Complexity Theory for Discrete Black-Box Optimization Heuristics

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.Comment: This survey article is to appear (in a slightly modified form) in the book "Theory of Randomized Search Heuristics in Discrete Search Spaces", which will be published by Springer in 2018. The book is edited by Benjamin Doerr and Frank Neumann. Missing numbers of pointers to other chapters of this book will be added as soon as possibl

Theoretical and Empirical Evaluation of Diversity-preserving Mechanisms in Evolutionary Algorithms: On the Rigorous Runtime Analysis of Diversity-preserving Mechanisms in Evolutionary Algorithms

Evolutionary algorithms (EAs) simulate the natural evolution of species by iteratively applying evolutionary operators such as mutation, recombination, and selection to a set of solutions for a given problem. One of the major advantages of these algorithms is that they can be easily implemented when the optimisation problem is not well understood, and the design of problem-specific algorithms cannot be performed due to lack of time, knowledge, or expertise to design problem-specific algorithms. Also, EAs can be used as a first step to get insights when the problem is just a black box to the developer/programmer. In these cases, by evaluating candidate solutions it is possible to gain knowledge on the problem at hand. EAs are well suited to dealing with multimodal problems due to their use of a population. A diverse population can explore several hills in the fitness landscape simultaneously and offer several good solutions to the user, a feature desirable for decision making, multi-objective optimisation and dynamic optimisation. However, a major difficulty when applying EAs is that the population may converge to a sub-optimal individual before the fitness landscape is explored properly. Many diversity-preserving mechanisms have been developed to reduce the risk of such premature convergence and given such a variety of mechanisms to choose from, it is often not clear which mechanism is the best choice for a particular problem. We study the (expected/average) time for such algorithms to find satisfactory solutions for multimodal and multi-objective problems and to extract guidelines for the informed design of efficient and effective EAs. The resulting runtime bounds are used to predict and to judge the performance of algorithms for arbitrary problem sizes, further used to clarify important design issues from a theoretical perspective. We combine theoretical research with empirical applications to test the theoretical recommendations for their practicality, and to engage in rapid knowledge transfer from theory to practice. With this approach, we provide a better understanding of the working principles of EAs with diversity-preserving mechanisms. We provide theoretical foundations and we explain when and why certain diversity mechanisms are effective, and when they are not. It thus contributes to the informed design of better EAs

Evolutionary computation for digital art

PowerPoint presentationAneta Neumann, Frank Neuman