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Balancing Constraints and Objectives by Considering Problem Types in constrained multiobjective optimization

IEEE TRANSACTIONS ON CYBERNETICS 2023年第1期53卷 88-101页

作者： Xiang, Yi Yang, Xiaowei Huang, Han Wang, Jiahai South China Univ Technol Sch Software Engn Guangzhou 510006 Peoples R China South China Univ Technol Sch Software Engn Minist Educ Guangzhou 510006 Peoples R China South China Univ Technol Minist Educ Key Lab Big Data & Intelligent Robot Guangzhou 510006 Peoples R China Sun Yat Sen Univ Sch Comp Sci & Engn Guangzhou 510275 Peoples R China

constrained multiobjective optimization problems widely exist in real-world applications. To handle them, the balance between constraints and objectives is crucial, but remains challenging due to non-negligible impacts of problem types. In our context, the problem types refer particularly to those determined by the relationship between the constrained Pareto-optimal front (PF) and the unconstrained PF. Unfortunately, there has been little awareness on how to achieve this balance when faced with different types of problems. In this article, we propose a new constraint handling technique (CHT) by taking into account potential problem types. Specifically, inspired by the prior work, problems are classified into three primary types: 1) I;2) II;and 3) III, with the constrained PF being made up of the entire, part and none of the unconstrained counterpart, respectively. Clearly, any problem must be one of the three types. For each possible type, there exists a tailored mechanism being used to handle the relationships between constraints and objectives (i.e., constraint priority, objective priority, or the switch between them). It is worth mentioning that exact problem types are not required because we just consider their possibilities in the new CHT. Conceptually, we show that the new CHT can make a tradeoff among different types of problems. This argument is confirmed by experimental studies performed on 38 benchmark problems, whose types are known, and a real-world problem (with unknown types) in search-based software engineering. Results demonstrate that within both decomposition-based and nondecomposition-based frameworks, the new CHT can indeed achieve a good tradeoff among different problem types, being better than several state-of-the-art CHTs.

关键词： optimization Switches Statistics Sociology Software engineering Taxonomy Software algorithms constrained multiobjective optimization constraint handling techniques (CHTs) problem types tradeoff model (ToM)

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AN INFEASIBLE ELITIST BASED PARTICLE SWARM optimization FOR constrained multiobjective optimization AND ITS CONVERGENCE

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INTERNATIONAL JOURNAL OF PATTERN RECOGNITION AND ARTIFICIAL INTELLIGENCE 2010年第3期24卷 381-400页

作者： Wei, Jingxuan Wang, Yuping Xidian Univ Sch Comp Sci & Technol Xian 710071 Peoples R China

In this paper, an infeasible elitist based particle swarm optimization is proposed for solving constrained optimization problems. Firstly, an infeasible elitist preservation strategy is proposed, which keeps some infeasible solutions with smaller rank values at the early stage of evolution regardless of how large the constraint violations are, and keep some infeasible solutions with smaller constraint violations and rank values at the later stage of evolution. In this manner, the true Pareto front will be found easier. Secondly, in order to find a set of diversity and uniformly distributed Pareto optimal solutions, a new crowding distance function is designed. It can assign large function values not only for the particles located in the sparse regions of the objective space but also for the crowded particles located near to the boundary of the Pareto front as well. Thirdly, a new mutation operator with two phases is proposed. In the first phase, the particles whose constraint violations are less than the threshold value will be used to compute the total force, then the force will be used as a mutation direction, being helpful to find the better solutions along this direction. In order to guarantee the convergence of the algorithm, the second phase of mutation is proposed. Finally, the convergence of the algorithm is proved. The comparative study shows that the proposed algorithm can generate widespread and uniformly distributed Pareto fronts and outperforms those compared algorithms.

