS-CMOP9	3.6413e-1 (7.29e-2) -	4.3406e-2 (2.36e-3)	1.2787e-1 (1.09e-2) -	2.0192e-1 (7.50e-4)
Empty Cell	+ / - / =	13/20/4		14/18/5

Also, we compare MACA with a variant called MACA without ML, which removes the manifold learning-based exploration operation. The statistical results on three test suites are listed in Table 6. From Table 6, it can be observed that MACA outperforms MACA without ML on 25 out of 37 problems. This indicates that the manifold learning-based exploration strategy is beneficial in improving the quality of the solution set obtained.
此外，我们还将 MACA 与一种名为 "无 ML MACA "的变体进行了比较，后者取消了基于流形学习的探索操作。表 6 列出了三个测试套件的统计结果。从表 6 可以看出，在 37 个问题中，MACA 在 25 个问题上的表现优于不带 ML 的 MACA。这表明基于流形学习的探索策略有利于提高所获得解集的质量。

Table 6. The IGD and HV results of MACA and MACA without ML.
表 6.MACA 和无 ML MACA 的 IGD 和 HV 结果。

Type 类型	Problem https://linux.do/t/topic/111737	MACA without ML MACA 无 ML	MACA	MACA without ML MACA 无 ML	MACA
Empty Cell	Empty Cell	IGD		HV
I	MW2	3.7066e-3 (3.66e-5) +	4.2988e-3 (1.18e-4)	5.8237e-1 (5.14e-5) +	5.7275e-1 (8.97e-3)
	MW4	4.0763e-2 (3.82e-4) +	4.6969e-2 (5.60e-4)	8.4180e-1 (5.93e-4) +	8.3376e-1 (1.00e-3)
	MW14	1.0190e-1 (3.74e-3) =	1.0256e-1 (2.08e-3)	4.6728e-1 (5.56e-3) =	4.6510e-1 (1.98e-3)
II	MW1	1.6175e-3 (1.41e-5) +	1.6978e-3 (1.38e-5)	4.8981e-1 (1.02e-4) =	4.8984e-1 (3.01e-5)
	MW5	5.2463e-3 (1.92e-3) -	1.4621e-3 (3.95e-4)	3.2170e-1 (1.07e-3) -	3.2419e-1 (1.75e-4)
	MW6	2.7397e-3 (2.45e-5) +	2.7802e-3 (2.24e-5)	3.2848e-1 (1.59e-5) +	3.2747e-1 (3.03e-3)
	MW8	4.3000e-2 (5.50e-4) =	4.3553e-2 (1.59e-3)	5.5200e-1 (9.18e-4) +	5.4549e-1 (4.37e-3)
III	MW3	4.6608e-3 (1.73e-4) =	4.8136e-3 (1.36e-4)	5.4472e-1 (3.99e-4) +	5.4403e-1 (2.27e-4)
	MW7	5.4041e-3 (4.52e-4) -	4.9111e-3 (3.12e-4)	4.1239e-1 (3.08e-4) =	4.1232e-1 (1.33e-4)
	MW10	3.4135e-3 (7.02e-5) +	3.4974e-3 (4.59e-5)	4.5468e-1 (1.67e-4) +	4.5436e-1 (1.41e-4)
	MW13	1.1140e-2 (6.03e-4) +	1.1763e-2 (3.50e-4)	4.7652e-1 (1.18e-4) +	4.7252e-1 (5.62e-3)
IV	MW9	4.8843e-3 (1.50e-4) +	5.1316e-3 (1.78e-4)	3.9800e-1 (1.44e-3) =	3.9718e-1 (2.11e-3)
	MW11	6.9715e-3 (9.94e-4) -	6.2865e-3 (2.36e-4)	4.4664e-1 (7.33e-4) =	4.4671e-1 (3.75e-4)
	MW12	5.0844e-3 (1.82e-4) =	5.0738e-3 (1.11e-4)	6.0418e-1 (2.47e-4) =	6.0388e-1 (3.45e-4)
I	LIR-CMOP5	9.4511e-2 (5.34e-2) -	7.6642e-3 (3.82e-4)	2.6465e-1 (1.17e-2) -	2.8992e-1 (4.13e-4)
	LIR-CMOP6	2.1460e-1 (1.10e-1) -	7.3227e-3 (1.78e-4)	1.1421e-1 (3.09e-2) -	1.9584e-1 (1.48e-4)
	LIR-CMOP13	9.3082e-2 (9.46e-4) +	1.0450e-1 (2.46e-3)	5.5649e-1 (1.35e-3) +	5.3273e-1 (2.89e-3)
	LIR-CMOP14	9.4888e-2 (9.37e-4) +	9.8236e-2 (9.30e-4)	5.5617e-1 (1.49e-3) +	5.4659e-1 (1.97e-3)
II	LIR-CMOP9	3.8206e-1 (4.36e-2) -	5.4303e-2 (1.13e-1)	4.4057e-1 (2.95e-2) =	4.8464e-1 (6.03e-2)
II	LIR-CMOP10	1.0211e-1 (3.07e-2) -	5.8874e-3 (3.16e-4)	6.6195e-1 (1.46e-2) -	7.0579e-1 (2.68e-4)
III	LIR-CMOP11	1.1825e-1 (4.27e-2) -	3.0086e-3 (5.78e-4)	6.2169e-1 (3.29e-2) -	6.9350e-1 (2.29e-4)
III	LIR-CMOP12	1.1930e-1 (2.32e-2) =	7.0618e-2 (5.59e-2)	5.6793e-1 (1.32e-2) =	5.8310e-1 (3.51e-2)
IV	LIR-CMOP1	2.4058e-1 (6.32e-2) =	2.1356e-1 (2.21e-2)	1.3688e-1 (2.49e-2) -	1.7610e-1 (1.91e-2)
	LIR-CMOP2	6.7588e-1 (7.00e-4) -	6.6563e-1 (9.81e-3)	8.7388e-1 (1.99e-4) =	8.7392e-1 (2.85e-4)
	LIR-CMOP3	2.2261e-1 (8.05e-2) -	1.5679e-1 (5.52e-2)	1.2618e-1 (2.21e-2) -	1.6587e-1 (2.09e-2)
	LIR-CMOP4	2.7851e-1 (1.80e-1) -	1.4503e-1 (4.54e-2)	1.9478e-1 (6.58e-2) -	2.8996e-1 (9.08e-3)
	LIR-CMOP7	3.5330e-2 (2.20e-2) -	7.5630e-3 (2.78e-4)	2.8498e-1 (5.05e-3) -	2.9378e-1 (2.62e-4)
	LIR-CMOP8	2.4923e-1 (1.72e-1) -	7.5117e-3 (2.87e-4)	2.4855e-1 (2.99e-2) -	2.9406e-1 (1.75e-4)
	DAS-CMOP1	2.2771e-2 (5.91e-3) -	6.2524e-3 (6.58e-4)	1.9825e-1 (7.13e-3) -	2.0830e-1 (1.94e-3)
	DAS-CMOP2	1.6021e-2 (7.02e-3) -	5.7026e-3 (3.64e-4)	3.4898e-1 (4.07e-3) -	3.5317e-1 (3.84e-4)
	DAS-CMOP3	9.2458e-2 (2.35e-2) -	2.2298e-2 (1.28e-3)	2.9780e-1 (5.33e-3) -	3.0419e-1 (1.46e-2)
	DAS-CMOP4	7.6911e-3 (1.50e-3) =	9.8662e-3 (3.73e-3)	1.9479e-1 (3.96e-3) =	1.9542e-1 (3.33e-3)
	DAS-CMOP5	4.9934e-3 (4.68e-4) +	1.0333e-2 (2.06e-3)	3.4952e-1 (2.71e-4) =	3.4915e-1 (5.42e-4)
	DAS-CMOP6	2.2301e-2 (3.20e-3) =	2.3838e-2 (2.92e-3)	3.0876e-1 (2.30e-3) =	3.0895e-1 (2.15e-3)
	DAS-CMOP7	3.3632e-2 (1.77e-3) +	4.4674e-2 (2.86e-3)	2.8574e-1 (7.24e-4) +	2.8295e-1 (1.36e-3)
	DAS-CMOP8	4.2187e-2 (2.10e-3) +	5.5823e-2 (3.96e-3)	2.0447e-1 (7.80e-4) +	2.0142e-1 (4.14e-3)
	DAS-CMOP9	1.2685e-1 (1.23e-1) -	5.3803e-2 (5.28e-3)	1.8138e-1 (2.85e-2) -	1.9953e-1 (9.22e-4)
Empty Cell	+ / - / =	12/17/8		11/14/12

