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A constrained multimodal multi-objective evolutionary algorithm based on adaptive epsilon method and two-level environmental selection

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SWARM AND EVOLUTIONARY COMPUTATION 2025年 93卷

作者： Wang, Fengxia Huang, Min Yang, Shengxiang Wang, Xingwei Northeastern Univ Coll Informat Sci & Engn Shenyang 110819 Liaoning Peoples R China Montfort Univ Sch Comp Sci & Informat Leicester LE1 9BH England Northeastern Univ Coll Comp Sci & Engn Shenyang 110169 Liaoning Peoples R China

constrained multimodal multi-objective optimization problems (CMMOPs) commonly arise in practical problems in which multiple Pareto optimal sets (POSs) correspond to one Pareto optimal front (POF). The existence of constraints and multimodal characteristics makes it challenging to design effective algorithms that promote diversity in the decision space and convergence in the objective space. Therefore, this paper proposes a novel constrained multimodal multi-objective evolutionary algorithm, namely CM-MOEA, to address CMMOPs. In CM-MOEA, an adaptive epsilon-constrained method is designed to utilize promising infeasible solutions, promoting exploration in the search space. Then, a diversity-based offspring generation method is performed to select diverse solutions for mutation, searching for more equivalent POSs. Furthermore, the two-level environmental selection strategy that combines local and global environmental selection is developed to guarantee diversity and convergence of solutions. Finally, we design an archive update strategy that stores well-distributed excellent solutions, which more effectively approach the true POF. The proposed CM-MOEA is compared with several state-of-the-art algorithms on 17 test problems. The experimental results demonstrate that the proposed CM-MOEA has significant advantages in solving CMMOPs.

关键词： constrained multimodal multi-objective optimization epsilon-constrained method Two-level environmental selection Archive update

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Development of A Multi-objective Robust Probabilistic Programming Mathematical Model Using Improved ε- Aconstrained method in Gas Distribution Network Projects Implementation

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INTERNATIONAL JOURNAL OF ENGINEERING 2025年第4期38卷 848-858页

作者： Najafabadi, A. Rostami Nikbakht, M. Amindoust, A. Karbasian, M. Islamic Azad Univ Dept Ind Engn Najafabad Branch Najafabad Iran Islamic Azad Univ Modern Mfg Technol Res Ctr Dept Ind Engn Najafabad Branch Najafabad Iran

Optimizing time, Acost, quality and risk are the four main factorsAinAgas project management. By considering the significance of gas projects in the country's gas development as one of the top sources of revenue and economic advantages, efficient and targeted management in gas distribution network projects implementation is believed essential. The objective of this research is to develop a multi- objective robust probabilistic programming model using the improved epsilon- Aconstraint method while adhering to the objectives of minimizing time, Acost and risk functions and maximizing the expected quality function. To thisAend, a sample size of 36 experts and specialists in the gas industry has been selected. The results of this research enable gas project managers to select the best solution among various scenarios based on effective management with different values between the four objective functions of time, cost, quality and risk, thus achieving strong optimization. Additionally, the results indicate that the two objective functions derived from the Improved epsilon- Aconstraint method have 99% confidence in better exploring the problem-solving space in producing superior solutions and Pareto charts of the risk and quality objective functions exhibit the optimal solution. The findings demonstrate the accuracy of the model and the efficiency of the proposed method.

关键词： epsilon-constrained method Gas Distribution Networks Robust Probabilistic Programming

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ε-constrained Differential Evolution Using an Adaptive ε-Level Control method

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IEEE TRANSACTIONS ON SYSTEMS MAN CYBERNETICS-SYSTEMS 2022年第2期52卷 769-785页

作者： Zhang, Chunjiang Qin, A. K. Shen, Weiming Gao, Liang Tan, Kay Chen Li, Xinyu Huazhong Univ Sci & Technol State Key Lab Digital Mfg Equipment & Technol Wuhan 430074 Peoples R China Swinburne Univ Technol Dept Comp Sci & Software Engn Hawthorn Vic 3122 Australia City Univ Hong Kong Dept Comp Sci Hong Kong Peoples R China

