A Two-stage Hypervolume Contribution Approximation Method Based on R2 Indicator

Abstract

Hypervolume-based multi-objective evolutionary algorithms (HV-MOEAs) are one of the popular algorithm classes in the evolutionary multi-objective optimization (EMO) community. HV-MOEAs, which can directly optimize the HV of a solution set, are useful in various applications. However, the computation time of HV-MOEAs is very long for many-objective problems since the calculation of the hypervolume contribution (HVC) is computationally expensive. Therefore, a number of approximation methods for the HVC calculation were proposed to reduce its time cost. An R2-based hypervolume contribution approximation (R2-HVC) method was proposed for HVC approximation. However, for HV-MOEAs, the point is to find the worst solution, instead of accurately approximating the HVC of each solution. In this paper, a novel method (i.e., two-stage R2-HVC) is proposed for improving the ability of R2-HVC to correctly identify the worst solution (i.e., the solution with the smallest HVC value) in a solution set. In the proposed method, some candidate solutions are selected based on rough HVC approximation in the first stage, and they are carefully evaluated in the second stage. It is shown through computational experiments that the proposed method performs much better than the original R2-HVC method.