4.4. Performance of MACA on CMOPs with fraudulent constrains
4.4.MACA 在有欺诈约束的 CMOP 上的性能

To evaluate the performance of MACA in addressing CMOPs with fraudulent constraints (FCPs) characterized by non-monotonous constraint violations and the frequent occurrence of randomly generated individuals in infeasible regions below the CPFs, we conduct experiments on the FCP series [25]. Within this series, constraint violations exhibit non-monotonic patterns, and randomly generated individuals frequently fall in infeasible regions below the CPFs. Table 7 shows the IGD results of MACA in comparison with C-TAEA, PPS, ToP, and ICMA. It is evident that C-TAEA, PPS, and ToP struggle to locate the CPF on FCP1–4, managing to find only a few CPF solutions on FCP5. This implies that CMOPs with deceptive characteristics indeed pose challenges to these algorithms. In contrast, ICMA, which was tailored specifically for the FCP test suite, demonstrates superior performance in handling deceptive problems. Compared with ICMA, MACA exhibits promising performance, outperforming ICMA on FCP 1, FCP3 and FCP4. While MACA achieves better results compared to C-TAEA, PPS, and ToP, it still gets trapped in local optima on the remaining two test functions.
为了评估 MACA 在处理具有欺诈性约束（FCP）的 CMOP 时的性能，我们在 FCP 系列[25]上进行了实验，这些欺诈性约束（FCP）的特点是非单调的约束违反以及随机生成的个体经常出现在 CPF 以下的不可行区域。在这个系列中，违反约束条件的情况呈现出非单调模式，随机生成的个体经常出现在 CPFs 以下的不可行区域。表 7 显示了 MACA 与 C-TAEA、PPS、ToP 和 ICMA 相比的 IGD 结果。很明显，C-TAEA、PPS 和 ToP 都很难在 FCP1-4 上找到 CPF，只能在 FCP5 上找到几个 CPF 解。这意味着具有欺骗性特征的 CMOP 确实给这些算法带来了挑战。相比之下，专为 FCP 测试套件量身定制的 ICMA 在处理欺骗性问题方面表现出色。与 ICMA 相比，MACA 在 FCP 1、FCP3 和 FCP4 上的表现优于 ICMA，表现令人鼓舞。虽然与 C-TAEA、PPS 和 ToP 相比，MACA 取得了更好的结果，但它在其余两个测试功能上仍然陷入了局部最优状态。

Table 7. Comparison results of the IGD averages of compared algorithms on FCP series.
表 7.比较算法的 IGD 平均值在 FCP 系列上的比较结果。

Pro.	C-TAEA	PPS	ToP	ICMA	MACA
FCP1	NaN (NaN) NaN (无)	NaN (NaN) NaN (无)	NaN (NaN) NaN (无)	4.1969e-3 (8.95e-5) -	4.0230e-3 (7.87e-5)
FCP2	NaN (NaN) NaN (无)	NaN (NaN) NaN (无)	NaN (NaN) NaN (无)	3.4663e-3 (2.06e-4) +	1.5605e-1 (1.33e-1)
FCP3	NaN (NaN) NaN (无)	NaN (NaN) NaN (无)	NaN (NaN) NaN (无)	4.5782e-3 (9.90e-5) -	4.3962e-3 (1.46e-4)
FCP4	NaN (NaN) NaN (无)	NaN (NaN) NaN (无)	NaN (NaN) NaN (无)	3.2360e-3 (1.48e-4) -	3.2271e-3 (3.37e-4)
FCP5	2.0167e-1 (4.78e-2) -	1.7095e-1 (1.87e-2) =	1.6702e-1 (1.01e-3) =	5.8171e-3 (1.49e-3) +	1.6581e-1 (9.87e-3)
+ / - / =	0/1/0	0/0/1	1/0/0	2/3/0

5. Conclusion 5.结论

In this study, we propose a novel manifold-assisted coevolutionary algorithm for constrained multi-objective optimization. Our approach incorporates guided feasible search and manifold learning-based exploration and employs a two populations coevolutionary strategy to address the challenges posed by CMOPs. The guided feasible search mechanism facilitates exploration towards the PS, aiding in the discovery of feasible regions and escaping local optima. By utilizing manifold learning, our algorithm effectively spreads the population along the PS manifold, ensuring comprehensive coverage of the final CPF, particularly in segments partitioned due to constraints. The interaction between the primary and secondary populations, coupled with the basic constraint-handling technique CDP, enhances the algorithm's ability to find and cover the global CPF comprehensively. Our proposed algorithm demonstrates a promising ability to handle FCPs, excelling in certain benchmark problems. The combination of guided feasible search and manifold learning-based exploration provides a robust framework for navigating complex feasible regions and achieving better optimization results. Despite its strengths, our study also highlights areas for improvement. The computational complexity and potential scalability issues of the algorithm may affect its applicability in large-scale problems. To address these challenges, future work will focus on exploring additional techniques, such as leveraging parallel computing and implementing algorithmic simplifications, and new optimization methods [64,65] to improve the algorithm's efficiency and robustness. By addressing these issues, we aim to enhance the performance of MACA on more complex CMOPs and ensure its practical applicability to a broader range of optimization applications. For real-world implementation, practical steps are suggested, such as conducting pilot studies in specific industries, collaborating with domain experts, and integrating the algorithm into existing optimization frameworks.
在本研究中，我们提出了一种用于约束多目标优化的新型流形辅助协同进化算法。我们的方法结合了引导可行搜索和基于流形学习的探索，并采用双种群协同进化策略来应对 CMOP 带来的挑战。引导可行搜索机制促进了对 PS 的探索，有助于发现可行区域并摆脱局部最优状态。通过利用流形学习，我们的算法能有效地将种群沿着 PS 流形扩散，确保最终 CPF 的全面覆盖，尤其是在因限制而分割的区段。主种群和次种群之间的相互作用，再加上基本的约束处理技术 CDP，增强了算法找到并全面覆盖全局 CPF 的能力。我们提出的算法在某些基准问题上表现出了处理 FCP 的出色能力。引导可行搜索和基于流形学习的探索相结合，为导航复杂可行区域和实现更好的优化结果提供了一个稳健的框架。尽管有这些优势，我们的研究也强调了需要改进的地方。该算法的计算复杂性和潜在的可扩展性问题可能会影响其在大规模问题中的适用性。为了应对这些挑战，未来的工作将侧重于探索更多技术，如利用并行计算和实现算法简化，以及新的优化方法[64,65]，以提高算法的效率和鲁棒性。通过解决这些问题，我们旨在提高 MACA 在更复杂的 CMOP 上的性能，并确保其实际适用于更广泛的优化应用。对于现实世界的实施，我们提出了切实可行的步骤，如在特定行业开展试点研究、与领域专家合作以及将算法集成到现有优化框架中。

