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constrained multi-objective optimization assisted by convergence and diversity auxiliary tasks

ENGINEERING APPLICATIONS OF ARTIFICIAL INTELLIGENCE 2025年 139卷

作者： Dang, Qianlong Shang, Wutao Huang, Zhengxin Yang, Shuai Northwest A&F Univ Coll Sci Yangling 712100 Peoples R China Northwest A&F Univ Joint Lab Algorithm & Simulat Low Altitude Aircraf Yangling 712100 Peoples R China Youjiang Med Univ Nationalities Dept Math & Comp Sci Baise 533000 Peoples R China Anhui Agr Univ Sch Informat & Artificial Intelligence Hefei 230036 Peoples R China

In the field of constrained multi-objective optimization, constructing auxiliary tasks can guide the algorithm to achieve efficient search. Different forms of auxiliary tasks have their own advantages, and a reasonable combination can effectively improve the performance of the algorithm. Inspired by this, a constrained multi-objective optimization Evolutionary Algorithm based on Convergence and Diversity auxiliary Tasks (CMOEA-CDT) is proposed. This algorithm achieves efficient search through simultaneous optimization and knowledge transfer of the main task, convergence auxiliary task, and diversity auxiliary task. Specifically, the main task is to find feasible Pareto front, which improves the global exploration and local exploitation of the algorithm through knowledge transfer from the convergence and diversity auxiliary tasks. In addition, the convergence auxiliary task helps the main task population traverse infeasible obstacles by ignoring constraints to achieve global search. The diversity auxiliary task aims to provide local diversity to the regions around the main task population to exploit promising search regions. The convergence and diversity of the algorithm are significantly improved by knowledge transfer between the convergence auxiliary task, diversity auxiliary task, and main task. CMOEA-CDT is compared with five state-of-the-art constrained multi-objective evolutionary optimization algorithms on 37 benchmark problems and a disc brake engineering design problem. The experimental results indicate that the proposed CMOEA-CDT respectively obtains 19 and 20 best results on the two indicators, and achieves the best performance on disc brake engineering design problem.

关键词： constrained multi-objective optimization Convergence auxiliary task Diversity auxiliary task Knowledge transfer Disc brake engineering design problem

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Optimizing Police Patrol Strategies in Real-World Scenarios: A Modified PPS-MOEA/D Approach for constrained multi-objective optimization

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APPLIED SCIENCES-BASEL 2025年第7期15卷 3651-3651页

作者： Sui, Jinguang Chen, Peng Jiang, Huan Peoples Publ Secur Univ China Sch Informat Network Secur Beijing 100038 Peoples R China Peoples Publ Secur Univ China Sch Criminol Beijing 100038 Peoples R China Beijing Technol & Business Univ Sch Comp & Artificial Intelligence Beijing 100048 Peoples R China

This study addresses the realistic constrained multi-objective optimization problem of police patrols by constructing a mathematical model tailored to the actual operational context of police patrols in China. To solve this problem, a modified PPS-MOEA/D algorithm is proposed and its performance is systematically evaluated against several state-of-the-art constrained multi-objective Evolutionary Algorithms (CMOEAs). The results demonstrate the superiority of the proposed approach in terms of the solution quality and computational efficiency. Furthermore, the optimal solution set is discussed and visualized on a map, providing decision makers with practical and actionable insights that align with real-world patrol requirements. This research not only advances the theoretical framework for police patrol optimization, but also offers a practical tool for enhancing the effectiveness and efficiency of law enforcement operations in urban environments.

关键词： police patrol optimization constrained multi-objective optimization PPS-MOEA/D CMOEAs

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constrained multi-objective optimization via neural network and cooperative populations

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APPLIED SOFT COMPUTING 2025年 176卷

作者： Cao, Jie Wang, Yiyuan Zhang, Jianlin Chen, Zuohan Lanzhou Univ Technol Sch Comp & Commun LanGongPing 287 Lanzhou 730050 Gansu Peoples R China Lanzhou Univ Technol Gansu Engn Res Ctr Mfg Informat LanGongPing 287 Lanzhou 730050 Gansu Peoples R China Lanzhou City Univ Sch Informat Engn JieFang 11 Lanzhou 730050 Gansu Peoples R China

