Optimizing Linear Functions with Randomized Search Heuristics - The Robustness of Mutation

Abstract

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.