关键词： constrained multiobjective optimization particle swarm optimization infeasible elitist

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An Instance Space Analysis of constrained multiobjective optimization Problems

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IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION 2023年第5期27卷 1427-1439页

作者： Alsouly, Hanan Kirley, Michael Munoz, Mario Andres Univ Melbourne Sch Comp & Informat Melbourne Vic 3010 Australia ARC Ctr Optimisat Technol Integrated Methodol & A Melbourne Vic 3010 Australia Imam Mohammad Ibn Saud Islamic Univ Coll Comp & Informat Sci Riyadh 11564 Saudi Arabia

constrained multiobjective optimization problems (CMOPs) are generally more challenging than unconstrained problems. This in part can be attributed to the infeasible region generated by the constraint functions, the interaction between constraints and objectives, or both. In this article, we explore the relationship between the performance of constrained multiobjective evolutionary algorithms (CMOEAs) and the instance characteristics of CMOP using instance space analysis (ISA). To do this, we extend recent work on Landscape Analysis features for characterizing CMOPs. Specifically, we introduce new features to describe the multiobjective-violation landscape, formed by the interaction between constraint violation and multiobjective fitness. The detailed evaluation of the algorithm footprints, spanning eight CMOP benchmark suites and 15 CMOEAs, demonstrates that ISA effectively captures the strength and weakness of the CMOEAs. We conclude that two characteristics, the isolation of nondominate set and the correlation between constraints and objectives evolvability, have the greatest impact on algorithm performance. However, the current benchmarks problems lack of diversity to represent the real-world problems and to fully reveal the efficacy of CMOEAs evaluated.

关键词： Algorithm selection constrained multiobjective optimization evolutionary algorithm landscape analysis problem characterization

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A METHOD FOR constrained multiobjective optimization BASED ON SQP TECHNIQUES

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SIAM JOURNAL ON optimization 2016年第4期26卷 2091-2119页

作者： Fliege, Jorg Vaz, A. Ismael F. Univ Southampton Sch Math Southampton SO17 1BJ Hants England Univ Minho ALGORITMI Res Ctr Campus Gualtar P-4710057 Braga Portugal

We propose a method for constrained and unconstrained nonlinear multiobjective optimization problems that is based on an SQP-type approach. The proposed algorithm maintains a list of nondominated points that is improved both for spread along the Pareto front and optimality by solving single-objective constrained optimization problems. These single-objective problems are derived as SQP problems based on the given nondominated points. Under appropriate differentiability assumptions we discuss convergence to local optimal Pareto points. We provide numerical results for a set of unconstrained and constrained multiobjective optimization problems in the form of performance and data profiles, where several performance metrics are used. The numerical results confirm the superiority of the proposed algorithm against a state-of-the-art multiobjective solver and a classical scalarization approach, both in the quality of the approximated Pareto front and in the computational effort necessary to compute the approximation.

关键词： sequential quadratic programming constrained multiobjective optimization Pareto front

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Separations and Optimality of constrained multiobjective optimization via Improvement Sets

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JOURNAL OF optimization THEORY AND APPLICATIONS 2018年第3期178卷 794-823页

作者： Chen, Jiawei Huang, La Li, Shengjie Southwest Univ Sch Math & Stat Chongqing 400715 Peoples R China Chongqing Univ Coll Comp Sci Chongqing 400044 Peoples R China Chongqing Univ Coll Math & Stat Chongqing 401331 Peoples R China

In this paper, we investigate the separations and optimality conditions for the optimal solution defined by the improvement set of a constrained multiobjective optimization problem. We introduce a vector-valued regular weak separation function and a scalar weak separation function via a nonlinear scalarization function defined in terms of an improvement set. The nonlinear separation between the image of the multiobjective optimization problem and an improvement set in the image space is established by the scalar weak separation function. Saddle point type optimality conditions for the optimal solution of the multiobjective optimization problem are established, respectively, by the nonlinear and linear separation methods. We also obtain the relationships between the optimal solution and approximate efficient solution of the multiobjective optimization problem. Finally, sufficient and necessary conditions for the (regular) linear separation between the approximate image of the multiobjective optimization problem and a convex cone are also presented.

