A clustering-based differential evolution algorithm for solving multimodal multi-objective optimization problems

https://doi.org/10.1016/j.swevo.2020.100788Get rights and content

Abstract

Multimodal Multi-objective Optimization Problems (MMOPs) refer to the problems that have multiple Pareto-optimal solution sets in decision space corresponding to the same or similar Pareto-optimal front in objective space. These problems require the optimization algorithm to locate multiple Pareto Sets (PSs). This paper proposes a differential evolution algorithm based on the clustering technique and an elite selection mechanism to solve MMOPs. In this algorithm, a Clustering-based Special Crowding Distance (CSCD) method is designed to calculate the comprehensive crowding degree in decision and objective spaces. Subsequently, a distance-based elite selection mechanism (DBESM) is introduced to determine the learning exemplars of various individuals. New individuals are generated around the exemplars to obtain a well-distributed population in both decision and objective spaces. To test the performance of the proposed algorithm, extensive experiments on the suit of CEC'2019 benchmark functions have been conducted. The results indicate that the proposed method has superior performance compared with other commonly used algorithms.

Introduction

Multi-objective optimization aims to find a set of non-dominated solutions among multiple conflicting objectives [1], [2], [3]. The non-dominated solutions in the decision space are called the Pareto Set (PS) and its mapping vector in the objective space is called the Pareto Front (PF). When dealing with multi-objective optimization problems (MOPs), multi-objective evolutionary algorithms (MOEAs) are commonly used due to their ability of finding multiple solutions in one single run. However, most MOEAs only focus on finding PF in objective space, while the information in decision space is generally ignored. These algorithms perform poorly on handling multimodal multi-objective optimization problems (MMOPs) [4], which refer to the problems that have multiple PSs in decision space corresponding to the same or similar PF in objective space as shown in Fig. 1. It is not easy to solve MMOPs as they require the optimization algorithm to find multiple PSs.

To solve MMOPs, various niching techniques [5], [6], [7] have been incorporated into MOEAs in the past few years [8], [9], [10]. Niching techniques were inspired by the way that organisms evolve in nature and they were originally introduced to solve single-objective multi-modal problems. These techniques aim to form stable niches to prevent the solutions from converging to a local area. Crowding [11,12], fitness share [13,14], clustering [15,16], and clearing [17,18] are commonly used niching techniques for multimodal problems and they can maintain the diversity of the population in decision space. However, the distributions of the population in both decision space and objective space need to be improved when solving MMOPs. To this end, a Multimodal Multi-Objective Differential Evolution algorithm using the Clustering-based Special Crowding Distance method and elite selection mechanism (MMODE_CSCD) is proposed in this paper. In this algorithm, the clustering-based special crowding distance (CSCD) is embedded into the non-dominated sorting scheme [1] to obtain well-distributed solutions in both decision space and objective space. The non-dominated sorting scheme first divides the population into multiple non-dominated levels, and all solutions corresponding to the same level are grouped into multiple classes by k-means [19] clustering algorithm in decision space. In this manner, the individuals in the same class come from the same PS. The crowding distances are calculated and assigned to each individual by using the neighborhood relationships within the same class. Moreover, a distance-based elite selection mechanism (DBESM) is designed to improve the distribution of the population on all PSs. The main contributions of this work are summarized as follows:

  • (1)

    A clustering-based special crowding distance method is proposed to seek the neighborhood relationship and measure the comprehensive crowding distance of each solution in both decision and objective spaces. The crowding distance is regarded as an indicator in the process of exemplar selection and environmental selection.

  • (2)

    A new distance-based elite selection mechanism which comprehensively considers the diversity and convergence of the population is introduced to determine the learning exemplar for each individual.

The rest of this paper is arranged as follows. Section 2 reviews the basic DE and related works. The proposed algorithm is introduced in Section 3. Section 4 presents an empirical evaluation of MMODE_CSCD in comparison with several state-of-the-art algorithms. Detailed summary and future works are given in Section 5.

Section snippets

Differential evolution

Differential evolution (DE) [20] is a popular population-based optimization algorithm because of its simple structure and high searching efficiency. Mutation, crossover, and selection are the three main operators in the evolution process of DE. To be specific, at the generation G = 0, an initial population consisting of NP individuals are generated randomly in the search domain, and each individual is called target individual and denoted by xi0=(xi,10,xi,20,,xi,D0), i = 1, 2,…, NP, where D

The proposed method

In this section, the detailed descriptions of the CSCD and DBESM are explained, and then the main framework of MMODE_CSCD is presented.

Experimental setups

In this section, the effectiveness and the superiority of MMODE_CSCD are verified on the CEC’2019 multimodal multi-objective optimization benchmark suite [37]. The benchmark suite contains 22 test functions with different characteristics, thus they can verify the performance of the proposed method systematically. In addition, this benchmark suite provides four performance indicators: the reciprocal of Pareto Sets Proximity (1/PSP, rPSP [38]), the reciprocal of Hypervolume (1/HV, rHV [37]),

Conclusions and future works

In this study, a new clustering-based differential evolution algorithm is proposed to address multimodal multi-objective optimization problems. The CSCD method adopts the k-means algorithm to overcome the shortcomings of SCD. The k-means algorithm is used to divide the individuals in the same non-dominated level into multiple classes. Thus one individual and its neighbors can be from the same PS. By doing this, more accurate CSCD values are obtained. To improve population diversity, the DBESM

CRediT authorship contribution statement

Jing Liang: Conceptualization, Investigation, Writing - review & editing. Kangjia Qiao: Software, Methodology, Writing - original draft. Caitong Yue: Software, Visualization. Kunjie Yu: Validation, Visualization. Boyang Qu: Investigation, Methodology, Writing - review & editing. Ruohao Xu: Software, Validation. Zhimeng Li: Methodology, Data curation. Yi Hu: Validation, Visualization.

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.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (61922072, 61976237, 61876169, 61806179, 61673404).

References (39)

  • H.A. Abbass et al.

    PDE: a Pareto-frontier differential evolution approach for multi-objective optimization problems

  • J.J. Liang et al.

    Multimodal multi-objective optimization: a preliminary study

  • S. Biswas et al.

    Inducing niching behavior in differential evolution through local information sharing

    IEEE Trans. Evol. Comput.

    (2015)
  • Y. Zhang et al.

    Toward fast niching evolutionary algorithms: a locality sensitive hashing-based approach

    IEEE Trans. Evol. Comput.

    (2017)
  • Y. Xu

    A niching particle swarm segmentation of infrared images

  • Y. Liu et al.

    A double-niched evolutionary algorithm and its behavior on polygon-based problems

  • B.Y. Qu et al.

    A niching multi-objective harmony search algorithm for multimodal multi-objective problems

  • O.J. Mengshoel et al.

    The crowding approach to niching in genetic algorithms

    Evol. Comput.

    (2008)
  • B. Sareni et al.

    Fitness sharing and niching methods revisited

    IEEE Trans. Evol. Comput.

    (1998)
  • Cited by (79)

    View all citing articles on Scopus
    View full text