CRediT authorship contribution statement
CRediT 作者贡献声明

Weiwei Zhang: Methodology, Formal analysis. Jiaxin Yang: Writing – original draft, Software. Guoqing Li: Validation, Resources. Weizheng Zhang: Supervision, Project administration. Gary G. Yen: Writing – review & editing, Supervision, Conceptualization.
Weiwei Zhang：方法论、形式分析杨佳欣写作 - 原稿、软件。李国庆：验证、资源。张维政监督、项目管理。Gary G. Yen：写作--审阅和编辑、监督、构思。

Declaration of competing interest
利益冲突声明

We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.
我们谨此确认，本出版物不存在任何已知的利益冲突，本工作也未获得任何可能影响其结果的重大资金支持。

We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us.
我们确认手稿已由所有署名作者阅读并批准，没有其他符合作者标准但未列名的人员。我们还确认，手稿中列出的作者顺序已得到我们所有人的认可。

Acknowledgment 鸣谢

This work is supported by the Natural Science Foundation of Henan Province of China (242300421413), the National Natural Science Foundation of China (No. 62106230), the China Postdoctoral Science Foundation (2021T140616, 2021M692920), Zhejiang Provincial Natural Science Foundation of China under Grant (No. Q23F030020), and the Fundamental Research Funds for the Provincial Universities of Zhejiang (SJLY2023010), and the Humanities and Social Science Fund of Ministry of Education.
本研究得到河南省自然科学基金（242300421413）、国家自然科学基金（62106230）、中国博士后科学基金（2021T140616、2021M692920）、浙江省自然科学基金（Q23F030020）、浙江省高校基本科研业务费（SJLY2023010）和教育部人文社科基金的资助。