constrained multi-objective optimization problems are widely used in practical scenarios such as intelligent manufacturing and network communication. These problems are often made intractable by constraints, and achieving a balance between convergence, diversity, and feasibility becomes increasingly challenging. To address this issue, a constrained multi-objective evolutionary algorithm named NNCP is proposed, which is based on the neural network and, three cooperative populations. Specifically, the neural network is employed to accelerate the population's convergence by utilizing neuron weights to capture neighborhood information. Among the three populations, the first population uses self-organizing mapping and curvature estimation to approximate the Pareto front, the second population utilizes non-dominance sorting and an angle selection mechanism to identify high-quality infeasible solutions, thereby enhancing diversity, and the third population adopts an adaptive penalty mechanism to improve feasibility. These populations work cooperatively to identify promising infeasible solutions and navigate infeasible regions to approximate the Pareto front. Finally, five state-of-the-art constrained multi-objective optimization algorithms are compared with NNCP. Out of the total 47 test problems, NNCP outperforms the best-performing baseline algorithm on more than 35 problems, highlighting its superior convergence and diversity capabilities.

关键词： constrained multi-objective optimization Neural network Three cooperative populations Self-organizing mapping Neuron weights

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A Constraint Priority Decision framework for constrained multi-objective optimization with complex feasible regions

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APPLIED SOFT COMPUTING 2025年 172卷

作者： Yan, Pengguo Tian, Ye Wang, Jiesheng Liu, Yu Univ Sci & Technol Liaoning Sch Elect & Informat Engn Anshan 114051 Peoples R China Anhui Univ Sch Comp Sci & Technol Hefei 230601 Peoples R China

constrained multi-objective optimization problems (CMOPs) present significant challenges due to the simultaneous consideration of objectives and constraints, which becomes particularly arduous when the feasible regions are exceedingly complex. Most of the existing algorithms fail to obtain high-quality solutions for the CMOPs with complex feasible regions. To address this issue, this paper proposes a Constraint Priority Decision framework applied to multi-stage evolutionary algorithms, which incorporates constraints sequentially throughout the solution process to facilitate the retention of optimal diversity and feasibility within the population. Specifically, the proposed framework employs a traditional multi-objective evolutionary algorithm as the optimizer and decomposes various constraints of the CMOP. These constraints are introduced independently into the optimizer, generating an index value for each respective constraint. Following this, a judgment matrix is constructed based on these indices to grade the constraints, thus facilitating the establishment of a priority sequence for multiple constraints. Furthermore, a two-stage strategy is implemented in this study. After incorporating all constraints into the algorithm, the epsilon-constrained method is utilized to impose constraints on the entire problem to increase the genetic diversity of the population while maintaining the feasibility of the population. The experimental results derived from four popular benchmark suites and six real-world applications indicate that the proposed framework surpassed multiple state-of-the-art constrained multi-objective evolutionary algorithms in addressing CMOPs with complex feasible regions.

关键词： Constraint-handling priority constrained multi-objective optimization Evolutionary algorithm Analytic hierarchy process

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Growing neural gas network based environment selection strategy for constrained multi-objective optimization

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INFORMATION SCIENCES 2025年 698卷

作者： Bai, Weiting Dang, Qianlong Wu, Jingxiang Gao, Xiaochuan Zhang, Guanghui Northwest A&F Univ Coll Sci Yangling 712100 Peoples R China Northwest A&F Univ Joint Lab Algorithm & Simulat Low Altitude Aircraf Yangling 712100 Peoples R China Hebei Agr Univ Sch Informat Sci & Technol Baoding 071001 Peoples R China

When solving constrained multi-objective problems, it is necessary to consider the satisfaction of constraints and the optimization of objective functions at the same time. However, emphasizing constraints may lead to the loss of valuable potential solutions, while focusing solely on the objective functions may cause the population to deviate from the feasible regions. Therefore, how to better handle the relationship between constraints and objective functions is a great challenge, and traditional algorithms often face the problems of local optima and poor diversity. To address these problems, this paper proposes a growing neural gas network based environment selection strategy for constrained multi-objective evolutionary algorithm (MTCMEA-GNG), which includes the main task, unconstrained auxiliary task, and topology auxiliary task. In the main task, the growing neural gas network can adaptively simulate the topology of the population and find more valuable potential solutions through connections between nodes, which makes the population gradually converge to the feasible regions. Moreover, the unconstrained auxiliary task can help the main task cross larger infeasible regions, which enhances the convergence. In addition, the topology auxiliary task utilizes the growing neural gas network to provide more valuable potential solutions for the main task, which helps the main task explore smaller infeasible regions and enhances the diversity. These two auxiliary tasks work together to solve the balance between convergence and diversity. Besides, the growing neural gas network is embedded into the environment selection strategy to select more valuable potential solutions through the dynamically adjusted node connections. Compared with the existing advanced constrained multi-objective evolutionary algorithms, the experimental results on three test suites and seven engineering application problems demonstrate that MTCMEA-GNG has strong competitiveness and superiority.

