Elsevier

Information Sciences

Volume 554, April 2021, Pages 236-255
Information Sciences

A many-objective evolutionary algorithm with reference points-based strengthened dominance relation

https://doi.org/10.1016/j.ins.2020.12.025Get rights and content

Abstract

The main issues for the optimization of many-objective evolutionary are about two aspects: the balance between convergence and diversity, and increasing the selection pressure toward the true Pareto-optimal front. To overcome these difficulties, a new Reference Points-based Strengthened dominance relation (RPS-dominance) is proposed and integrated into NSGA-II, named RPS-NSGA-II. It introduces a reference point set and convergence metric Cov to distinguish Pareto-equipment solutions and further stratifies them. The performance of RPS-NSGA-II is evaluated by the WFG and MaF series benchmark problems. Extensive experimental results demonstrate that RPS-NSGA-II has the competitiveness and frequently better results when compared against the main existing algorithm (five recently proposed decomposition-based MOEAs) on 90 commonly-used benchmark problems involving up to 20 objectives.

Introduction

The Many-objective Optimization Problems (MaOPs) refer to the optimization tasks involving m (i.e., m > 3) conflicting objectives to be optimized concurrently [47]. Generally, a MaOP with only box constraints can be stated as follows:MinimizeF(x)=(f1(x),,fm(x))T,subjecttoxΩwhere Ω=Πi=1n[ui,li]Rn, Ω is the decision (variable) space. x=(x1,,xn)T,xΩ is a candidate solution. F:Ωm is a vector of m conflicting objective functions, and m is called the objective space. MaOPs usually have many optimal solutions which can be called non-dominated solutions. Some important definitions are displayed. There are two candidate solutions x1 and x2 that x1 dominates x2 (denoted as x1x2) can be expressed as only fi(x1)fi(x2) for every i{1,,m} and fj(x1)<fj(x2) for at least one j{1,,m}. If there is no xΩ such that xx, x* is called Pareto optimal solution. The set of all the Pareto optimal solutions is defined as the Pareto optimal set (PS), PS=xΩ|x'Ω,x'x, and the set of all the Pareto optimal objective vectors is the Pareto optimal front (PF*), PF={F(x)m|xPS} [47].

During the last two decades, various multi-objective evolutionary algorithms (MOEAs) have been verified to be suitable for multi-objective optimization problems. These algorithms can be divided into three categories roughly: performance indicator-based algorithms [34], [38] dominance relation-based algorithms [13], [32] decomposition-based algorithm [26], [46]. However, recent studies on MOEAs have proven that most of them are confronted with massive difficulties in solving MaOPs [47]. One primary reason for the failure can be attributed to the loss of selection pressure toward the PF*. With the increase of objectives, most population members of MOEAs become non-dominated with one another. So, the selection mechanism based on Pareto-dominance is challenging to distinguish them. The search capability of the MOEAs also deteriorates sharply [19]. The other reason can be regarded as the difficulty in maintaining a balance between the population diversity and convergence in a high-dimensional objective space [43]. Diversity and convergence are both desired by MOEAs, but most of them can not achieve simultaneously [14].

This paper aims to overcome the two challenges and focuses on the dominance relation and decomposition-based MOEAs. Motivated by the potential benefits of combining dominance relation with decomposition-based approach (i.e. MOEA/DD [21] NSGA-III [12] GWS-PLS [5] et al.), a new reference points-based strengthened dominance relation is proposed, which introduces the idea of decomposition. And it is embedded into NSGA-II [13] to build a new algorithm for solving unconstrained (with box constraints only) MaOPs. The main contributions of the proposed algorithm can be summarized as follows:

  • To increase the selection pressure toward the PF*, a new dominance relation is proposed. It is termed reference points-based strengthened dominance (RPS-dominance).

  • To strike a balance between convergence and diversity in the high-dimensional objective space, the density of reference point and a convergence metric Cov are introduced. For calculating the density of reference point, the niche technology which is based on the angles between the candidate solutions and reference directions is employed.

  • The RPS-dominance is merged into NSGA-II to build a new MaOEA, i.e., RPS dominance-based NSGA-II (RPS-NSGA-II). The effectiveness of RPS-NSGA-II is evaluated against WFG [18] and MaF [10] benchmark suites. And the results demonstrate that our proposal is efficacy for dealing with problems characterized by concave, convex, and mixed.

The rest of this paper is organized as follows. First, related works are reviewed in Section 2. Section 3 is devoted to the detailed description of the proposed dominance relation and algorithm. The experimental setup is shown in Section 4. The obtained experimental results and discussions are displayed in Section 5. And the conclusions are drawn in Section 6.

Section snippets

The dominance relation

As mentioned above, traditional dominance relations (i.e. CDAS [32] (1-k)-dominance [15] et al.) have an inferior performance on MaOPs with the increase of objectives. In the high-dimensional solution space, candidate solutions are almost independent and difficult to distinguish. Consequently, the selection pressure toward the PF* is exponential reduction. To enhance selection pressure, some scholars were making efforts to modify the old dominance relations [14], [20], [33] and some put forward

Generate reference points set

Most MOEAs use the Systematic approach to generate a set of reference points [12]. On a unit simplex for M objectives, the number of the reference points is:H=M+p-1M-1where p is the number of divisions considered along each objective coordinate axis. Deb et al. [12] pointed out that as long as p ≥ M is not chosen, no intermediate point will be created by the Systematic appoach. Therefore, the two-layer approach was to be proposed and widely used in many MaOEAs [12], [21]. An example is

Experimental settings

This section is devoted to the experimental design for investigating the performance of RPS-NSGA-II. It describes the benchmark problems and the performance metrics used in empirical studies first. Afterwards, the comparison algorithms and the parameter settings are displayed. Finally, the non-parametric statistical testing approach is introduced.

