Theoretical Analyses of Evolutionary Algorithms on Time-Linkage OneMax with General Weights

TL;DR

This paper provides a theoretical analysis of evolutionary algorithms on time-linkage OneMax problems with general weights, revealing limitations in convergence depending on the sign and magnitude of the weights.

Contribution

It extends the understanding of time-linkage effects by analyzing general weights, showing how they influence the convergence of RLS and (1+1)EA.

Findings

Negative weights hinder convergence with high probability.

Positive weights greater than one allow some chance of successful convergence.

Small positive weights have less impact on the algorithms' ability to find the global optimum.

Abstract

Evolutionary computation has shown its superiority in dynamic optimization, but for the (dynamic) time-linkage problems, some theoretical studies have revealed the possible weakness of evolutionary computation. Since the theoretically analyzed time-linkage problem only considers the influence of an extremely strong negative time-linkage effect, it remains unclear whether the weakness also appears in problems with more general time-linkage effects. Besides, understanding in depth the relationship between time-linkage effect and algorithmic features is important to build up our knowledge of what algorithmic features are good at what kinds of problems. In this paper, we analyze the general time-linkage effect and consider the time-linkage OneMax with general weights whose absolute values reflect the strength and whose sign reflects the positive or negative influence. We prove that except for some small and positive time-linkage effects (that is, for weights $0$ and $1$), randomized local search (RLS) and (1+1)EA cannot converge to the global optimum with a positive probability. More precisely, for the negative time-linkage effect (for negative weights), both algorithms cannot efficiently reach the global optimum and the probability of failing to converge to the global optimum is at least $1-o(1)$. For the not so small positive time-linkage effect (positive weights greater than $1$), such a probability is at most $c+o(1)$ where $c$ is a constant strictly less than $1$.