Evolutionary algorithms and swarm intelligence algorithms have been widely used for constrained optimization problems for decades and numerous techniques for constraint handling have been proposed. The epsilon-constrained method is a very effective one. In the literature, the epsilon value was usually controlled via an exponential function, which is not competent for solving certain types of constrained optimization problems, e.g., whose global optima are located near the boundary of the feasible and infeasible regions. To solve this problem, this article proposes a new adaptive epsilon control method and incorporate it into a basic differential evolution (DE) algorithm: (DE/rand/1/exp). Based on the information of constraint violation in the current population, the adaptive method controls the value of epsilon through a simple heuristic rule. Compared with the traditional exponential function-based control methods, the proposed adaptive method can prevent the algorithm from being trapped into local optima while retaining the obtained near-optimal candidate solutions in the infeasible region for generating promising searching paths. Besides, we set the crossover rate (CR) as a more reasonable value for DE/rand/1/exp, which can enhance the efficiency significantly. The well-known 2006 IEEE Congress on Evolutionary Computation (CEC 2006) competition on real-parameter single-objective constrained optimization benchmark is adopted to evaluate the effectiveness of the proposed adaptive epsilon-constrained DE. Fifteen constrained engineering optimization problems are collected from the literature to test the proposed algorithm. Moreover, the adaptive epsilon control method is extended to an adaptive algorithm to solve the benchmark problems from CEC 2017. The comparison results confirm the superiority of the proposed method.

关键词： epsilon-constrained method constrained optimization problem differential evolutionary engineering optimization

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A modified brain storm optimization algorithm with a special operator to solve constrained optimization problems

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APPLIED INTELLIGENCE 2020年第12期50卷 4145-4161页

作者： Cervantes-Castillo, Adriana Mezura-Montes, Efren Univ Veracruz Artificial Intelligence Res Ctr Sebastian Camacho 5 Xalapa 91000 Veracruz Mexico

This paper presents a novel approach based on the combination of the Modified Brain Storm Optimization algorithm (MBSO) with a simplified version of the Constraint Consensus method as special operator to solve constrained numerical optimization problems. Regarding the special operator, which aims to reach the feasible region of the search space, the consensus vector becomes the feasibility vector computed by the hardest constraint in turn for a current infeasible solution;then the operations to mix the other feasibility vectors are avoided. This new combined algorithm, named as MBSO-R+V, solves a suit of eighteen test problems in ten and thirty dimensions. From a set of experiments related to the location and frequency of application of the constraint consensus method within MBSO, a suitable design of the combined approach is presented. This proposal shows encouraging final results while being compared against state-of-the-art algorithms, showing that it is viable to add special operators to improve the capabilities of swarm-intelligence algorithms when dealing with continuous constrained search spaces.

关键词： Brain storm optimization algorithm constrained numerical optimization problems Constraint-consensus method Feasibility vectors epsilon-constrained method

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Study on guaranteed output constraints in the long term joint optimal scheduling for the hydropower station group

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ENERGY 2019年 185卷 1210-1224页

作者： He, Zhongzheng Zhou, Jianzhong Xie, Mengfei Jia, Benjun Bao, Zhengfeng Qin, Hui Zhang, Hairong Huazhong Univ Sci & Technol Sch Hydropower & Informat Engn Wuhan 430074 Hubei Peoples R China Hubei Key Lab Digital Valley Sci & Technol Wuhan 430074 Hubei Peoples R China Kunming Power Exchange Ctr Co Ltd Kunming 650011 Yunnan Peoples R China China Yangtze Power Co Ltd Three Gorges Cascade Dispatching & Commun Ctr Yichang 443133 Peoples R China China Yangtze Power Co Ltd Dept Water Resources Management Yichang 443133 Peoples R China

