A two-archive model based evolutionary algorithm for multimodal multi-objective optimization problems
Introduction
Owning multiple optimization objectives is very common in many optimization tasks. In addition, different objectives have the characters of confliction and interconnection. These problems are usually called multi-objective problems (MOPs), which can be expressed by the following equation: where x represents a set of solutions in an n-dimensional space, which is also defined as the decision space (DS). M is the number of optimization objectives. f() represents a set of solutions in an M-dimensional space corresponding to solutions in the DS. The M-dimensional space is also defined as the objective space (OS). expresses the constraint of MOP.
For two solutions x and x from the DS, it can be stated that x dominates x (x x) if the relationship of x and x is as follows:
Eq. (2) is also defined as the Pareto dominance relationship in MOPs. By comparing all the solutions in the DS via Eq. (2), the best trade-off solution set, which is also called as Pareto-optimal set (PS), can be maintained. The solutions in the OS corresponding to PS are defined as Pareto front (PF).
We normally treat MOPs with only one PS and PF. However, some practical problems possess several equivalent Pareto optimal subsets (ePSs) corresponding to one PF. As shown in Fig. 1, and are two different solutions that belong to ePS and ePS in the DS, respectively. However, in the OS, and are corresponding to the same point O. Such case is indicated as a multimodal multi-objective optimization problem (MMOP) [1]. The evolutionary algorithm designed for solving MMOPs is called multimodal multi-objective evolutionary algorithm (MMOEA). In multimodal multi-objective optimization (MMO), effectiveness and efficiency are two important goals that need to be pursued. Effectiveness means achieving good convergence and diversity both in the DS and OS when solving MMOPs. It is unsuccessful if an MMOEA obtains only one ePS in the evolutionary process. MMOEAs should approximate both the ideal PS and PF of an MMOP simultaneously. Efficiency means solving MMOPs with relatively low computational cost. The efficiency of MMOEAs is mainly reflected by the computational complexity and runtime in this paper.
MMO plays an important role in real world problems. Finding more efficient solutions is conducive to providing diverse decisions for the decision maker (DM). As shown in Fig. 2, someone from place A plans a trip to place B. 6 paths can be chosen for this travel. In addition, each of Path 4, Path 5, and Path 6 will pass by convenience store, while other paths contain no convenience store. Each of Path 1 and Path 6 will pass by 5 schools and take 2 h. Path 2 and Path 5 individually will pass by 3 schools and will take 2.5 h. Path 3 and Path 4 individually will pass by 1 school and take 3 h. From Fig. 2, it can be found that Path 1, 2, 3 and Path 4, 5, 6 are two ePSs corresponding to the same PF. Since DMs have different requirements for convenience store, MMOEA can help DMs to choose the ideal path from Path 1, 6, Path 2, 5, and Path 3, 4, for his/her trip. Otherwise, MMO can help analyze characters of practical problems [2], achieve robust solutions in engineering, and feature selection in classification [3], etc.
As described in Fig. 1, MMOP is the special case of MOP. Many researches [4], [5], [6], [7], [8], [9] indicate that multi-objective evolutionary algorithms (MOEAs) are useful tools for solving MOPs. Cheng et al. [4] proposed a reference vector guided MOEA to achieve the preferred solutions of the whole PF. Gong et al. [5] designed a preference-based algorithm with interval parameters, which can interact with a decision maker during the evolutionary process. In [6], a knee point based selection strategy was used to enhance the convergence pressure of MOEA. Rong et al. [7] proposed a new multidirectional prediction (MDP) mechanism for solving dynamic MOPs. However, the above-mentioned MOEAs pay more attention to obtain a better approximation to the ideal PF. The distribution of solutions in the DS is often neglected.
MOPs with complex PS distribution bring troubles to most existing MOEAs. Researchers have made various attempts, e.g. clustering-based scheme [10], one-by-one selection strategy [11], neighborhood-based strategy [12], and probabilistic model-based method [13] to further explore the DS. These attempts achieved some success in dealing with MOPs with different types of PS. However, when it comes to MMOPs, most existing MOEAs do not work very well [14], [15]. Because finding all the ePSs in the DS is not an explicit goal in the common MOEAs, the exploration will terminate even if only one ePS is found. It is obviously not enough for problems with multiple ePSs.
In order to achieve more multimodal solutions, some researches such as ring topology-based mechanisms [15], [16], local information-based particle swarm optimizer (PSO) [17], [18], mutation-based differential evolution (DE) [14], [19], clustering-based mechanisms [20], [21], and decomposition-based methods [22] were proposed. However, researchers usually use a single evolutionary operator (e.g. PSO, DE) as the offspring generation mechanism to produce excellent solutions when solving MMOPs. Researches about taking advantage of different kinds of evolutionary operators for improving MMOEAs are still lacking. Moreover, the existing diversity maintenance strategies in the DS are always affected by hyperparameters such as niche radius and neighborhood parameters. Furthermore, the existing MMO algorithms always rely on large scale population to locate as many multimodal solutions as possible, which will require a significant amount of computing resources.