关键词： constrained multi-objective optimization multitasking optimization Growing neural gas network Environment selection Engineering application

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constrained multi-objective optimization With Constraint Priority

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IEEE TRANSACTIONS ON EMERGING TOPICS IN COMPUTATIONAL INTELLIGENCE 2025年

作者： Zhou, Xinyu Zhu, Yanjun Wang, Yizhi Sun, Ruiqing Zou, Juan Jiangxi Normal Univ Sch Comp & Informat Engn Nanchang 330022 Peoples R China Ganzhou Open Univ Ganzhou 341000 Peoples R China Natl Univ Def Technol Coll Comp Changsha 410073 Peoples R China Xiangtan Univ Key Lab Intelligent Comp & Informat Proc Minist Educ China Xiangtan 411105 Peoples R China

constrained multi-objective problems (CMOPs) are tricky, because it is difficult to handle multiple objectives and constraints simultaneously. Most existing algorithms perform well on CMOPs with a single constraint or multiple simple constraints, but their performance on CMOPs with multiple complex constraints often deteriorates. Generally, for CMOPs, it is much easier to deal with a single constraint than with multiple constraints. Therefore, this paper proposes a strategy for identifying the constraint priority, and then handles the constraints one by one according to the priority. Specifically, for each constraint, an associated population is first established to exclusively consider this constraint, and then the corresponding infeasibility rates of all the constraints are calculated to capture the constraint priority. After that, based on the constraint priority, the roulette selection method is used to determine which constraint should be handled first, which ensures the constraint with high priority has more chances to be handled. Experimental results on five benchmark suites and three real-world problems show that our approach has significant advantages over five state-of-the-art algorithms.

关键词： optimization Convergence Linear programming Classification algorithms Benchmark testing Sun Fans Evolutionary computation Vectors Training Evolutionary algorithm constraint priority constrained multi-objective optimization

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Two-stage bidirectional coevolutionary algorithm for constrained multi-objective optimization

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SWARM AND EVOLUTIONARY COMPUTATION 2025年 92卷

作者： Zhao, Shulin Hao, Xingxing Chen, Li Yu, Tingfeng Li, Xingyu Liu, Wei Northwest Univ Sch Informat Sci & Technol Xian 710127 Peoples R China Leiden Univ Leiden Inst Adv Comp Sci NL-2333 CA Leiden Netherlands

objective optimization and constraint satisfaction are two primary and conflicting tasks in solving constrained multi-objective optimization problems (CMOPs). To better trade off them, this paper proposes a two-stage bidirectional coevolutionary algorithm, termed C-TBCEA, for constrained multi-objective optimization. It consists of two stages, with each concentrating on specific targets, i.e., the first stage primarily focuses on objective optimization while the second stage focuses on constraint satisfaction by employing different evolutionary strategies at each stage. Via the synergy of the two stages, a dynamic trade-off between objective optimization and constraint satisfaction can be achieved, thus overcoming the distinctive challenges that may be encountered at different stages of evolution. In addition, to take advantage of both feasible and infeasible solutions, we employ two populations, i.e., the main population that stores the non-dominated feasible solutions and the archive population that maintains the informative infeasible solutions, to prompt the bidirectional coevolution of them. To validate the effectiveness of the proposed C-TBCEA, experiments are carried out on 6 CMOP test suites and 17 real-world CMOPs. The results demonstrate that the proposed algorithm is very competitive with 9 state-of-the-art constrained multi-objective optimization evolutionary algorithms (CMOEAs).

关键词： constrained multi-objective optimization Two-stage Bidirectional coevolution Evolutionary algorithms

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A knowledge driven two-stage co-evolutionary algorithm for constrained multi-objective optimization

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EXPERT SYSTEMS WITH APPLICATIONS 2025年 274卷

作者： Zhang, Wei Liu, Jianchang Li, Lin Liu, Yuanchao Wang, Honghai Northeastern Univ Natl Frontiers Sci Ctr Ind Intelligent & Syst Opti Shenyang Peoples R China Northeastern Univ Coll Informat Sci & Engn Shenyang Peoples R China