Data availability 数据可用性

Data will be made available on request.
数据将应要求提供。

References

[1]
J. Liang, et al.
A survey on evolutionary constrained multiobjective optimization
IEEE Trans. Evol. Comput., 27 (2) (Apr. 2023), pp. 201-221
Crossref View in Scopus Google Scholar
[2]
T. Sang-To, H. Le-Minh, M. Wahab, C. Thanh
A new metaheuristic algorithm: Shrimp and Goby association search algorithm and its application for damage identification in large-scale and complex structures
Advances in Engineering Software, 176 (103363) (2023)
Google Scholar
[3]
K. Qiao, J. Liang, K. Yu, M. Yuan, B. Qu, C. Yue
Self-adaptive resources allocation-based differential evolution for constrained evolutionary optimization
Knowl. Based Syst., 235 (Jan. 2022)
Art. no. 107653
Google Scholar
[4]
X. Zuo, C. Chen, W. Tan, M. Zhou
Vehicle scheduling of an urban bus line via an improved multiobjective genetic algorithm
IEEE Trans. Intell. Transp. Syst., 16 (2) (Apr. 2015), pp. 1030-1041
View in Scopus Google Scholar
[5]
X. Yu, X. Yu, Y. Lu, G.G. Yen, M. Cai
Differential evolution mutation operators for constrained multi-objective optimization
Appl. Soft Comput., 67 (Jun. 2018), pp. 452-466
View PDF View article View in Scopus Google Scholar
[6]
Z. Liu, F. Wu, J. Liu, Y. Qin and K. Li, “Constrained multiobjective optimization with escape and expansion forces,” IEEE Trans. Evol. Comput., 10.1109/TEVC.2023.3270483.
Google Scholar
[7]
J. Liang, et al.
Utilizing the relationship between unconstrained and constrained pareto fronts for constrained multiobjective optimization
IEEE Trans. Cybern., 53 (6) (Jun 2023), pp. 3873-3886, 10.1109/TCYB.2022.3163759
View in Scopus Google Scholar
[8]
J. Zou et al., “A multi-population evolutionary algorithm using new cooperative mechanism for solving multi-objective problems with multi-constraint,” IEEE Trans. Evol. Comput., 10.1109/TEVC.2023.3260306.
Google Scholar
[9]
K. Deb, A. Pratap, S. Agarwal, T. Meyarivan
A fast and elitist multiobjective genetic algorithm: NSGA-II
IEEE Trans. Evol. Comput., 6 (2) (Apr. 2002), pp. 182-197
View in Scopus Google Scholar
[10]
Y.G. Woldesenbet, G.G. Yen, B.G. Tessema
Constraint handling in multi-objective evolutionary optimization
IEEE Trans. Evol. Comput., 13 (3) (Jun. 2009), pp. 514-525, 10.1109/TEVC.2008.2009032
View in Scopus Google Scholar
[11]
T. Takahama, S. Sakai
Constrained optimization by the ϵ constrained differential evolution with gradient-based mutation and feasible elites
Proc. IEEE Int. Conf. Evol. Comput., Vancouver, BC, Canada (2006), pp. 1-8
View in Scopus Google Scholar
[12]
Y. Li, H. Minh, M. Cao, X. Qian, M. Wahab
An integrated surrogate model-driven and improved termite life cycle optimizer for damage identification in dams
Mech. Syst. Signal. Process., 208 (110986) (2024)
Google Scholar
[13]
T.P. Runarsson, X. Yao
Stochastic ranking for constrained evolutionary optimization
IEEE Trans. Evol. Comput., 4 (3) (Sep. 2000), pp. 284-294
View in Scopus Google Scholar
[14]
Y. Tian, T. Zhang, J. Xiao, X. Zhang, Y. Jin
A coevolutionary framework for constrained multiobjective optimization problems
IEEE Trans. Evol. Comput., 25 (1) (Feb. 2021), pp. 102-116, 10.1109/TEVC.2020.3004012
View in Scopus Google Scholar
[15]
K. Li, R. Chen, G. Fu, X. Yao
Two-archive evolutionary algorithm for constrained multiobjective optimization
IEEE Trans. Evol. Comput., 23 (2) (Apr. 2019), pp. 303-315
Crossref View in Scopus Google Scholar
[16]
Z. Fan, et al.
Push and pull search for solving constrained multi-objective optimization problems
Swarm Evol. Comput., 44 (Feb. 2019), pp. 665-679
View PDF View article View in Scopus Google Scholar
[17]
Y. Tian, Y. Zhang, Y. Su, X. Zhang, K.C. Tan, Y. Jin
Balancing objective optimization and constraint satisfaction in constrained evolutionary multiobjective optimization
IEEE Trans. Cybern., 52 (9) (Sept. 2022), pp. 9559-9572, 10.1109/TCYB.2020.3021138
View in Scopus Google Scholar
[18]
M. Xia, M. Dong
A novel two-archive evolutionary algorithm for constrained multi-objective optimization with small feasible regions
Knowl. Based Syst., 237 (Feb. 2022)
Art. no. 107693
Google Scholar
[19]
F. Jimenez, A.F. Gomez-Skarmeta, G. Sanchez, K. Deb
An evolutionary algorithm for constrained multi-objective optimization
Proc. IEEE Congr. Evol. Comput. (CEC), 2, Honolulu, HI, USA (2002), pp. 1133-1138, 10.1109/CEC.2002.1004402
View in Scopus Google Scholar
[20]
Z. Sun, et al.
An evolutionary algorithm with constraint relaxation strategy for highly constrained multiobjective optimization
IEEE Trans. Cybern., 53 (5) (May. 2023), pp. 3190-3204, 10.1109/TCYB.2022.3151974
View in Scopus Google Scholar
[21]
Z. Ma, Y. Wang
Shift-based penalty for evolutionary constrained multiobjective optimization and its application
IEEE Trans. Cybern. (May 25, 2021), 10.1109/TCYB.2021.3069814
Google Scholar
[22]
Q. Zhu, Q. Zhang, Q. Lin
A constrained multiobjective evolutionary algorithm with detect-and-escape strategy
IEEE Trans. Evol. Comput., 24 (5) (Oct. 2020), pp. 938-947
Crossref View in Scopus Google Scholar
[23]
B. Qu, P.N. Suganthan
Constrained multi-objective optimization algorithm with diversity enhanced differential evolution
Proc. IEEE Congr. Evol. Comput. (CEC), Barcelona, Spain (2010), pp. 1-5, 10.1109/CEC.2010.5585947
Google Scholar
[24]
Q. Zhang, A. Zhou, Y. Jin
RM-MEDA: A regularity model-based multiobjective estimation of distribution algorithm
IEEE Trans. Evol. Comput., 12 (1) (Feb. 2008), pp. 41-63, 10.1109/TEVC.2007.894202
View in Scopus Google Scholar
[25]
J. Yuan, H. Liu, Y. Ong, Z. He
Indicator-Based Evolutionary Algorithm for Solving Constrained Multiobjective Optimization Problems
IEEE Trans. Evol. Comput., 26 (2) (2022), pp. 379-391
Google Scholar
[26]
M.A. Jan, Q. Zhang
MOEA/D for constrained multiobjective optimization: Some preliminary experimental results
Proc. U.K. Workshop Comput. Intell. (UKCI), Colchester, U.K. (2010), pp. 1-6
Crossref Google Scholar
[27]
M.A. Jan, N. Tairan, R.A. Khanum
Threshold based dynamic and adaptive penalty functions for constrained multiobjective optimization
Proc. 1st Int. Conf. Artif. Intell. Model. Simul., Kota Kinabalu, Malaysia (2013), pp. 49-54
Crossref View in Scopus Google Scholar
[28]
H. Zhang, Y. Zhu, X. Yan
Multi-hive artificial bee colony algorithm for constrained multi-objective optimization
Proc. IEEE Congr. Evol. Comput. (CEC), Brisbane, QLD, Australia (2012), pp. 1-8, 10.1109/CEC.2012.6256499
View PDF View article Google Scholar
[29]
Z. Fan, et al.
A learning guided parameter setting for constrained multi-objective optimization
Proc. 1st Int. Conf. Ind. Artif. Intell. (IAI), Shenyang, China (2019), pp. 1-6
Google Scholar
[30]
W. Zhang, G.G. Yen, Z. He
Constrained optimization via artificial immune system
IEEE Trans. Cybern., 44 (2) (Feb. 2014), pp. 185-198, 10.1109/TCYB.2013.2250956
View in Scopus Google Scholar
[31]
H. Geng, M. Zhang, L. Huang, X. Wang
Infeasible elitists and stochastic ranking selection in constrained evolutionary multi-objective optimization
Proc. Asia–Pacific Conf. Simul. Evol. Learn. (2006), pp. 336-344
Crossref View in Scopus Google Scholar
[32]
W. Ying, W. He, Y. Huang, D. Li, Y. Wu
An adaptive stochastic ranking mechanism in MOEA/D for constrained multiobjective optimization
Proc. Int. Conf. Inf. Syst. Artif. Intell. (ISAI), Hong Kong (2016), pp. 514-518
View in Scopus Google Scholar
[33]
Z.Z. Liu, Y. Wang
Handling constrained multiobjective optimization problems with constraints in both the decision and objective spaces
IEEE Trans. Evol. Comput., 23 (5) (Oct. 2019), pp. 870-884
Crossref Google Scholar
[34]
Y. Xiang, X.Y. ang, H. Huang, J. Wang
Balancing constraints and objectives by considering problem types in constrained multiobjective optimization
IEEE Trans. Cybern. (Jul. 8, 2021), 10.1109/TCYB.2021.3089633
Google Scholar
[35]
K. Zhang, Z. Xu, G.G. Yen, L. Zhang
Two-stage multi-objective evolution strategy for constrained multi-objective optimization
IEEE Trans. Evol. Comput. (2022), 10.1109/TEVC.2022.3202723
Google Scholar
[36]
J. Wang, Y. Li, Q. Zhang, Z. Zhang, S. Gao
Cooperative multiobjective evolutionary algorithm with propulsive population for constrained multiobjective optimization
IEEE Trans. Syst., Man, Cybern., Syst (Apr. 9, 2021), 10.1109/TSMC.2021.3069986
Google Scholar
[37]
K. Yu, J. Liang, B. Qu, Y. Luo, C. Yue
Dynamic selection preference-assisted constrained multiobjective differential evolution
IEEE Trans. Syst., Man, Cybern., Syst (Mar. 9, 2021), 10.1109/TSMC.2021.3061698
Google Scholar
[38]
X. Li, S. Zeng, S. Qin, K. Liu
Constrained optimization problem solved by dynamic constrained NSGA-III multiobjective optimizational techniques
Proc. IEEE Congr. Evol. Comput. (CEC)Sendai, Japan (2015), pp. 2923-2928, 10.1109/CEC.2015.7257252
View in Scopus Google Scholar
[39]
K. Qiao, et al.
Dynamic auxiliary task-based evolutionary multitasking for constrained multiobjective optimization
IEEE Trans. Evol. Comput., 27 (3) (June 2023), pp. 642-656, 10.1109/TEVC.2022.3175065
View in Scopus Google Scholar
[40]
R. Jiao, S. Zeng, C. Li, W. Pedrycz
Evolutionary constrained multi-objective optimization using NSGA-II with dynamic constraint handling
Proc. IEEE Congr. Evol. Comput. (CEC), Wellington, New Zealand (2019), pp. 1634-1641, 10.1109/CEC.2019.8790172
View in Scopus Google Scholar
[41]
F. Ming, W. Gong, L. Wang and L. Gao, “Constrained multi-objective optimization via multitasking and knowledge transfer,” IEEE Trans. Evol. Comput., 10.1109/TEVC.2022.3230822.
Google Scholar
[42]
W. Zhang, S. Wang, A. Zhou, H. Zhang
A practical regularity model based evolutionary algorithm for multiobjective optimization
Appl. Soft Comput., 129 (Nov. 2022)
Art. no. 109614
Google Scholar
[43]
Y. Zhang, G. Dai, L. Peng, M. Wang
HMOEDA_LLE: A hybrid multi-objective estimation of distribution algorithm combining locally linear embedding
Proc. IEEE Congr. Evol. Comput. (CEC), Beijing, China (2014), pp. 707-714, 10.1109/CEC.2014.6900248
View in Scopus Google Scholar
[44]
P. Tino, I. Nabney
Hierarchical GTM: constructing localized nonlinear projection manifolds in a principled way
IEEE Trans. Pattern Anal. Mach. Intell. (PAMI), 24 (5) (May 2002), pp. 639-656, 10.1109/34.1000238
View in Scopus Google Scholar
[45]
Z. Wang, H. Hong, K. Ye, G. Zhang, M. Jiang, K.C. Tan
Manifold interpolation for large-scale multiobjective optimization via generative adversarial networks
IEEE Trans. Neural Netw. Learn. Syst. (TNNLS), 34 (8) (Aug. 2023), pp. 4631-4645, 10.1109/TNNLS.2021.3113158
View in Scopus Google Scholar
[46]
H. Jain, 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 Trans. Evol. Comput., 18 (4) (Aug. 2014), pp. 602-622
View in Scopus Google Scholar
[47]
Z. Fan, Y. Fang, W. Li, X. Cai, C. Wei, E. Goodman
MOEA/D with angle-based constrained dominance principle for constrained multi-objective optimization problems
Appl. Soft Comput., 74 (Jan. 2019), pp. 621-633
View PDF View article View in Scopus Google Scholar
[48]
C. He, R. Cheng, Y. Tian, X. Zhang, K.C. Tan, Y. Jin
Paired offspring generation for constrained large-scale multiobjective optimization
IEEE Trans. Evol. Comput., 25 (3) (Jun. 2021), pp. 448-462
Crossref View in Scopus Google Scholar
[49]
E. Zitzler, M. Laumanns, L. Thiele
SPEA2: Improving the strength Pareto evolutionary algorithm for multiobjective optimization
Proc. 5th Conf. Evol. Methods Design Optim. Control Appl. Ind. Prob. (2001), pp. 95-100
View in Scopus Google Scholar
[50]
C. He, et al.
Accelerating large-scale multiobjective optimization via problem reformulation
IEEE Trans. Evol. Comput., 23 (6) (Dec. 2019), pp. 949-961
Crossref View in Scopus Google Scholar
[51]
C. He, R. Cheng, Y. Tian, X. Zhang
Iterated problem reformulation for evolutionary large-scale multiobjective optimization
Proc. IEEE Congr. Evol. Comput. (CEC), Glasgow, UK (2020), pp. 1-8, 10.1109/CEC48606.2020.9185553
Google Scholar
[52]
L. Li, C. He, R. Cheng, L. Pan
Large-scale multiobjective optimization via problem decomposition and reformulation
Proc. IEEE Congr. Evol. Comput. (CEC), Kraków, Poland (2021), pp. 2149-2155, 10.1109/CEC45853.2021.9504820
View in Scopus Google Scholar
[53]
H. Gu, H. Wang, Y. Jin
Effects of pareto set on the performance of problem reformulation-based large-scale multiobjective optimization algorithms
Proc. IEEE Congr. Evol. Comput. (CEC), Chicago, IL, USA (2023), pp. 1-8, 10.1109/CEC53210.2023.10254122
Google Scholar
[54]
S. Qin, C. Sun, Y. Jin, Y. Tan, J. Fieldsend
Large-scale evolutionary multiobjective optimization assisted by directed sampling
IEEE Trans. Evol. Comput., 25 (4) (Aug. 2021), pp. 724-738, 10.1109/TEVC.2021.3063606
View in Scopus Google Scholar
[55]
Y. Tian, R. Cheng, X. Zhang, Y. Jin
PlatEMO: A MATLAB platform for evolutionary multi-objective optimization
IEEE Comput. Intell. Mag., 12 (4) (Nov. 2017), pp. 73-87
View in Scopus Google Scholar
[56]
Z. Ma, Y. Wang
Evolutionary constrained multiobjective optimization: Test suite construction and performance comparisons
IEEE Trans. Evol. Comput., 23 (6) (Dec. 2019), pp. 972-986
Crossref View in Scopus Google Scholar
[57]
Z. Fan, et al.
An improved epsilon constraint-handling method in MOEA/D for CMOPs with large infeasible regions
Soft. comput., 23 (23) (2019), pp. 12491-12510
Crossref View in Scopus Google Scholar
[58]
Z. Fan, et al.
Difficulty adjustable and scalable constrained multiobjective test problem toolkit
Evol. Comput., 28 (3) (Sep. 2020), pp. 339-378
Crossref Google Scholar
[59]
Y. Tian, X. Xiang, X. Zhang, R. Cheng, Y. Jin
Sampling reference points on the Pareto fronts of benchmark multi-objective optimization problems
Proc. IEEE Congr. Evol. Comput. (CEC), Rio de Janeiro, Brazil (2018), pp. 1-6, 10.1109/CEC.2018.8477730
Google Scholar
[60]
L. While, P. Hingston, L. Barone, S. Huband
A faster algorithm for calculating hypervolume
IEEE Trans. Evol. Comput., 10 (1) (Feb. 2006), pp. 29-38
View in Scopus Google Scholar
[61]
F. Ming, W. Gong, Y. Jin
Even search in a promising region for constrained multi-objective optimization
IEEE-CAA J. Automatic., 11 (2) (Feb. 2024), pp. 474-486, 10.1109/JAS.2023.123792
View in Scopus Google Scholar
[62]
R. Jiao, B. Xue, M. Zhang
A multiform optimization framework for constrained multiobjective optimization
IEEE Trans. Cybern., 53 (8) (Aug. 2023), pp. 5165-5177, 10.1109/TCYB.2022.3178132
View in Scopus Google Scholar
[63]
R. Jiao, S. Zeng, C. Li, Y.S. Ong
Two-type weight adjustments in MOEA/D for highly constrained many-objective optimization
Inf. Sci., 578 (Nov. 2021), pp. 592-614
View PDF View article View in Scopus Google Scholar
[64]
J.O. Agushaka, et al.
Greater cane rat algorithm (GCRA): A nature-inspired metaheuristic for optimization problems
Heliyon., 10 (11) (2024)
Google Scholar
[65]
M. Ghasemi, et al.
Geyser Inspired Algorithm: A New Geological-inspired Meta-heuristic for Real-parameter and Constrained Engineering Optimization
J. Bionic. Eng., 21 (2024), pp. 374-408
Crossref View in Scopus Google Scholar

