Exploring equivalent sparse optimal solutions of large-scale multi-modal multi-objective optimization problems using cooperative evolution

A multi-modal multi-objective optimization problem is a special kind of multi-objective optimization problem with multiple Pareto subsets. In this paper, we propose an efficient multi-modal multi-objective optimization algorithm based on the widely used MOEA/D algorithm. In our proposed algorithm, each weight vector has its own sub-population. With a clearing mechanism and a greedy removal strategy, our proposed algorithm can effectively preserve equivalent Pareto optimal solutions (i.e., different Pareto optimal solutions with same objective values). Experimental results show that our proposed algorithm can effectively preserve the diversity of solutions in the decision space when handling large-scale multi-modal multi-objective optimization problems.

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.

Multimodal multiobjective optimization problems, characterized by multiple solutions mapping to identical objective vectors, are ubiquitous in real-world applications. Despite their prevalence, most existing multimodal multiobjective evolutionary algorithms (MMOEAs) predominantly focus on identifying global Pareto sets, often overlooking the equally significant local Pareto sets. While some algorithms attempt to address local Pareto sets, their performance in the objective space remains suboptimal. The inherent challenge lies in the fact that a single strategy cannot effectively tackle problems with and without local Pareto fronts. This study proposes a novel approach that first detects the presence of local Pareto fronts using a neural network, thereby enabling adaptive adjustments to the algorithm’s selection strategy and search scope. Based on this detection mechanism, we design a microscale searching multimodal multiobjective evolutionary algorithm (MMOEA_MS). Through extensive experiments on twenty-two benchmark problems, MMOEA_MS demonstrates superior performance in identifying local Pareto fronts and outperforms existing algorithms in the objective space. This study highlights the effectiveness of MMOEA_MS in solving multimodal multiobjective optimization problems with diverse Pareto front characteristics, thereby addressing key limitations of current methodologies.

Multi-objective sand cat swarm optimization based on adaptive clustering for solving multimodal multi-objective optimization problems

A ring-hierarchy-based evolutionary algorithm for multimodal multi-objective optimization

In recent years, numerous efficient and effective multimodal multi-objective evolutionary algorithms (MMOEAs) have been developed to address multimodal multi-objective optimization problems (MMOPs) involving multiple equivalent sets of Pareto optimal solutions to be found simultaneously. However, the Pareto optimal solutions may have various contracting or expending shapes, and have random locations in the decision space. In addition, uniform decision distribution does not imply good objective distribution. Therefore, many existing MMOEAs are very difficult to guide the individuals converged to every Pareto subregion with good distribution in both the decision space and the objective space. In this paper, we present a multi-population evolutionary algorithm to search for the equivalent global Pareto optimal solutions. The original population should be divided into two groups of subpopulations with equal size. The first subpopulation is designed to search for the optimal solutions in objective space. At the same time. the second subpopulation focus to obtain high-quality optimal solutions in the decision space. The multi-population strategy is helpful to improve the decision and objective distributions simultaneously, and address the MMOPs effectively. The proposed algorithm is compared against five state-of-the-art MMOEAs. The experimental results indicate the proposed algorithm provides better performance than competing MMOEAs on IEEE CEC 2019 MMOPs test suite.

Dual-space distribution metric-based evolutionary algorithm for multimodal multi-objective optimization

Multimodal multiobjective optimization problems (MMOPs) have attracted wide attention in recent years. This kind of problem is very challenging since they need to locate different Pareto-optimal solution sets (PSs) that correspond to the same Pareto front. To resolve it, this article proposes a novel multimodal multiobjective differential evolution (DE) algorithm with species conservation, which develops a new way of locating different PSs. Specifically, the proposed algorithm adopts species conservation to determine different PSs in known areas, while it uses a variant of DE as the basic optimizer to explore new areas. There are three operators in species conservation: 1) species division; 2) seed determination; and 3) seed conservation. Species division mainly partitions the joint population of parents and children into various species in the decision space for retaining different PSs. Seed determination selects superior solutions from each species as seeds that need to be kept in the next generation. Seed conservation is to ensure that all species seeds are retained in the new generation by substituting no promising solutions with them, thereby guarantee not missing some known areas that may contain different PSs. Besides, the DE variant is utilized to produce diverse solutions to find new areas in the decision space where PSs may exist. The comparative experiments with ten state-of-the-art algorithms have been performed on the CEC 2019 MMOPs test set and two real-world problems. The experimental results have verified that the proposed algorithm has a competitive performance for MMOPs.

Recently, the large-scale optimization problems have become a common research topic in the field of evolutionary computation. It is hard to find optimal solutions when solving large-scale multi-objective optimization problems (LSMOPs), due to the ineffectiveness of existing operators. In the other word, the search ability of most existing MOEAs on solving LSMOPs is still weak. To address this issue, an efficient competitive swarm optimizer with a strong exploration ability, denoted as E-CSO, is presented in this paper, which designs a novel three-particle-based particle updating strategy to improve the search efficiency. The experimental results validate the high efficiency and effectiveness of our proposed approach when solving various LSMOPs.

