The analysis of randomized search heuristics on classes of functions is fundamental for 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 the simple (1+1) EA on the class of linear functions. We improve the 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 mutations probabilities p are derived, which imply expected polynomial optimization time as long as p=O((ln n)/n) and which are tight if p=c/n for a constant c. 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 large neighborhood explored by the mutation operator does not disrupt the search.
Lipics : Leibniz International Proceedings in Informatics, 2012, p. 420-431
Randomized Search Heuristics; Evolutionary Algorithms; Linear Functions; Running Time Analysis
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Leibniz International Proceedings in Informatics
29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)Symposium on Theoretical Aspects of Computer Science
Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany