While most theoretical run time analyses of discrete randomized search
heuristics focused on finite search spaces, we consider the search space
Zn. This is a further generalization of the search space of
multi-valued decision variables {0,…,r−1}n.
We consider as fitness functions the distance to the (unique) non-zero
optimum a (based on the L1-metric) and the \ooea which mutates by applying
a step-operator on each component that is determined to be varied. For changing
by ±1, we show that the expected optimization time is Θ(n⋅(∣a∣∞+log(∣a∣H))). In particular, the time is linear in the
maximum value of the optimum a. 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 O(n⋅log2(∣a∣1)⋅(log(log(∣a∣1)))1+ϵ).
Furthermore, we show that RLS with step size adaptation achieves an
optimization time of Θ(n⋅log(∣a∣1)).
We conclude with an empirical analysis, comparing the above algorithms also
with a variant of CMA-ES for discrete search spaces