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[1] [1]
J. Liang, et al.
A survey on evolutionary constrained multiobjective optimization
IEEE Trans. Evol. Comput., 27 (2) (Apr. 2023), pp. 201-221
Crossref View in Scopus Google Scholar

[2] [2]
T. Sang-To, H. Le-Minh, M. Wahab, C. Thanh
A new metaheuristic algorithm: Shrimp and Goby association search algorithm and its application for damage identification in large-scale and complex structures
Advances in Engineering Software, 176 (103363) (2023)
Google Scholar

[3] [3]
K. Qiao, J. Liang, K. Yu, M. Yuan, B. Qu, C. Yue
Self-adaptive resources allocation-based differential evolution for constrained evolutionary optimization
Knowl. Based Syst., 235 (Jan. 2022)
Art. no. 107653
Google Scholar

[4] [4]
X. Zuo, C. Chen, W. Tan, M. Zhou
Vehicle scheduling of an urban bus line via an improved multiobjective genetic algorithm
IEEE Trans. Intell. Transp. Syst., 16 (2) (Apr. 2015), pp. 1030-1041
View in Scopus Google Scholar

[5] [5]
X. Yu, X. Yu, Y. Lu, G.G. Yen, M. Cai
Differential evolution mutation operators for constrained multi-objective optimization
Appl. Soft Comput., 67 (Jun. 2018), pp. 452-466
View PDF View article View in Scopus Google Scholar

[6] [6]
Z. Liu, F. Wu, J. Liu, Y. Qin and K. Li, “Constrained multiobjective optimization with escape and expansion forces,” IEEE Trans. Evol. Comput., 10.1109/TEVC.2023.3270483.
Google Scholar

