PCA-assisted reproduction for continuous multi-objective optimization with complicated Pareto optimal set

Abstract

Abstract In terms of generating offspring solutions, simulated binary crossover (SBX) and differential evolution (DE) are two of the most representative reproduction operators in evolutionary multi-objective algorithms (EMOAs). However, they are found as less effective on multi-objective problems (MOPs) with complicated Pareto optimal set (PS). Under mild conditions, the PS of an MOP is an ( m − 1 ) -dimensional piecewise continuous manifolds where m is the number of objectives. Inspired from this regularity property, this study proposes a simple yet effective reproduction operator, namely, PCA-assisted reproduction (PCA-ar). Specifically, the PCA-ar first applies principal component analysis (PCA) method to construct a new decision space with reduced number of dimensions based on the information extracted from several well converged solutions. The PS is then estimated by a hyperplane in the new decision space. To this end, new offspring are sampled from the estimated PS, and then re-converted to the original decision space for fitness calculation. In order to systematically examine the effectiveness of the PCA-ar operator, we integrate it into NSGA-II and MOEA/D, and compare the derived algorithms, nNSGA-II and nMOEA/D, with their original versions (NSGA-II with SBX operator and MOEA/D with DE operator) as well as the regularity model based multi-objective estimated distribution algorithm (RM-MEDA) on the modified DTLZ benchmarks with up to 8 objectives. Experimental results show that nNSGA-II and nMOEA/D outperform the competitor EMOAs for most of problems, which indicate that the PCA-ar is effective. Lastly, the PCA-ar is also demonstrated to have good scalability to the number of decision variables.