Theoretical Perspective of Convergence Complexity of Evolutionary Algorithms Adopting Optimal Mixing

TL;DR

This paper provides a theoretical analysis of the convergence complexity of optimal mixing evolutionary algorithms (OMEAs), deriving population size bounds and analyzing their behavior under different mask scenarios to enhance understanding of their efficiency.

Contribution

It introduces a theoretical framework for understanding the convergence and population requirements of OMEAs with one-layer and two-layer masks, including asymptotic bounds and empirical insights.

Findings

Population size is derived from initial supply for one-layer masks.

NFE is asymptotically bounded by evaluating the probability of evaluations.

Two-layer masks require larger populations proportional to cross competition and initial supply.

Abstract

The optimal mixing evolutionary algorithms (OMEAs) have recently drawn much attention for their robustness, small size of required population, and efficiency in terms of number of function evaluations (NFE). In this paper, the performances and behaviors of OMEAs are studied by investigating the mechanism of optimal mixing (OM), the variation operator in OMEAs, under two scenarios -- one-layer and two-layer masks. For the case of one-layer masks, the required population size is derived from the viewpoint of initial supply, while the convergence time is derived by analyzing the progress of sub-solution growth. NFE is then asymptotically bounded with rational probability by estimating the probability of performing evaluations. For the case of two-layer masks, empirical results indicate that the required population size is proportional to both the degree of cross competition and the results from the one-layer-mask case. The derived models also indicate that population sizing is decided by initial supply when disjoint masks are adopted, that the high selection pressure imposed by OM makes the composition of sub-problems impact little on NFE, and that the population size requirement for two-layer masks increases with the reverse-growth probability.