In recent years, constrained multi-objective optimization problems (CMOPs) have received wide attention. However, most solving methods for CMOPs still cannot balance objectives and constraints well since constraints make CMOPs have the complicated constrained Pareto front (CPF). This implies that utilizing the related CPF information may be helpful to solve CMOPs. Inspired by the human ability that summarizes the information into different knowledge to address different problems, a knowledge driven two-stage co-evolutionary algorithm (KTCOEA) for CMOPs is developed, including knowledge generation and knowledge application stages. The knowledge generation stage aims at generating two kinds of knowledge: explicit knowledge and implicit knowledge. To this end, main and auxiliary populations co-evolve towards the CPF and unconstrained Pareto front in turn, and then two kinds of knowledge are generated by storing non-dominated solutions and analyzing the correlation of two populations, respectively. Based on the implicit knowledge, reasonable evolution strategies and environmental selection manners are designed for the knowledge application stage, where the explicit knowledge acts as the main population. In this way, the main population can find the complete CPF (i.e., well balance objectives and constraints). In addition, to ensure the generated knowledge quality, a dynamic cooperation mechanism is proposed, which can dynamically adjust the focus of two populations on objectives and constraints based on the performance requirement and evolution status. Experimental results on five benchmark test suites and five real-world applications demonstrate that KTCOEA is better than seven state-of-the-art algorithms on most test problems.

关键词： constrained multi-objective optimization Co-evolutionary algorithm Knowledge Two-stage Dynamic cooperation strategy

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A constrained multi-objective optimization algorithm with adaptive dual-stage search strategy utilizing the relationship between different Pareto fronts

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SWARM AND EVOLUTIONARY COMPUTATION 2025年 95卷

作者： Su, Kai He, Zhihui Wang, Feng Wuhan Univ Sch Comp Sci Wuhan 430072 Peoples R China

multi-stage dual-population constrained evolutionary algorithms (MDCMOEAs) demonstrate competitive performance in solving constrained multi-objective optimization problems (CMOPs). In these algorithms, the main population addresses the original problem, while the auxiliary population solves the helper problem across multiple stages, including both unconstrained and constrained stages. However, MDCMOEAs face challenges in effectively searching the constrained Pareto front (CPF) that overlaps with the unconstrained Pareto front (UPF), particularly when feasible regions are small or disconnected. This difficulty arises because the auxiliary population considers constraints in some stages, making it susceptible to becoming trapped in local feasible regions. To overcome this challenge, this paper proposes an algorithm with an adaptive dual-stage search strategy (ADSSCMO). First, an improved epsilon-constraint method is developed for the main population to tackle the original CMOPs. Second, an adaptive dual-stage search strategy is designed for the auxiliary population. This strategy dynamically evaluates the relationship between UPF and CPF and determines whether to solve the unconstrained or constrained problem. Extensive experiments on four test suites and seven real-world problems demonstrate that the proposed algorithm is more competitive than seven state-of-the-art CMOEAs.

关键词： constrained multi-objective optimization Evolutionary algorithm Dual-population Adaptive strategy

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constrained multi-objective optimization With Deep Reinforcement Learning Assisted Operator Selection

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IEEE/CAA Journal of Automatica Sinica 2024年第4期11卷 919-931页

作者： Fei Ming Wenyin Gong Ling Wang Yaochu Jin the School of Computer Science China University of GeosciencesWuhan 430074China the Department of Automation Tsinghua UniversityBeijing 100084China the Faculty of Technology Bielefeld UniversityNorth Rhine-Westphalia33619 BielefeldGermany

Solving constrained multi-objective optimization problems with evolutionary algorithms has attracted considerable *** constrained multi-objective optimization evolutionary algorithms(CMOEAs)have been developed with the use of different algorithmic strategies,evolutionary operators,and constraint-handling *** performance of CMOEAs may be heavily dependent on the operators used,however,it is usually difficult to select suitable operators for the problem at ***,improving operator selection is promising and necessary for *** work proposes an online operator selection framework assisted by Deep Reinforcement *** dynamics of the population,including convergence,diversity,and feasibility,are regarded as the state;the candidate operators are considered as actions;and the improvement of the population state is treated as the *** using a Q-network to learn a policy to estimate the Q-values of all actions,the proposed approach can adaptively select an operator that maximizes the improvement of the population according to the current state and thereby improve the algorithmic *** framework is embedded into four popular CMOEAs and assessed on 42 benchmark *** experimental results reveal that the proposed Deep Reinforcement Learning-assisted operator selection significantly improves the performance of these CMOEAs and the resulting algorithm obtains better versatility compared to nine state-of-the-art CMOEAs.

关键词： constrained multi-objective optimization deep Qlearning deep reinforcement learning(DRL) evolutionary algorithms evolutionary operator selection

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