关键词： constrained multiobjective optimization Image space analysis Improvement set Nonlinear separation Optimality condition

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A novel tri-stage with reward-switching mechanism for constrained multiobjective optimization problems

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COMPLEX & INTELLIGENT SYSTEMS 2024年第3期10卷 4625-4655页

作者： Qu, Jiqing Li, Xuefeng Xiao, Hui Tongji Univ Coll Elect & Informat Engn 4800 Caoan Highway Shanghai 201804 Peoples R China Tongji Univ Frontiers Sci Ctr Intelligent Autonomous Syst 55 Hechuan Rd Shanghai 201804 Peoples R China

The effective exploitation of infeasible solutions plays a crucial role in addressing constrained multiobjective optimization problems (CMOPs). However, existing constrained multiobjective optimization evolutionary algorithms (CMOEAs) encounter challenges in effectively balancing objective optimization and constraint satisfaction, particularly when tackling problems with complex infeasible regions. Subsequent to the prior exploration, this paper proposes a novel tri-stage with reward-switching mechanism framework (TSRSM), including the push, pull, and repush stages. Each stage consists of two coevolutionary populations, namely Pop 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Pop}}_1$$\end{document} and Pop 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Pop}}_2$$\end{document} . Throughout the three stages, Pop 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Pop}}_1$$\end{document} is tasked with converging to the constrained Pareto front (CPF). However, Pop 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Pop}}_2$$\end{document} is assigned with distinct tasks: (i) converging to the unconstrained Pareto front (UPF) in the push stage;(ii) utilizing constraint relaxation technique to discover the CPF in the pull

关键词： constrained multiobjective optimization Evolutionary algorithm Tri-stage Reward-switching mechanism

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A two-stage bidirectional coevolution algorithm with reverse search for constrained multiobjective optimization

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COMPLEX & INTELLIGENT SYSTEMS 2024年第4期10卷 4973-4988页

作者： Liu, Cancan Wang, Yujia Xue, Yunfeng Shanghai Univ Engn Sci Sch Elect & Elect Engn Shanghai 201620 Peoples R China Shanghai Aerosp Elect Technol Inst Shanghai 201108 Peoples R China

constrained multiobjective optimization problems (CMOPs) are widespread in reality. The presence of constraints complicates the feasible region of the original problem and increases the difficulty of problem solving. There are not only feasible regions, but also large areas of infeasible regions in the objective space of CMOPs. Inspired by this, this paper proposes a bidirectional coevolution method with reverse search (BCRS) combined with a two-stage approach. In the first stage of evolution, constraints are ignored and the population is pushed toward promising regions. In the second stage, evolution is divided into two parts, i.e., the main population evolves toward the constrained Pareto front (CPF) within the feasible region, while the reverse population approaches the CPF from the infeasible region. Then a solution exchange strategy similar to weak cooperation is used between the two populations. The experimental results on benchmark functions and real-world problems show that the proposed algorithm exhibits superior or at least competitive performance compared to other state-of-the-art algorithms. It demonstrates BCRS is an effective algorithm for addressing CMOPs.

关键词： constrained multiobjective optimization Evolutionary algorithm Two-stage Bidirectional coevolution Reverse search

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A Hybrid Constraint Handling Mechanism with Differential Evolution for constrained multiobjective optimization

A Hybrid Constraint Handling Mechanism with Differential Evo...

