A Blend of Markov-Chain and Drift Analysis

Abstract

In their seminal article [Theo. Comp. Sci. 276(2002):51–82] Droste, Jansen, and Wegener present the first theoretical analysis of the expected runtime of a basic direct-search heuristic with a global search operator, namely the (1+1) Evolutionary Algorithm (EA), for the class of linear functions over the search space {0,1}ⁿ. In a rather long and involved proof they show that, for any linear function, the expected runtime of the EA is O(nlogn), i.e., that there are two constants c and n′ such that, for n ≥ n′, the expected number of iterations until a global optimum is generated is bound above by c·nlogn. However, neither c nor n′ are specified – they would be pretty large. Here we reconsider this optimization scenario to demonstrate the potential of an analytical method that makes use not only of the drift (w.r.t. a potential function, here the number of bits set correctly), but also of the distribution of the evolving candidate solution over the search space {0,1}ⁿ: An invariance property of this distribution is proved, which is then used to derive a significantly better lower bound on the drift. Finally, this better estimate of the drift results in an upper bound on the expected number of iterations of 3.8 nlog₂ n + 7.6log₂ n for n ≥ 2.