A Blend of Markov-Chain and Drift Analysis

In their seminal article [Theo. Comp. Sci. 2762002:51---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 n log n , i.e., that there are two constants c and n ' such that, for n ï ź n ', the expected number of iterations until a global optimum is generated is bound above by c ï ź n log n . 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 n log₂ n +7.6log₂ n for n ï ź2.