A Many-Objective Problem Where Crossover Is Provably Indispensable

Abstract

This paper addresses theory in evolutionary multiobjective optimisation (EMO) and focuses on the role of crossover operators in many-objective optimisation. The advantages of using crossover are hardly understood and rigorous runtime analyses with crossover are lagging far behind its use in practice, specifically in the case of more than two objectives. We present a many-objective problem class together with a theoretical runtime analysis of the widely used NSGA-III to demonstrate that crossover can yield an exponential speedup on the runtime. In particular, this algorithm can find the Pareto set in expected polynomial time when using crossover while without crossover it requires exponential time to even find a single Pareto-optimal point. To our knowledge, this is the first rigorous runtime analysis in many-objective optimisation demonstrating an exponential performance gap when using crossover for more than two objectives.