Long-term joint scheduling of hydropower station group (LJSHSG) is a constrained optimization problem, which suffers from a variety of complex and time-state coupling constraints. It is beset with difficulties to solve LJSHSG problem owing to these equality constraints, rigid constraints and flexible constraints of hydropower station group (HSG). The new hybrid constraint handling method combining epsilon-constraint (EC) and penalty functions (PF) (named EC-PF) is proposed to deal with these constraints in this paper. In the proposed EC-PF, all equality constraints are forced to satisfy according to the equation;EC handle all rigid constraints with minimizing the value of constraint violation and constraint relaxation rule;and the unique flexible constraint, guaranteed output constraint, is processed by PF. Then the proposed EC-PF is compared with the superiority of feasible solutions (SF), stochastic ranking (SR), PF, EC and lexicographic method (LM) in the application to LJSHSG problems;the experimental results verify the superiority of the proposed method. Moreover, the conclusions that it is necessary to deal with rigid constraints from flexible constraints differently in LJSHSG problem are obtained. On this basis, in view of the defect that guaranteed output constraints are difficult to satisfy in LJSHSG, the LJSHSG with the cooperative mode to handle guaranteed output constraints (named LJSHSG-cm) is puts forward. The comparative experimental results of LJSHSG and LJSHSG-cm show that LJSHSG-cm can increase the power generation by 0.3% while increasing the satisfaction rate of guaranteed output constraints by 78.56% compared with LJSHSG for long sequence calculation from 1959 to 2014. These are fully illustrated that the hybrid constraint handling method EC-PF and LJSHSG-cm with the cooperative mode to handle guaranteed output constraints are valid and reliable practical tools in solving LJSHSG. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Long-term scheduling Hydropower station group Constraint handling Guaranteed output constraints epsilon-constrained method Penalty function methods

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Backtracking Search Algorithm with three constraint handling methods for constrained optimization problems

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EXPERT SYSTEMS WITH APPLICATIONS 2015年第21期42卷 7831-7845页

作者： Zhang, Chunjiang Lin, Qun Gao, Liang Li, Xinyu Huazhong Univ Sci & Technol State Key Lab Digital Mfg Equipment & Technol Wuhan 430074 Hubei Peoples R China

A new evolutionary algorithm, Backtracking Search Algorithm (BSA), is applied to solve constrained optimization problems. Three constraint handling methods are combined with BSA for constrained optimization problems;namely feasibility and dominance (FAD) rules, epsilon-constrained method with fixed control way of epsilon value and a proposed epsilon-constrained method with self-adaptive control way of epsilon value. The proposed method controls epsilon value according to the properties of current population. This kind of epsilon value enables algorithm to sufficiently search boundaries between infeasible regions and feasible regions. It can avoid low search efficiency and premature convergence which happens in fixed control method and FAD rules. The comparison of the above three algorithms demonstrates BSA combined epsilon-constrained method with self-adaptive control way of epsilon value (BSA-SA epsilon) is the best one. The proposed BSA-SA epsilon also outperforms other five classic and the latest constrained optimization algorithms. Then, BSA-SA epsilon has been applied to four engineering optimization instances, and the comparison with other algorithms has proven its advantages. Finally, BSA-SA epsilon is used to solve the car side impact design optimization problem, which illustrates the wide application prospects of the proposed BSA-SA epsilon. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Backtracking Search Algorithm constrained optimization problem Feasibility and dominance rules epsilon-constrained method Engineering optimization

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Bias-specified robust design optimization and its analytical solutions

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COMPUTERS & INDUSTRIAL ENGINEERING 2005年第1期48卷 129-140页

作者： Shin, S Cho, BR Clemson Univ Dept Ind Engn Adv Qual Engn Lab Clemson SC 29634 USA

Robust design has received consistent attention from researchers and practitioners for years, and a number of methodologies for robust design optimization have been reported in the research community. However. the majority of these existing methodologies ignore the case where the customer may tolerate and specify an upper bound on process bias. This paper proposes a bias-specified robust design method and formulates a nonlinear program that minimizes process variability subject to customer-specified constraints on the process bias using the e-constrained method. This paper then derives the Karush-Khun-Tucker conditions and provides a solution procedure based on the Lagrangean method. A numerical example is provided for illustration. (C) 2004 Elsevier Ltd. All rights reserved.