[7] [7]
J. Liang, et al.
Utilizing the relationship between unconstrained and constrained pareto fronts for constrained multiobjective optimization
IEEE Trans. Cybern., 53 (6) (Jun 2023), pp. 3873-3886, 10.1109/TCYB.2022.3163759
View in Scopus Google Scholar

[8] [8]
J. Zou et al., “A multi-population evolutionary algorithm using new cooperative mechanism for solving multi-objective problems with multi-constraint,” IEEE Trans. Evol. Comput., 10.1109/TEVC.2023.3260306.
Google Scholar

[9] [9]
K. Deb, A. Pratap, S. Agarwal, T. Meyarivan
A fast and elitist multiobjective genetic algorithm: NSGA-II
IEEE Trans. Evol. Comput., 6 (2) (Apr. 2002), pp. 182-197
View in Scopus Google Scholar

[10] [10]
Y.G. Woldesenbet, G.G. Yen, B.G. Tessema
Constraint handling in multi-objective evolutionary optimization
IEEE Trans. Evol. Comput., 13 (3) (Jun. 2009), pp. 514-525, 10.1109/TEVC.2008.2009032
View in Scopus Google Scholar

[11] [11]
T. Takahama, S. Sakai
Constrained optimization by the ϵ constrained differential evolution with gradient-based mutation and feasible elites
Proc. IEEE Int. Conf. Evol. Comput., Vancouver, BC, Canada (2006), pp. 1-8
View in Scopus Google Scholar

[12] [12]
Y. Li, H. Minh, M. Cao, X. Qian, M. Wahab
An integrated surrogate model-driven and improved termite life cycle optimizer for damage identification in dams
Mech. Syst. Signal. Process., 208 (110986) (2024)
Google Scholar

[13] [13]
T.P. Runarsson, X. Yao
Stochastic ranking for constrained evolutionary optimization
IEEE Trans. Evol. Comput., 4 (3) (Sep. 2000), pp. 284-294
View in Scopus Google Scholar

[14] [14]
Y. Tian, T. Zhang, J. Xiao, X. Zhang, Y. Jin
A coevolutionary framework for constrained multiobjective optimization problems
IEEE Trans. Evol. Comput., 25 (1) (Feb. 2021), pp. 102-116, 10.1109/TEVC.2020.3004012
View in Scopus Google Scholar

[15] [15]
K. Li, R. Chen, G. Fu, X. Yao
Two-archive evolutionary algorithm for constrained multiobjective optimization
IEEE Trans. Evol. Comput., 23 (2) (Apr. 2019), pp. 303-315
Crossref View in Scopus Google Scholar

[16] [16]
Z. Fan, et al.
Push and pull search for solving constrained multi-objective optimization problems
Swarm Evol. Comput., 44 (Feb. 2019), pp. 665-679
View PDF View article View in Scopus Google Scholar

[17] [17]
Y. Tian, Y. Zhang, Y. Su, X. Zhang, K.C. Tan, Y. Jin
Balancing objective optimization and constraint satisfaction in constrained evolutionary multiobjective optimization
IEEE Trans. Cybern., 52 (9) (Sept. 2022), pp. 9559-9572, 10.1109/TCYB.2020.3021138
View in Scopus Google Scholar

[18] [18]
M. Xia, M. Dong
A novel two-archive evolutionary algorithm for constrained multi-objective optimization with small feasible regions
Knowl. Based Syst., 237 (Feb. 2022)
Art. no. 107693
Google Scholar

[19] [19]
F. Jimenez, A.F. Gomez-Skarmeta, G. Sanchez, K. Deb
An evolutionary algorithm for constrained multi-objective optimization
Proc. IEEE Congr. Evol. Comput. (CEC), 2, Honolulu, HI, USA (2002), pp. 1133-1138, 10.1109/CEC.2002.1004402
View in Scopus Google Scholar

[20] [20]
Z. Sun, et al.
An evolutionary algorithm with constraint relaxation strategy for highly constrained multiobjective optimization
IEEE Trans. Cybern., 53 (5) (May. 2023), pp. 3190-3204, 10.1109/TCYB.2022.3151974
View in Scopus Google Scholar

[21] [21]
Z. Ma, Y. Wang
Shift-based penalty for evolutionary constrained multiobjective optimization and its application
IEEE Trans. Cybern. (May 25, 2021), 10.1109/TCYB.2021.3069814
Google Scholar

[22] [22]
Q. Zhu, Q. Zhang, Q. Lin
A constrained multiobjective evolutionary algorithm with detect-and-escape strategy
IEEE Trans. Evol. Comput., 24 (5) (Oct. 2020), pp. 938-947
Crossref View in Scopus Google Scholar

[23] [23]
B. Qu, P.N. Suganthan
Constrained multi-objective optimization algorithm with diversity enhanced differential evolution
Proc. IEEE Congr. Evol. Comput. (CEC), Barcelona, Spain (2010), pp. 1-5, 10.1109/CEC.2010.5585947
Google Scholar

[24] [24]
Q. Zhang, A. Zhou, Y. Jin
RM-MEDA: A regularity model-based multiobjective estimation of distribution algorithm
IEEE Trans. Evol. Comput., 12 (1) (Feb. 2008), pp. 41-63, 10.1109/TEVC.2007.894202
View in Scopus Google Scholar

[25] [25]
J. Yuan, H. Liu, Y. Ong, Z. He
Indicator-Based Evolutionary Algorithm for Solving Constrained Multiobjective Optimization Problems
IEEE Trans. Evol. Comput., 26 (2) (2022), pp. 379-391
Google Scholar

[26] [26]
M.A. Jan, Q. Zhang
MOEA/D for constrained multiobjective optimization: Some preliminary experimental results
Proc. U.K. Workshop Comput. Intell. (UKCI), Colchester, U.K. (2010), pp. 1-6
Crossref Google Scholar

[27] [27]
M.A. Jan, N. Tairan, R.A. Khanum
Threshold based dynamic and adaptive penalty functions for constrained multiobjective optimization
Proc. 1st Int. Conf. Artif. Intell. Model. Simul., Kota Kinabalu, Malaysia (2013), pp. 49-54
Crossref View in Scopus Google Scholar

[28] [28]
H. Zhang, Y. Zhu, X. Yan
Multi-hive artificial bee colony algorithm for constrained multi-objective optimization
Proc. IEEE Congr. Evol. Comput. (CEC), Brisbane, QLD, Australia (2012), pp. 1-8, 10.1109/CEC.2012.6256499
View PDF View article Google Scholar

[29] [29]
Z. Fan, et al.
A learning guided parameter setting for constrained multi-objective optimization
Proc. 1st Int. Conf. Ind. Artif. Intell. (IAI), Shenyang, China (2019), pp. 1-6
Google Scholar

[30] [30]
W. Zhang, G.G. Yen, Z. He
Constrained optimization via artificial immune system
IEEE Trans. Cybern., 44 (2) (Feb. 2014), pp. 185-198, 10.1109/TCYB.2013.2250956
View in Scopus Google Scholar

[31] [31]
H. Geng, M. Zhang, L. Huang, X. Wang
Infeasible elitists and stochastic ranking selection in constrained evolutionary multi-objective optimization
Proc. Asia–Pacific Conf. Simul. Evol. Learn. (2006), pp. 336-344
Crossref View in Scopus Google Scholar

[32] [32]
W. Ying, W. He, Y. Huang, D. Li, Y. Wu
An adaptive stochastic ranking mechanism in MOEA/D for constrained multiobjective optimization
Proc. Int. Conf. Inf. Syst. Artif. Intell. (ISAI), Hong Kong (2016), pp. 514-518
View in Scopus Google Scholar

