An Immune-Inspired Approach with Interval Allocation in Solving Multimodal Multi-Objective Optimization Problems with Local Pareto Sets

In practical engineering,multi-objective optimization often encounters situations where multiple Pareto sets(PS)in the decision space correspond to the same Pareto front(PF)in the objective space,known as Multi-Modal Multi-Objective Optimization Problems(MMOP).Locating multiple equivalent global PSs poses a significant challenge in real-world applications,especially considering the existence of local PSs.Effectively identifying and locating both global and local PSs is a major challenge.To tackle this issue,we introduce an immune-inspired reproduction strategy designed to produce more offspring in less crowded,promising regions and regulate the number of offspring in areas that have been thoroughly explored.This approach achieves a balanced trade-off between exploration and exploitation.Furthermore,we present an interval allocation strategy that adaptively assigns fitness levels to each antibody.This strategy ensures a broader survival margin for solutions in their initial stages and progressively amplifies the differences in individual fitness values as the population matures,thus fostering better population convergence.Additionally,we incorporate a multi-population mechanism that precisely manages each subpopulation through the interval allocation strategy,ensuring the preservation of both global and local PSs.Experimental results on 21 test problems,encompassing both global and local PSs,are compared with eight state-of-the-art multimodal multi-objective optimization algorithms.The results demonstrate the effectiveness of our proposed algorithm in simultaneously identifying global Pareto sets and locally high-quality PSs.

Coevolutionary Framework for Generalized Multimodal Multi-Objective Optimization

Most multimodal multi-objective evolutionary algorithms(MMEAs)aim to find all global Pareto optimal sets(PSs)for a multimodal multi-objective optimization problem(MMOP).However,in real-world problems,decision makers(DMs)may be also interested in local PSs.Also,searching for both global and local PSs is more general in view of dealing with MMOPs,which can be seen as generalized MMOPs.Moreover,most state-of-theart MMEAs exhibit poor convergence on high-dimension MMOPs and are unable to deal with constrained MMOPs.To address the above issues,we present a novel multimodal multiobjective coevolutionary algorithm(Co MMEA)to better produce both global and local PSs,and simultaneously,to improve the convergence performance in dealing with high-dimension MMOPs.Specifically,the Co MMEA introduces two archives to the search process,and coevolves them simultaneously through effective knowledge transfer.The convergence archive assists the Co MMEA to quickly approach the Pareto optimal front.The knowledge of the converged solutions is then transferred to the diversity archive which utilizes the local convergence indicator and the-dominance-based method to obtain global and local PSs effectively.Experimental results show that Co MMEA is competitive compared to seven state-of-the-art MMEAs on fifty-four complex MMOPs.

Zoning Search With Adaptive Resource Allocating Method for Balanced and Imbalanced Multimodal Multi-Objective Optimization

Maintaining population diversity is an important task in the multimodal multi-objective optimization.Although the zoning search(ZS)can improve the diversity in the decision space,assigning the same computational costs to each search subspace may be wasteful when computational resources are limited,especially on imbalanced problems.To alleviate the above-mentioned issue,a zoning search with adaptive resource allocating(ZS-ARA)method is proposed in the current study.In the proposed ZS-ARA,the entire search space is divided into many subspaces to preserve the diversity in the decision space and to reduce the problem complexity.Moreover,the computational resources can be automatically allocated among all the subspaces.The ZS-ARA is compared with seven algorithms on two different types of multimodal multi-objective problems(MMOPs),namely,balanced and imbalanced MMOPs.The results indicate that,similarly to the ZS,the ZS-ARA achieves high performance with the balanced MMOPs.Also,it can greatly assist a“regular”algorithm in improving its performance on the imbalanced MMOPs,and is capable of allocating the limited computational resources dynamically.

Immune Optimization Approach for Dynamic Constrained Multi-Objective Multimodal Optimization Problems

This work investigates one immune optimization approach for dynamic constrained multi-objective multimodal optimization in terms of biological immune inspirations and the concept of constraint dominance. Such approach includes mainly three functional modules, environmental detection, population initialization and immune evolution. The first, inspired by the function of immune surveillance, is designed to detect the change of such kind of problem and to decide the type of a new environment;the second generates an initial population for the current environment, relying upon the result of detection;the last evolves two sub-populations along multiple directions and searches those excellent and diverse candidates. Experimental results show that the proposed approach can adaptively track the environmental change and effectively find the global Pareto-optimal front in each environment.

Exploring a Promising Region and Enhancing Decision Space Diversity for Multimodal Multi-Objective Optimization

During the past decade,research efforts have been gradually directed to the widely existing yet less noticed multimodal multi-objective optimization problems(MMOPs)in the multi-objective optimization community.Recently,researchers have begun to investigate enhancing the decision space diversity and preserving valuable dominated solutions to overcome the shortage caused by a preference for objective space convergence.However,many existing methods still have limitations,such as giving unduly high priorities to convergence and insufficient ability to enhance decision space diversity.To overcome these shortcomings,this article aims to explore a promising region(PR)and enhance the decision space diversity for handling MMOPs.Unlike traditional methods,we propose the use of non-dominated solutions to determine a limited region in the PR in the decision space,where the Pareto sets(PSs)are included,and explore this region to assist in solving MMOPs.Furthermore,we develop a novel neighbor distance measure that is more suitable for the complex geometry of PSs in the decision space than the crowding distance.Based on the above methods,we propose a novel dual-population-based coevolutionary algorithm.Experimental studies on three benchmark test suites demonstrates that our proposed methods can achieve promising performance and versatility on different MMOPs.The effectiveness of the proposed neighbor distance has also been justified through comparisons with crowding distance methods.

A Dynamic Resource Allocation Strategy with Reinforcement Learning for Multimodal Multi-objective Optimization

Many isolation approaches, such as zoning search, have been proposed to preserve the diversity in the decision space of multimodal multi-objective optimization(MMO). However, these approaches allocate the same computing resources for subspaces with different difficulties and evolution states. In order to solve this issue, this paper proposes a dynamic resource allocation strategy(DRAS)with reinforcement learning for multimodal multi-objective optimization problems(MMOPs). In DRAS, relative contribution and improvement are utilized to define the aptitude of subspaces, which can capture the potentials of subspaces accurately. Moreover, the reinforcement learning method is used to dynamically allocate computing resources for each subspace. In addition, the proposed DRAS is applied to zoning searches. Experimental results demonstrate that DRAS can effectively assist zoning search in finding more and better distributed equivalent Pareto optimal solutions in the decision space.

基于分区搜索和强化学习的多模态多目标头脑风暴优化算法

维持种群多样性和提高算法搜索效率是多模态多目标优化亟需解决的两大问题。为解决以上问题,提出了一种基于分区搜索和强化学习的多模态多目标头脑风暴优化算法(MMBSO-ZSRL)。在MMBSO-ZSRL中,首先将决策空间分解为多个子空间以降低搜索难度和维持种群多样性;然后,使用SARSA(state-action-reward-state-action)算法来平衡头脑风暴算法的全局探索和局部开发能力;并使用特殊拥挤距离来挑选个体来指导种群进化。为了验证所提算法的性能,选取六种先进的多模态多目标优化算法来进行比较,并选取IEEE CEC2019多模态多目标问题基准测试集来对所有比较算法的性能进行测试。实验结果表明,MMBSO-ZSRL的整体性能要显著优于其他六种比较算法。MMBSO-ZSRL不仅可以找到多样性和逼近性更好的帕累托前沿,而且可以在决策空间找到更多的帕累托最优解。