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Tight Bounds on the Optimization Time of a Randomized Search Heuristic on Linear Functions†

Published online by Cambridge University Press:  09 January 2013

CARSTEN WITT
CARSTEN WITT*
Affiliation:
DTU Informatics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark (e-mail: [email protected])
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Abstract

The analysis of randomized search heuristics on classes of functions is fundamental to the understanding of the underlying stochastic process and the development of suitable proof techniques. Recently, remarkable progress has been made in bounding the expected optimization time of a simple evolutionary algorithm, called (1+1) EA, on the class of linear functions. We improve the previously best known bound in this setting from (1.39+o(1))en ln n to en ln n+O(n) in expectation and with high probability, which is tight up to lower-order terms. Moreover, upper and lower bounds for arbitrary mutation probabilities p are derived, which imply expected polynomial optimization time as long as p = O((ln n)/n) and p = Ω(n−C) for a constant C > 0, and which are tight if p = c/n for a constant c > 0. As a consequence, the standard mutation probability p = 1/n is optimal for all linear functions, and the (1+1) EA is found to be an optimal mutation-based algorithm. Furthermore, the algorithm turns out to be surprisingly robust since the large neighbourhood explored by the mutation operator does not disrupt the search.

Keywords

Primary 68W20Secondary 68Q25
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 2 , March 2013 , pp. 294 - 318
Copyright
Copyright © Cambridge University Press 2013

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Footnotes

†

An extended abstract of this paper appeared at STACS'12 [27].

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