关键词： robust design epsilon-constrained method Lagrangean method Karush-Khun-Tucker conditions response surface methodology

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A bi-objective approach to discrete cost-bottleneck location problems

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ANNALS OF OPERATIONS RESEARCH 2018年第1-2期267卷 179-201页

作者： Gadegaard, Sune Lauth Klose, Andreas Nielsen, Lars Relund Aarhus Univ Sch Business & Social Sci Dept Econ & Business Econ Fuglesangs Alle 4 DK-8210 Aarhus V Denmark Aarhus Univ Dept Math Sci & Technol Ny Munkegade 118 DK-8000 Aarhus C Denmark

This paper considers a family of bi-objective discrete facility location problems with a cost objective and a bottleneck objective. A special case is, for instance, a bi-objective version of the (vertex) p-centdian problem. We show that bi-objective facility location problems of this type can be solved efficiently by means of an -constraint method that solves at most minisum problems, where n is the number of customer points and m the number of potential facility sites. Additionally, we compare the approach to a lexicographic -constrained method that only returns efficient solutions and to a two-phase method relying on the perpendicular search method. We report extensive computational results obtained from several classes of facility location problems. The proposed algorithm compares very favorably to both the lexicographic -constrained method and to the two phase method.

关键词： Discrete facility location Bi-objective optimization epsilon-constrained method Lexicographic optimization

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An Improved C²oDE based on Stochastic Ranking 40

An Improved C<SUP>2</SUP>oDE based on Stochastic Ranking

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40th Chinese Control Conference (CCC)

作者： Si, Chengyong Shen, Jianqiang Wang, Chaofeng Wang, Lei Univ Shanghai Sci & Technol Sino German Coll Shanghai 200093 Peoples R China Tongji Univ Coll Elect & Informat Engn Shanghai 201804 Peoples R China

ISBN: (纸本)9789881563804

How to balance diversity and convergence as well as constraint violation and objective function in constrained Optimization Problems (COPs) is an important and key issue when designing a new constrained Optimization Evolutionary Algorithms (COEAs). C(2)oDE set a good example on this point, and obtains promising results. In C(2)oDE, the two Constraint Handling Techniques (CHTs) are adopted fixed in two phases. To verify whether there are some other more efficient combinations, Stochastic Ranking (SR) is adopted to balance the combination. Experimental results show a good performance, which also give some inspiration for other research.

关键词： constrained optimization Stochastic ranking Deb's feasibility-based rule epsilon-constrained method

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Long-Term Scheduling of Large-Scale Cascade Hydropower Stations Using Improved Differential Evolution Algorithm

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WATER 2018年第4期10卷 383-383页

作者： Wen, Xiaohao Zhou, Jianzhong He, Zhongzheng Wang, Chao Huazhong Univ Sci & Technol Sch Hydropower & Informat Engn Wuhan 430074 Hubei Peoples R China China Three Gorges Corp Beijing 100038 Peoples R China Hubei Key Lab Digital Valley Sci & Technol Wuhan 430074 Hubei Peoples R China China Inst Water Resources & Hydropower Res Beijing 100038 Peoples R China

Long-term scheduling of large cascade hydropower stations (LSLCHS) is a complex problem of high dimension, nonlinearity, coupling and complex constraint. In view of the above problem, we present an improved differential evolution (iLSHADE) algorithm based on LSHADE, a state-of-the-art evolutionary algorithm. iLSHADE uses new mutation strategies "current to pbest/2-rand" to obtain wider search range and accelerate convergence with the preventing individual repeated failure evolution (PIRFE) strategy. The handling of complicated constraints strategy of e-constrained method is presented to handle outflow, water level and output constraints in the cascade reservoir operation. Numerical experiments of 10 benchmark functions have been done, showing that iLSHADE has stable convergence and high efficiency. Furthermore, we demonstrate the performance of the iLSHADE algorithm by comparing it with other improved differential evolution algorithms for LSLCHS in four large hydropower stations of the Jinsha River. With the applications of iLSHADE in reservoir operation, LSLCHS can obtain more power generation benefit than other alternatives in dry, normal, and wet years. The results of numerical experiments and case studies show that the iLSHADE has a distinct optimization effect and good stability, and it is a valid and reliable tool to solve LSLCHS problem.

关键词： energy hydropower stations differential evolution algorithm optimal scheduling epsilon-constrained method

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