In order to address the above-mentioned shortcomings in solving MMOPs, a Two-Archive model based MMOEA is proposed in this paper. This study aims to develop an effective and efficient MMO algorithm by using different evolutionary operators with a Two-Archive model and diversity maintenance strategies without presetting niching parameters. We also expand the proposed algorithm to deal with practical feature selection problems.
The main contributions of the present work can be summarized as follows:
(1) Based on PSO and DE, two offspring generation mechanisms with a Two-Archive model are proposed for pursuing high-quality Pareto-optimal solutions on MMOPs.
(2) Two diversity maintenance strategies named niching local search scheme and reverse vector mutation strategy are designed to explore and exploit the DS and OS. Moreover, these two strategies contain no niching parameters.
(3) Based on the Two-Archive model and diversity maintenance strategies, a novel MMO algorithm is proposed, which can obtain good convergence and diversity on MMOPs.
(4) Comparison experiments with several state-of-the-art MMOEAs have been conducted to evaluate the proposed algorithm on 22 MMOPs. Experimental results validate that the proposed algorithm can well balance the effectiveness and efficiency with a relatively smaller population size on different MMOPs.
(5) The proposed algorithm is expanded to deal with 9 practical feature selection problems. Experimental results show the superiority of the proposed algorithm in terms of selected feature number, accuracy, equivalent feature combinations number, and efficiency on different feature selection problems.
We organize the remainder of this paper as follows. Firstly, Section 2 introduces the related works. After that, Section 3 describes the proposed algorithm. Experimental setup, test functions, and indicators for the evaluation of the proposed algorithm are presented in Section 4. Section 5 introduces several state-of-the-art MMOEAs for experimental studies and comparisons. In addition, the application of the proposed algorithm in feature selection problems is also given in this section. Finally, The conclusions and future work are given in Section 6.
Section snippets
Prior works on multimodal multi-objective evolutionary algorithm
Many evolutionary algorithms have been proposed for solving MMOPs in recent years. We can roughly divide the existing MMOEAs into the following categories.
The first category is Pareto dominance-based method [1], [14], [15], [20], [21], [23], [24], [25], [26], [27]. This method aims at increasing the selection pressure towards the PF as well as improving solution distribution in the DS. Omni-Optimizer [23] is a generic optimizer applicable to various types of problems based on nondominated
Procedure of MMOHEA
The Two-Archive method [38] is a low-complexity algorithm, which separates the two evolutionary goals (i.e. convergence, diversity) directly and divides solutions into two archives. Based on this idea, different updating rules can be designed to pursue different goals [39], [40]. Inspired by this, the MMOHEA is proposed. We use the Two-Archive model to combine the competitive PSO and DE operator for building the hybrid evolutionary algorithm, where solutions obtained by these two operators form
Test functions
In order to test the proposed MMOHEA, 22 MMO test functions are used in this study. Most of these test functions have been used by many researches such as [1], [15], [23], [29], [46], [47], [48], [49], [50], [51]. These test problems contain MMF1 and MMF4 from [1], MMF5MMF10 from [15], SYM-PART-simple and SYM-PART-rotated from [29], Omni-test from [23], and other 11 benchmarks (MMF2, MMF3, and MMF11MMF19) from [46]. The characteristics of these MMOPs are indicated in Table 1.
In Table 1, ‘’
Effectiveness verification of MMOHEA
Aiming at demonstrating the effectiveness of each component in MMOHEA, experiments of algorithms with different strategies of MMOHEA are conducted on 22 test functions in this section.
The comparison algorithms comprise CMOPSO; CMOPSO with ENS-SCD (CE); CMOPSO with ENS-SCD and niching local search (CEN); Reverse vector mutation strategy based offspring generation method with ENS-SCD (RE); The proposed MMOHEA. For a fair comparison, we set the environment selection methods of CMOPSO as
Conclusions
In this paper, an efficient Two-Archive model based hybrid evolutionary MMO algorithm, termed MMOHEA, is proposed. Our anticipation is finding a method that can achieve equilibrium between effectiveness and efficiency in solving MMOPs. In the proposed MMOHEA, improved competitive PSO with niching local search and DE with reverse vector mutation are designed for pursuing evolutionary goals of convergence and diversity, respectively. In addition, we use these two improved offspring generation
CRediT authorship contribution statement
Yi Hu: Software, Visualization, Writing – original draft. Jie Wang: Methodology, Investigation, Funding acquisition, Writing – review & editing. Jing Liang: Methodology, Investigation, Funding acquisition, Writing – review & editing. Yanli Wang: Software, Validation. Usman Ashraf: Writing – review & editing. Caitong Yue: Writing – review & editing. Kunjie Yu: Software, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The work is supported by National Natural Science Foundation of China (61876169, 61922072, 62176238, 62106068, 62106230, 61976237, 61673404 and 61806179), China Postdoctoral Science Foundation (2021T140616, 2021M692920, 2020M682347), Training Plan for Young Backbone Teachers of Higher Vocational Schools in Henan Province, China (2019GZGG101), Science and Technology Project of Science and Technology Department of Henan Province, China (212102210149), Key Laboratory of Electric Drive and Control
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