Precision, Local Search and Unimodal Functions

We investigate the effects of precision on the efficiency of var- ious local search algorithms on 1-D unimodal functions. We present a (1 + 1)-EA with adaptive step size which finds the optimum in O(log n) steps, where n is the number of points used. We then consider binary and Gray representations with single bit mutations. The standard binary method does not guarantee locating the optimum, whereas using Gray code does so in O((log n)2) steps. A (1 + 1)-EA with a fixed mutation probability distribution is then presented which also runs in O((log n)2). Moreover, a recent result shows that this is optimal (up to some constant scaling factor), in that there exist unimodal functions for which a lower bound of Ω((log n)2) holds regardless of the choice of mutation dis- tribution. Finally, we show that it is not possible for a black box algorithms to efficiently optimise unimodal functions for two or more dimensions (in terms of the precision used).