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IEEE Congress on Evolutionary Computation (CEC)

作者： Hsieh, Min-Nan Chiang, Tsung-Che Fu, Li-Chen Natl Taiwan Univ Dept Comp Sci & Informat Engn Taipei Taiwan Natl Taiwan Univ Dept Elect Engn & Dept Comp Sci & Informat Engn Taipei Taiwan

ISBN: (纸本)9781424478354

In real-world applications, the optimization problems usually include some conflicting objectives and subject to many constraints. Much research has been done in the fields of multiobjective optimization and constrained optimization, but little focused on both topics simultaneously. In this study we present a hybrid constraint handling mechanism, which combines the epsilon-comparison method and penalty method. Unlike original epsilon-comparison method, we set an individual epsilon-value to each constraint and control it by the amount of violation. The penalty method deals with the region where constraint violation exceeds the epsilon-value and guides the search toward the epsilon-feasible region. The proposed algorithm is based on a well-known multiobjective evolutionary algorithm, NSGA-II, and introduces the operators in differential evolution (DE). A modified DE strategy, DE/better-to-best_feasible/1, is applied. The better individual is selected by tournament selection, and the best individual is selected from an archive. Performance of the proposed algorithm is compared with NSGA-II and an improved version with a self-adaptive fitness function. The proposed algorithm shows competitive results on sixteen public constrained multiobjective optimization problem instances.

关键词： multiobjective Evolutionary Algorithm Constraint Handling Differential Evolution constrained multiobjective optimization

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Large Language Model-Aided Evolutionary Search for constrained multiobjective optimization 20th

Large Language Model-Aided Evolutionary Search for Constrain...

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20th International Conference on Intelligent Computing (ICIC)

作者： Wang, Zeyi Liu, Songbai Chen, Jianyong Tan, Kay Chen Shenzhen Univ Coll Comp Sci & Software Engn Shenzhen Peoples R China Hong Kong Polytech Univ Dept Comp Hung Hom Hong Kong Peoples R China

ISBN: (纸本)9789819755806;9789819755813

Evolutionary algorithms excel in solving complex optimization problems, especially those with multiple objectives. However, their stochastic nature can sometimes hinder rapid convergence to the global optima, particularly in scenarios involving constraints. In this study, we employ a large language model (LLM) to enhance evolutionary search for solving constrained multiobjective optimization problems. Our aim is to speed up the convergence of the evolutionary population. To achieve this, we finetune the LLM through tailored prompt engineering, integrating information concerning both objective values and constraint violations of solutions. This process enables the LLM to grasp the relationship between well-performing and poorly performing solutions based on the provided input data. Solution's quality is assessed based on their constraint violations and objective-based performance. By leveraging the refined LLM, it can be used as a search operator to generate superior-quality solutions. Experimental evaluations across various test benchmarks illustrate that LLM-aided evolutionary search can significantly accelerate the population's convergence speed and stands out competitively against cutting-edge evolutionary algorithms.

关键词： constrained multiobjective optimization Large Language Model Evolutionary Algorithm

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Threshold Based Dynamic and Adaptive Penalty Functions for constrained multiobjective optimization 1

Threshold Based Dynamic and Adaptive Penalty Functions for C...

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1st International Conference on Artificial Intelligence, Modelling and Simulation (AIMS)

作者： Jan, Muhammad Asif Tairan, Nasser Khanum, Rashida Adeeb Kohat Univ Sci & Technol Dept Math Kohat Pakistan King Khalid Univ Coll Comp Sci Abha Saudi Arabia Univ Peshawar Jinnah Coll Women Peshawar Pakistan

ISBN: (纸本)9781479932511

Penalty functions are frequently used for dealing with constraints in constrained optimization. Among different types of penalty functions, dynamic and adaptive penalty functions seem effective, since the penalty coefficients in them are adjusted based on the current generation number (or number of solutions searched) and feedback from the search. In this paper, we propose dynamic and adaptive versions of our recently proposed threshold based penalty function. They are then implemented in the multiobjective evolutionary algorithm based on decomposition (MOEA/D) framework to solve constrained multiobjective optimization problems (CMOPs). This led to a new algorithm, denoted by CMOEA/D-DE-TDA. The performance of CMOEA/D-DE-TDA is tested on CTP-series test instances in terms of the HV-metric and SC-metric. The experimental results are compared with IDEA and NSGA-II, which show the effectiveness of the proposed algorithm.

关键词： constrained multiobjective optimization decomposition MOEA/D penalty function threshold

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