Evolutionary Search with Multiple Utopian Reference Points in Decomposition‐Based Multiobjective Optimization

Abstract

Decomposition-based multiobjective evolutionary algorithms (MOEA/Ds) have become increasingly popular in recent years. In these MOEA/Ds, evolutionary search is guided by the used weight vectors in decomposition function to approximate the Pareto front (PF). Generally, the decomposition function will be constructed by the weight vectors and the reference point, which play an important role to balance convergence and diversity during the evolutionary search. However, in most existing MOEA/Ds, only one ideal point is used as the reference point for all the evolutionary search, which is harmful to search the entire PF when tackling the problems with difficult-to-approximate PF boundaries. To address the above problem, this paper proposes an evolutionary search method with multiple utopian reference points in MOEA/Ds. Similar to the existing MOEA/Ds, each solution is associated with one weight vector, which provides an evolutionary search direction, while the novelty of our approach is to use multiple utopian reference points, which can provide evolutionary search directions for different weight vectors. Corner solutions are used to approximate the nadir point and then multiple utopian reference points for evolutionary search can be constructed based on the ideal point and the nadir point, which are uniformly distributed on the coordinate axis or planes. The use of these utopian points can prevent solutions to gather in the same region of PF and helps to strike a good balance of exploration and exploitation in the search space. The performance of our proposed algorithm is validated on tackling 16 recently proposed test problems with difficult-to-approximate PF boundaries and empirically compared to eight state-of-the-art multiobjective evolutionary algorithms. The experimental results demonstrate the superiority of the proposed algorithm on solving most of the test problems adopted.

Decomposition-Based Multiobjective Evolutionary Optimization with Adaptive Multiple Gaussian Process Models

“…Note that most of these algorithms adopt additional reference information (reference vectors, reference points, or weight vectors) during the environmental selection, which helps to maintain the diversity of the population. Due to the advantage with a better mathematical explanation, decomposition-based MOEAs have become very popular in recent years [21][22][23][24].…”

Design of optimal blind watermarking technique based on MOEA/D

“…The multi-objective evolutionary algorithm based on decomposition (MOEA/D) has been studied in [20][21][22][23][24]. Zhang and Li [20] first proposed the MOEA/D.…”

“…Wang et al [23] presented the Pareto adaptive scalarising method (MOEA/D-PaS) which is based on the selection of appropriate norm spaces in order to increase the searching ability. Lin et al [24] improved the effectiveness of the Pareto solution search by using multiple utopian reference points instead of one.…”

An Evolutionary Algorithm with Clustering‐Based Assisted Selection Strategy for Multimodal Multiobjective Optimization

“…Pareto optimal solutions (PS) are a set of nondominated solutions for an MOP, and Pareto optimal front (PF) represents their objective values [4][5][6]. To get the optimal solutions of an MOP, many multiobjective evolutionary optimization algorithms (MOEAs) are proposed to evolve a population with multiple solutions, aiming to approximate the true PF [7][8][9]. Due to the advantage of being able to optimize multiple solutions simultaneously, MOEAs have been a widely used approach to solve various kinds of optimization problems, that is, MOEAs based on Pareto dominance relation [10][11][12][13], MOEAs based on performance indicator [14][15][16], and MOEAs based on decomposition [17][18][19][20][21][22][23][24][25].…”