Complexity Theory for Discrete Black-Box Optimization Heuristics

TL;DR

This paper reviews black-box complexity models in discrete optimization, analyzing their bounds and how they inform the design of more effective evolutionary algorithms.

Contribution

It surveys existing black-box complexity models, discusses their implications, and explores how they can inspire new algorithmic strategies in evolutionary computation.

Findings

Different black-box complexity models have been developed for discrete optimization.

Bounds on performance for these models have been established.

Interplay between complexity theory and running time analysis can lead to improved algorithms.

Abstract

A predominant topic in the theory of evolutionary algorithms and, more generally, theory of randomized black-box optimization techniques is running time analysis. Running time analysis aims at understanding the performance of a given heuristic on a given problem by bounding the number of function evaluations that are needed by the heuristic to identify a solution of a desired quality. As in general algorithms theory, this running time perspective is most useful when it is complemented by a meaningful complexity theory that studies the limits of algorithmic solutions. In the context of discrete black-box optimization, several black-box complexity models have been developed to analyze the best possible performance that a black-box optimization algorithm can achieve on a given problem. The models differ in the classes of algorithms to which these lower bounds apply. This way, black-box complexity contributes to a better understanding of how certain algorithmic choices (such as the amount of memory used by a heuristic, its selective pressure, or properties of the strategies that it uses to create new solution candidates) influences performance. In this chapter we review the different black-box complexity models that have been proposed in the literature, survey the bounds that have been obtained for these models, and discuss how the interplay of running time analysis and black-box complexity can inspire new algorithmic solutions to well-researched problems in evolutionary computation. We also discuss in this chapter several interesting open questions for future work.