Analysis of Evolutionary Algorithms on Fitness Function With Time-Linkage Property

Abstract

In real-world applications, many optimization problems have the time-linkage property, that is, the objective function value relies on the current solution as well as the historical solutions. Although the rigorous theoretical analysis on evolutionary algorithms (EAs) has rapidly developed in recent two decades, it remains an open problem to theoretically understand the behaviors of EAs on time-linkage problems. This article takes the first step to rigorously analyze EAs for time-linkage functions. Based on the basic OneMax function, we propose a time-linkage function where the first bit value of the last time step is integrated but has a different preference from the current first bit. We prove that with probability <inline-formula> <tex-math notation= LaTeX >$1-o(1)$ </tex-math></inline-formula>, randomized local search and (1 + 1) EA cannot find the optimum, and with probability <inline-formula> <tex-math notation= LaTeX >$1-o(1)$ </tex-math></inline-formula>, <inline-formula> <tex-math notation= LaTeX >$(\\mu +1)$ </tex-math></inline-formula> EA is able to reach the optimum.