A many-objective evolutionary algorithm with reference points-based strengthened dominance relation
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:where is the decision (variable) space. is a candidate solution. is a vector of m conflicting objective functions, and 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 ) can be expressed as only for every and for at least one . If there is no such that , x* is called Pareto optimal solution. The set of all the Pareto optimal solutions is defined as the Pareto optimal set (PS), , and the set of all the Pareto optimal objective vectors is the Pareto optimal front (PF*), [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:
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To increase the selection pressure toward the PF*, a new dominance relation is proposed. It is termed reference points-based strengthened dominance (RPS-dominance).
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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.
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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:where 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].
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