Abstract
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 Θ((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 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).
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It was with great sorrow that we learned of the death of Ingo Wegener on 26 November 2008. Ingo brought the rest of us together to work on this problem. He is greatly missed by us all.
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Dietzfelbinger, M., Rowe, J.E., Wegener, I. et al. Precision, Local Search and Unimodal Functions. Algorithmica 59, 301–322 (2011). https://doi.org/10.1007/s00453-009-9352-x
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DOI: https://doi.org/10.1007/s00453-009-9352-x