Bivariate estimation-of-distribution algorithms can find an exponential number of optima

Finding a large set of optima in a multimodal optimization landscape is a challenging task. Classical population-based evolutionary algorithms typically converge only to a single solution. While this can be counteracted by applying niching strategies, the number of optima is nonetheless trivially bo...

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Bibliographic Details
Published in:	Theoretical computer science Vol. 971; p. 114074
Main Authors:	Doerr, Benjamin, Krejca, Martin S.
Format:	Journal Article
Language:	English
Published:	Elsevier B.V 06.09.2023 Elsevier
Subjects:	Computer Science Estimation-of-distribution algorithms Multimodal optimization Probabilistic model building Multimodal optimization Probabilistic model building Estimation-of-distribution algorithms multimodal optimization probabilistic model building estimation-of-distribution algorithms estimation-of-distribution algorithms probabilistic model building multimodal optimization
ISSN:	0304-3975, 1879-2294
Online Access:	Get full text
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Summary:	Finding a large set of optima in a multimodal optimization landscape is a challenging task. Classical population-based evolutionary algorithms typically converge only to a single solution. While this can be counteracted by applying niching strategies, the number of optima is nonetheless trivially bounded by the population size. Estimation-of-distribution algorithms (EDAs) are an alternative, maintaining a probabilistic model of the solution space instead of a population. Such a model is able to implicitly represent a solution set far larger than any realistic population size. To support the study of how optimization algorithms handle large sets of optima, we propose the test function EqualBlocksOneMax (EBOM). It has an easy fitness landscape with exponentially many optima. We show that the bivariate EDA mutual-information-maximizing input clustering, without any problem-specific modification, quickly generates a model that behaves very similarly to a theoretically ideal model for EBOM, which samples each of the exponentially many optima with the same maximal probability. We also prove via mathematical means that no univariate model can come close to having this property: If the probability to sample an optimum is at least inverse-polynomial, there is a Hamming ball of logarithmic radius such that, with high probability, each sample is in this ball.
ISSN:	0304-3975 1879-2294
DOI:	10.1016/j.tcs.2023.114074