Precision, Local Search and Unimodal Functions
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Colleges, School and Institutes
We investigate the effects of precision on the efficiency of various 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 (base-2) and reflected Gray code representations with single bit mutations. The standard binary method does not guarantee locating the optimum, whereas using the reflected Gray code does so in I similar to((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 Omega((log n)(2)) holds regardless of the choice of mutation distribution. For continuous multimodal functions, the algorithm also locates the global optimum in O((log n)(2)). Finally, we show that it is not possible for a black box algorithm to efficiently optimise unimodal functions for two or more dimensions (in terms of the precision used).
|Number of pages||22|
|Early online date||13 Aug 2009|
|Publication status||Published - 1 Mar 2011|