Performance analysis

The obtained experimental results of RPS dominance-based NSGA-II (RPS-NSGA-II) are shown in this section. It is compared with NSGA-III [12], MOEA/DD [21], RVEA [9], RPD-NSGAII [14], and NSGAII-SDR [37] on the WFG [18] and MaF [10] benchmark problems with up to 20 objectives. All the experiments and the Wilcoxon signed-rank test are implemented on the PlatEMO1 [36]. The results of the pairwise comparisons are presented using the sign

Conclusion and future work

This paper presents a new reference points-based strengthened dominance in NSGA-II for MaOPs, termed RPS-NSGA-II. Its purposes are to increase the pressure of solution selection and strike a balance between diversity and convergence. According to the RPS-dominance conditions, the Pareto non-dominance solutions are further distinguished. Meanwhile, the diversity and convergence of the population are emphasized by introducing the convergence metric Cov and a niche technology based on the angles

CRediT authorship contribution statement

Qinghua Gu: Conceptualization, Methodology, Investigation, Supervision, Project administration, Funding acquisition, Writing - original draft. Huayang Chen: Methodology, Software, Validation, Visualization, Writing - original draft, Writing - review & editing. Lu Chen: Methodology, Writing - original draft, Writing - review & editing. Xinhong Li: Writing - review & editing. Neal N. Xiong: 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

We would like to convey our gratitude to the editors and reviewers for their excellent advice to improve the quality of this work.

Funding sources

This work was supported by the National Natural Science Foundation of China [grant number 51774228]; Shaanxi Province Fund for Distinguished Young Scholars [grant number 2020JC-44].

References (50)

  • Biswas Subhodip, Das Swagatam, Suganthan Ponnuthurai, et al., Evolutionary multiobjective optimization in dynamic...
  • X. Cai et al.

    A decomposition-based many-objective evolutionary algorithm with two types of adjustments for direction vectors

    IEEE Trans. Cybern.

    (2018)
  • X. Cai et al.

    A constrained decomposition approach with grids for evolutionary multiobjective optimization

    IEEE Trans. Evol. Comput.

    (2018)
  • X. Cai et al.

    A grid weighted sum pareto local search for combinatorial multi- and many-objective optimization

    IEEE Trans. Cybern.

    (2019)
  • X. Cai et al.

    The collaborative local search based on dynamic-constrained decomposition with grids for combinatorial multiobjective optimization

    IEEE Trans. Cybern.

    (2019)
  • J. Carrasco et al.

    Recent trends in the use of statistical tests for comparing swarm and evolutionary computing algorithms: Practical guidelines and a critical review

    Swarm Evol. Comput.

    (2020)
  • R. Cheng et al.

    A reference vector guided evolutionary algorithm for many-objective optimization

    IEEE Trans. Evol. Comput.

    (2016)
  • Ran Cheng et al.

    A benchmark test suite for evolutionary many-objective optimization

    Complex Intelligent Syst.

    (2017)
  • ChengzhongL. et al.

    Evolutionary many objective optimization based on bidirectional decomposition

    J. Syst. Eng. Electron.

    (2019)
  • K. Deb et al.

    An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, Part I: solving problems with box constraints

    IEEE Trans. Evol. Comput.

    (2014)
  • K. Deb et al.

    A fast and elitist multiobjective genetic algorithm: NSGA-II

    IEEE Trans. Evol. Comput.

    (2002)
  • M. Elarbi et al.

    A new decomposition-based NSGA-II for many-objective optimization

    IEEE Trans. Syst., Man, Cybernetics: Syst.

    (2018)
  • M. Farina et al.

    A fuzzy definition of “optimality” for many-criteria optimization problems

    IEEE Trans. Syst. Man Cybern. Part A Syst. Hum.

    (2004)
  • Z. He et al.

    Fuzzy-based pareto optimality for many-objective evolutionary algorithms

    IEEE Trans. Evol. Comput.

    (2014)
  • S. Huband et al.

    A review of multiobjective test problems and a scalable test problem toolkit

    IEEE Trans. Evol. Comput.

    (2006)
  • Cited by (21)

    • A constrained multi-objective evolutionary algorithm based on decomposition with improved constrained dominance principle

      2022, Swarm and Evolutionary Computation
      Citation Excerpt :

      Then the Wilcoxon rank-sum test [49] is used to evaluate the results of MOEA/D-ICDP. In the Wilcoxon rank-sum test, the significance level is set at 0.05 and the p-value of the hypothesis is calculated [50]. When the p-value > 0.05, the two CMOEAs do not have a significant difference.

    • Multi-objective planning for voltage sag compensation of sparse distribution networks with unified power quality conditioner using improved NSGA-III optimization

      2022, Energy Reports
      Citation Excerpt :

      Sato H (2010) proposed a multi-objective optimization algorithm based on Controlling Dominance Area of Solutions (CDAS) and Self-Controlling Dominance Area of Solutions (S-CDAS) in succession, which carries out adaptive dominance based on fixed marker vector [13,14]. Gu (2021) proposed a many-objective evolutionary algorithm with reference points-based strengthened dominance relation, which used convergence degree Cov to strengthen the Pareto dominance [15]. However, these methods could not take account in all dimensions of Pareto front’s feature.

    View all citing articles on Scopus
    View full text