[33] [33]
Z.Z. Liu, Y. Wang
Handling constrained multiobjective optimization problems with constraints in both the decision and objective spaces
IEEE Trans. Evol. Comput., 23 (5) (Oct. 2019), pp. 870-884
Crossref Google Scholar

[34] [34]
Y. Xiang, X.Y. ang, H. Huang, J. Wang
Balancing constraints and objectives by considering problem types in constrained multiobjective optimization
IEEE Trans. Cybern. (Jul. 8, 2021), 10.1109/TCYB.2021.3089633
Google Scholar

[35] [35]
K. Zhang, Z. Xu, G.G. Yen, L. Zhang
Two-stage multi-objective evolution strategy for constrained multi-objective optimization
IEEE Trans. Evol. Comput. (2022), 10.1109/TEVC.2022.3202723
Google Scholar

[36] [36]
J. Wang, Y. Li, Q. Zhang, Z. Zhang, S. Gao
Cooperative multiobjective evolutionary algorithm with propulsive population for constrained multiobjective optimization
IEEE Trans. Syst., Man, Cybern., Syst (Apr. 9, 2021), 10.1109/TSMC.2021.3069986
Google Scholar

[37] [37]
K. Yu, J. Liang, B. Qu, Y. Luo, C. Yue
Dynamic selection preference-assisted constrained multiobjective differential evolution
IEEE Trans. Syst., Man, Cybern., Syst (Mar. 9, 2021), 10.1109/TSMC.2021.3061698
Google Scholar

[38] [38]
X. Li, S. Zeng, S. Qin, K. Liu
Constrained optimization problem solved by dynamic constrained NSGA-III multiobjective optimizational techniques
Proc. IEEE Congr. Evol. Comput. (CEC)Sendai, Japan (2015), pp. 2923-2928, 10.1109/CEC.2015.7257252
View in Scopus Google Scholar

[39] [39]
K. Qiao, et al.
Dynamic auxiliary task-based evolutionary multitasking for constrained multiobjective optimization
IEEE Trans. Evol. Comput., 27 (3) (June 2023), pp. 642-656, 10.1109/TEVC.2022.3175065
View in Scopus Google Scholar

[40] [40]
R. Jiao, S. Zeng, C. Li, W. Pedrycz
Evolutionary constrained multi-objective optimization using NSGA-II with dynamic constraint handling
Proc. IEEE Congr. Evol. Comput. (CEC), Wellington, New Zealand (2019), pp. 1634-1641, 10.1109/CEC.2019.8790172
View in Scopus Google Scholar

[41] [41]
F. Ming, W. Gong, L. Wang and L. Gao, “Constrained multi-objective optimization via multitasking and knowledge transfer,” IEEE Trans. Evol. Comput., 10.1109/TEVC.2022.3230822.
Google Scholar

[42] [42]
W. Zhang, S. Wang, A. Zhou, H. Zhang
A practical regularity model based evolutionary algorithm for multiobjective optimization
Appl. Soft Comput., 129 (Nov. 2022)
Art. no. 109614
Google Scholar

[43] [43]
Y. Zhang, G. Dai, L. Peng, M. Wang
HMOEDA_LLE: A hybrid multi-objective estimation of distribution algorithm combining locally linear embedding
Proc. IEEE Congr. Evol. Comput. (CEC), Beijing, China (2014), pp. 707-714, 10.1109/CEC.2014.6900248
View in Scopus Google Scholar

[44] [44]
P. Tino, I. Nabney
Hierarchical GTM: constructing localized nonlinear projection manifolds in a principled way
IEEE Trans. Pattern Anal. Mach. Intell. (PAMI), 24 (5) (May 2002), pp. 639-656, 10.1109/34.1000238
View in Scopus Google Scholar

[45] [45]
Z. Wang, H. Hong, K. Ye, G. Zhang, M. Jiang, K.C. Tan
Manifold interpolation for large-scale multiobjective optimization via generative adversarial networks
IEEE Trans. Neural Netw. Learn. Syst. (TNNLS), 34 (8) (Aug. 2023), pp. 4631-4645, 10.1109/TNNLS.2021.3113158
View in Scopus Google Scholar

[46] [46]
H. Jain, 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 Trans. Evol. Comput., 18 (4) (Aug. 2014), pp. 602-622
View in Scopus Google Scholar

[47] [47]
Z. Fan, Y. Fang, W. Li, X. Cai, C. Wei, E. Goodman
MOEA/D with angle-based constrained dominance principle for constrained multi-objective optimization problems
Appl. Soft Comput., 74 (Jan. 2019), pp. 621-633
View PDF View article View in Scopus Google Scholar

[48] [48]
C. He, R. Cheng, Y. Tian, X. Zhang, K.C. Tan, Y. Jin
Paired offspring generation for constrained large-scale multiobjective optimization
IEEE Trans. Evol. Comput., 25 (3) (Jun. 2021), pp. 448-462
Crossref View in Scopus Google Scholar

[49] [49]
E. Zitzler, M. Laumanns, L. Thiele
SPEA2: Improving the strength Pareto evolutionary algorithm for multiobjective optimization
Proc. 5th Conf. Evol. Methods Design Optim. Control Appl. Ind. Prob. (2001), pp. 95-100
View in Scopus Google Scholar

[50] [50]
C. He, et al.
Accelerating large-scale multiobjective optimization via problem reformulation
IEEE Trans. Evol. Comput., 23 (6) (Dec. 2019), pp. 949-961
Crossref View in Scopus Google Scholar

[51] [51]
C. He, R. Cheng, Y. Tian, X. Zhang
Iterated problem reformulation for evolutionary large-scale multiobjective optimization
Proc. IEEE Congr. Evol. Comput. (CEC), Glasgow, UK (2020), pp. 1-8, 10.1109/CEC48606.2020.9185553
Google Scholar

[52] [52]
L. Li, C. He, R. Cheng, L. Pan
Large-scale multiobjective optimization via problem decomposition and reformulation
Proc. IEEE Congr. Evol. Comput. (CEC), Kraków, Poland (2021), pp. 2149-2155, 10.1109/CEC45853.2021.9504820
View in Scopus Google Scholar

[53] [53]
H. Gu, H. Wang, Y. Jin
Effects of pareto set on the performance of problem reformulation-based large-scale multiobjective optimization algorithms
Proc. IEEE Congr. Evol. Comput. (CEC), Chicago, IL, USA (2023), pp. 1-8, 10.1109/CEC53210.2023.10254122
Google Scholar

[54] [54]
S. Qin, C. Sun, Y. Jin, Y. Tan, J. Fieldsend
Large-scale evolutionary multiobjective optimization assisted by directed sampling
IEEE Trans. Evol. Comput., 25 (4) (Aug. 2021), pp. 724-738, 10.1109/TEVC.2021.3063606
View in Scopus Google Scholar

[55] [55]
Y. Tian, R. Cheng, X. Zhang, Y. Jin
PlatEMO: A MATLAB platform for evolutionary multi-objective optimization
IEEE Comput. Intell. Mag., 12 (4) (Nov. 2017), pp. 73-87
View in Scopus Google Scholar

[56] [56]
Z. Ma, Y. Wang
Evolutionary constrained multiobjective optimization: Test suite construction and performance comparisons
IEEE Trans. Evol. Comput., 23 (6) (Dec. 2019), pp. 972-986
Crossref View in Scopus Google Scholar

[57] [57]
Z. Fan, et al.
An improved epsilon constraint-handling method in MOEA/D for CMOPs with large infeasible regions
Soft. comput., 23 (23) (2019), pp. 12491-12510
Crossref View in Scopus Google Scholar

