Publications

Convergence of a discretized self-adaptive evolutionary algorithm on multi-dimensional problems

Hart, William E.; Hart, William E.; Delaurentis, John M.

We consider the convergence properties of a non-elitist self-adaptive evolutionary strategy (ES) on multi-dimensional problems. In particular, we apply our recent convergence theory for a discretized (1,{lambda})-ES to design a related (1,{lambda})-ES that converges on a class of seperable, unimodal multi-dimensional problems. The distinguishing feature of self-adaptive evolutionary algorithms (EAs) is that the control parameters (like mutation step lengths) are evolved by the evolutionary algorithm. Thus the control parameters are adapted in an implicit manner that relies on the evolutionary dynamics to ensure that more effective control parameters are propagated during the search. Self-adaptation is a central feature of EAs like evolutionary stategies (ES) and evolutionary programming (EP), which are applied to continuous design spaces. Rudolph summarizes theoretical results concerning self-adaptive EAs and notes that the theoretical underpinnings for these methods are essentially unexplored. In particular, convergence theories that ensure convergence to a limit point on continuous spaces have only been developed by Rudolph, Hart, DeLaurentis and Ferguson, and Auger et al. In this paper, we illustrate how our analysis of a (1,{lambda})-ES for one-dimensional unimodal functions can be used to ensure convergence of a related ES on multidimensional functions. This (1,{lambda})-ES randomly selects a search dimension in each iteration, along which points generated. For a general class of separable functions, our analysis shows that the ES searches along each dimension independently, and thus this ES converges to the (global) minimum.