Run Time Bounds for Integer-Valued OneMax Functions

While most theoretical run time analyses of discrete randomized search heuristics focused on finite search spaces, we consider the search space . This is a further generalization of the search space of multi-valued decision variables . We consider as fitness functions the distance to the (unique) non-zero optimum (based on the -metric) and the \ooea which mutates by applying a step-operator on each component that is determined to be varied. For changing by , we show that the expected optimization time is . In particular, the time is linear in the maximum value of the optimum . Employing a different step operator which chooses a step size from a distribution so heavy-tailed that the expectation is infinite, we get an optimization time of . Furthermore, we show that RLS with step size adaptation achieves an optimization time of . We conclude with an empirical analysis, comparing the above algorithms also with a variant of CMA-ES for discrete search spaces.
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