[58] [58]
Z. Fan, et al.
Difficulty adjustable and scalable constrained multiobjective test problem toolkit
Evol. Comput., 28 (3) (Sep. 2020), pp. 339-378
Crossref Google Scholar

[59] [59]
Y. Tian, X. Xiang, X. Zhang, R. Cheng, Y. Jin
Sampling reference points on the Pareto fronts of benchmark multi-objective optimization problems
Proc. IEEE Congr. Evol. Comput. (CEC), Rio de Janeiro, Brazil (2018), pp. 1-6, 10.1109/CEC.2018.8477730
Google Scholar

[60] [60]
L. While, P. Hingston, L. Barone, S. Huband
A faster algorithm for calculating hypervolume
IEEE Trans. Evol. Comput., 10 (1) (Feb. 2006), pp. 29-38
View in Scopus Google Scholar

[61] [61]
F. Ming, W. Gong, Y. Jin
Even search in a promising region for constrained multi-objective optimization
IEEE-CAA J. Automatic., 11 (2) (Feb. 2024), pp. 474-486, 10.1109/JAS.2023.123792
View in Scopus Google Scholar

[62] [62]
R. Jiao, B. Xue, M. Zhang
A multiform optimization framework for constrained multiobjective optimization
IEEE Trans. Cybern., 53 (8) (Aug. 2023), pp. 5165-5177, 10.1109/TCYB.2022.3178132
View in Scopus Google Scholar

[63] [63]
R. Jiao, S. Zeng, C. Li, Y.S. Ong
Two-type weight adjustments in MOEA/D for highly constrained many-objective optimization
Inf. Sci., 578 (Nov. 2021), pp. 592-614
View PDF View article View in Scopus Google Scholar

[64] [64]
J.O. Agushaka, et al.
Greater cane rat algorithm (GCRA): A nature-inspired metaheuristic for optimization problems
Heliyon., 10 (11) (2024)
Google Scholar

[65] [65]
M. Ghasemi, et al.
Geyser Inspired Algorithm: A New Geological-inspired Meta-heuristic for Real-parameter and Constrained Engineering Optimization
J. Bionic. Eng., 21 (2024), pp. 374-408
Crossref View in Scopus Google Scholar

Outline 概要

Figures (17) 数字 (17)

Tables (10) 表格 (10)

Swarm and Evolutionary Computation
蜂群和进化计算

Abstract https://linux.do/t/topic/111737

Keywords https://linux.do/t/topic/111737

1. Introduction https://linux.do/t/topic/111737

2. Related work and motivation
2.相关工作和动机

2.1. Existing constraint handling techniques
2.1.现有的约束处理技术

2.2. Existing manifold learning techniques
2.2.现有的流形学习技术

2.3. Motivation 2.3.动机

3. Proposed algorithm 3.拟议算法

3.1. The procedure of the proposed algorithm
3.1.拟议算法的程序

3.2. Guided feasible search strategy
3.2.指导下的可行搜索策略

3.3. The manifold learning based exploration strategy
3.3.基于流形学习的探索策略

3.4. Analysis of MACA 3.4.分析 MACA

4. Experimental studies 4.实验研究

4.1. Experimental setting
4.1.实验环境

4.1.1. Test problems and performance metrics
4.1.1.测试问题和性能指标

4.1.2. Competing algorithms
4.1.2.竞争算法

4.1.3. Parameter settings
4.1.3.参数设置

4.2. Comparison between MACA and state-of-the-art CMOEAs
4.2.MACA 与最先进 CMOEA 的比较

4.3. Analysis of the effects of the individual components
4.3.各组成部分的效果分析

4.3.1. Effectiveness of the guided feasible search strategy
4.3.1.有指导的可行搜索策略的有效性

4.3.2. Effectiveness of the manifold learning based exploration strategy
4.3.2.基于流形学习的探索策略的有效性

4.4. Performance of MACA on CMOPs with fraudulent constrains
4.4.MACA 在有欺诈约束的 CMOP 上的性能

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

Acknowledgment 鸣谢

Data availability 数据可用性

References

Cited by (0)

Swarm and Evolutionary Computation蜂群和进化计算

Manifold-assisted coevolutionary algorithm for constrained multi-objective optimization约束多目标优化的歧义辅助协同进化算法

Abstract https://linux.do/t/topic/111737

Keywords https://linux.do/t/topic/111737

1. Introduction https://linux.do/t/topic/111737

2. Related work and motivation2.相关工作和动机

2.1. Existing constraint handling techniques2.1.现有的约束处理技术

2.2. Existing manifold learning techniques2.2.现有的流形学习技术

2.3. Motivation 2.3.动机

3. Proposed algorithm 3.拟议算法

3.1. The procedure of the proposed algorithm3.1.拟议算法的程序

3.2. Guided feasible search strategy3.2.指导下的可行搜索策略

3.3. The manifold learning based exploration strategy3.3.基于流形学习的探索策略

3.4. Analysis of MACA 3.4.分析 MACA

4. Experimental studies 4.实验研究

4.1. Experimental setting4.1.实验环境

4.1.1. Test problems and performance metrics4.1.1.测试问题和性能指标

4.1.2. Competing algorithms4.1.2.竞争算法

4.1.3. Parameter settings4.1.3.参数设置

4.2. Comparison between MACA and state-of-the-art CMOEAs4.2.MACA 与最先进 CMOEA 的比较

4.3. Analysis of the effects of the individual components4.3.各组成部分的效果分析

4.3.1. Effectiveness of the guided feasible search strategy4.3.1.有指导的可行搜索策略的有效性

4.3.2. Effectiveness of the manifold learning based exploration strategy4.3.2.基于流形学习的探索策略的有效性

4.4. Performance of MACA on CMOPs with fraudulent constrains4.4.MACA 在有欺诈约束的 CMOP 上的性能

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

Acknowledgment 鸣谢

Data availability 数据可用性

References

Swarm and Evolutionary Computation
蜂群和进化计算

Manifold-assisted coevolutionary algorithm for constrained multi-objective optimization
约束多目标优化的歧义辅助协同进化算法

2. Related work and motivation
2.相关工作和动机

2.1. Existing constraint handling techniques
2.1.现有的约束处理技术

2.2. Existing manifold learning techniques
2.2.现有的流形学习技术

3.1. The procedure of the proposed algorithm
3.1.拟议算法的程序

3.2. Guided feasible search strategy
3.2.指导下的可行搜索策略

3.3. The manifold learning based exploration strategy
3.3.基于流形学习的探索策略

4.1. Experimental setting
4.1.实验环境

4.1.1. Test problems and performance metrics
4.1.1.测试问题和性能指标

4.1.2. Competing algorithms
4.1.2.竞争算法

4.1.3. Parameter settings
4.1.3.参数设置

4.2. Comparison between MACA and state-of-the-art CMOEAs
4.2.MACA 与最先进 CMOEA 的比较

4.3. Analysis of the effects of the individual components
4.3.各组成部分的效果分析

4.3.1. Effectiveness of the guided feasible search strategy
4.3.1.有指导的可行搜索策略的有效性

4.3.2. Effectiveness of the manifold learning based exploration strategy
4.3.2.基于流形学习的探索策略的有效性

4.4. Performance of MACA on CMOPs with fraudulent constrains
4.4.MACA 在有欺诈约束的 CMOP 上的性能

CRediT authorship contribution statement
CRediT 作者贡献声明

Declaration of competing interest
利益冲突声明