In artificial intelligence, evolutionary algorithms (EAs) have shown to be effective and robust in solving difficult optimization problems. EAs are generic populationbased metaheuristic optimization algorithms. The mechanisms used in EAs are inspired by biological evolution: reproduction, mutation, recombination, and selection. The development of EAs can be classified into two categories: single objective and multi-objective optimization. In this thesis, both single objective and multi-objective evolutionary algorithms have been studied. For single objective optimization, various niching techniques are integrated with differential evolution (DE) and particle swarm optimization (PSO) for multi-modal optimization. Multi-modal optimization deals with optimization tasks that involve finding all or most of the global/local peaks in one single run. EAs in their original forms are usually designed for locating one single global solution. To promote and maintain formation of multiple stable subpopulations within a single population, we introduced a neighborhood mutation technique to enhance DE with ability of handling multi-modal problems. We also proposed a locally informed PSO to tackle multi-modal optimization. Beside these, several existing niching techniques from the literature were modified and improved by us.

Recently, multimodal multiobjective optimization problems (MMOPs) have received increasing attention. Their goal is to find a Pareto front and as many equivalent Pareto optimal solutions as possible. Although some evolutionary algorithms for them have been proposed, they mainly focus on the convergence rate in the decision space while ignoring solutions diversity. In this paper, we propose a new multiobjective fireworks algorithm for them, which is able to balance exploitation and exploration in the decision space. We first extend a latest single-objective fireworks algorithm to handle MMOPs. Then we make improvements by incorporating an adaptive strategy and special archive guidance into it, where special archives are established for each firework, and two strategies (i.e., explosion and random strategies) are adaptively selected to update the positions of sparks generated by fireworks with the guidance of special archives. Finally, we compare the proposed algorithm with eight state-of-the-art multimodal multiobjective algorithms on all 22 MMOPs from CEC2019 and several imbalanced distance minimization problems. Experimental results show that the proposed algorithm is superior to compared algorithms in solving them. Also, its runtime is less than its peers'.

Single-objective continuous optimization can be challenging, especially when dealing with multimodal problems. This work sheds light on the effects that multi-objective optimization may have in the single-objective space. For this purpose, we examine the inner mechanisms of the recently developed sophisticated local search procedure SOMOGSA. This method solves multimodal single-objective continuous optimization problems based on first expanding the problem with an additional objective (e.g., a sphere function) to the bi-objective domain and subsequently exploiting local structures of the resulting landscapes. Our study particularly focuses on the sensitivity of this multiobjectivization approach w.r.t. (1) the parametrization of the artificial second objective, as well as (2) the position of the initial starting points in the search space. As SOMOGSA is a modular framework for encapsulating local search, we integrate Nelder–Mead local search as optimizer in the respective module and compare the performance of the resulting hybrid local search to its original single-objective counterpart. We show that the SOMOGSA framework can significantly boost local search by multiobjectivization. Hence, combined with more sophisticated local search and metaheuristics, this may help solve highly multimodal optimization problems in the future.

Multimodal optimization is one of the most challenging issues in the field of optimization, which requires to detect and locate multiple global and local optima. Differential evolution (DE) is a well-known and powerful optimization algorithm with fast convergence capability. In this paper, we proposed a method to accurately solve high dimensional multimodal problems based on DE. Parallel sub-populations that are created using roaming algorithm, are randomly assigned with several chaotically improved strategies. Furthermore, a novel Hill-Valley method is proposed for detecting whether two points are in same species or not. Finally, our proposed approach is compared with well-known state-of-the-art niching algorithms and results show that our approach outperforms all of them.

Effectively solving multimodal multiobjective optimization problems (MMOPs) requires maintaining an optimal balance between the diversity and the convergence. Traditional algorithms often struggle with environmental selection adaptability, leading to suboptimal performance across diverse MMOPs. This article innovatively integrates the actor-critic reinforcement learning (RL) with the evolutionary algorithm, significantly improving environmental selection adaptability through synergistic online learning between its actor and critic components. An RL process that dynamically optimizes the niche size is established, which critically determines the trade-off between the diversity and convergence preferences. The process is formulated by defining: 1) convergence and diversity measures as the state; 2) niche size adjustment as the continuous action; and 3) state improvement as the reward. Two specialized fully connected neural networks are employed as the actor network for action generation and the critic network for value estimation, which collaboratively adapt to population states through real-time online learning. This adaptive niching technology, integrated with local convergence quality assessment, enables comprehensive evaluation of both diversity and potential convergence. Extensive experimental validation demonstrates the superior performance of the proposed algorithm against ten state-of-the-art algorithms across 48 benchmark problems and a real-world application. The results consistently show significant improvements in balance maintenance and overall optimization effectiveness compared to existing algorithms.

Balancing convergence and diversity in the decision space is essential in solving multimodal multi-objective optimization problems (MMOPs), which have multiple equivalent Pareto optimal sets (PSs) with the same Pareto optimal front (PF). For MMOPs with an imbalance between convergence and diversity in the decision space (MMOP-ICD), numerous efficient multimodal multiobjective evolutionary algorithms (MMEAs) avoid premature convergence and search for the imbalanced PS by relaxing the traditional convergence-first selection mechanism. Unfortunately, existing MMEAs suffer from convergence degradation due to excessive relaxation of the convergence-first selection mechanism. Therefore, this paper proposes an improved convergence-relaxed mechanism that includes an enhanced local convergence indicator and a two-stage mating selection. The enhanced local convergence indicator introduces the global convergence indicator into the local convergence indicator. The local convergence indicator can locate more equivalent PSs and prevent premature convergence caused by the global convergence indicator. The global convergence indicator can improve the convergence quality of the solution selected by the local convergence indicator. Then, the two-stage mating selection is used to enhance the diversity in the decision space and balance the improved convergence. Experimental results and statistical analysis show that the proposed algorithm is significantly superior to other state-of-the-art MMEAs.