Jansen, T., Zarges, C. (2013). Performance analysis of randomised search heuristics operating with a fixed budget. Theoretical Computer Science, 545, 39-58When for a difficult real-world optimisation problem no good problem-specific algorithm is available often randomised search heuristics are used. They are hoped to deliver good solutions in acceptable time. The theoretical analysis usually concentrates on the average time needed to find an optimal or approximately optimal solution. This matches neither the application in practice nor the empirical analysis since usually optimal solutions are not known and even if found cannot be recognised. More often the algorithms are stopped after some time. This motivates a theoretical analysis to concentrate on the quality of the best solution obtained after a pre-specified number of function evaluations called budget. Using this perspective two simple randomised search heuristics, random local search and the (1+1) evolutionary algorithm, are analysed on some well-known example problems. Upper and lower bounds on the expected quality of a solution for a fixed budget of function evaluations are proven. The analysis shows novel and challenging problems in the study of randomised search heuristics. It demonstrates the potential of this shift in perspective from expected run time to expected solution quality.authorsversionPeer reviewe

Jansen, T., Zarges, C. (2011). Analyzing different variants of immune inspired somatic contiguous hypermutations. Theoretical Computer Science, 412 (6), 517-533.Artificial immune systems can be applied to a variety of very different tasks including function optimization. There are even artificial immune systems tailored specifically for this task. In spite of their successful application there is little knowledge and hardly any theoretical investigation about how and why they perform well. Here rigorous analyses for a specific class of mutation operators introduced for function optimization called somatic contiguous hypermutation is presented. Different concrete instantiations of this operator are considered and shown to behave quite differently in general. While there are serious limitations to the performance of this type of operator even for simple optimization tasks it is proven that for some types of optimization problems it performs much better than standard bit mutations most often used in evolutionary algorithms. (C) 2010 Elsevier B.V. All rights reserved.Peer reviewe

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Extending previous analyses on function classes like linear functions, we analyze how the simple (1+1) evolutionary algorithm optimizes pseudo-Boolean functions that are strictly monotonic. These functions have the property that whenever only 0-bits are changed to 1, then the objective value strictly increases. Contrary to what one would expect, not all of these functions are easy to optimize. The choice of the constant c in the mutation probability p(n) = c/n can make a decisive difference. We show that if c < 1, then the (1+1) EA finds the optimum of every such function in Θ(n log n) iterations. For c = 1, we can still prove an upper bound of O(n(3/2)). However, for c ≥ 16, we present a strictly monotonic function such that the (1+1) EA with overwhelming probability needs 2(Ω(n)) iterations to find the optimum. This is the first time that we observe that a constant factor change of the mutation probability changes the runtime by more than a constant factor.