Black-Box Search by Unbiased Variation

The complexity theory for black-box algorithms, introduced by Droste, Jansen, and Wegener (Theory Comput. Syst. 39:525-544, 2006), describes common limits on the efficiency of a broad class of randomised search heuristics. There is an obvious trade-off between the generality of the black-box model and the strength of the bounds that can be proven in such a model. In particular, the original black-box model provides for well-known benchmark problems relatively small lower bounds, which seem unrealistic in certain cases and are typically not met by popular search heuristics. In this paper, we introduce a more restricted black-box model for optimisation of pseudo-Boolean functions which we claim captures the working principles of many randomised search heuristics including simulated annealing, evolutionary algorithms, randomised local search, and others. The key concept worked out is an unbiased variation operator. Considering this class of algorithms, significantly better lower bounds on the black-box complexity are proved, amongst them an Omega(nlogn) bound for functions with unique optimum. Moreover, a simple unimodal function and plateau functions are considered. We show that a simple (1+1) EA is able to match the runtime bounds in several cases.