While most theoretical run time analyses of discrete randomized search heuristics focused on finite search spaces, we consider the search space $\mathbb{Z}^n$. This is a further generalization of the search space of multi-valued decision variables $\{0,\ldots,r-1\}^n$. We consider as fitness functions the distance to the (unique) non-zero optimum $a$ (based on the $L_1$-metric) and the \ooea which mutates by applying a step-operator on each component that is determined to be varied. For changing by $\pm 1$, we show that the expected optimization time is $\Theta(n \cdot (|a|_{\infty} + \log(|a|_H)))$. 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 $O(n \cdot \log^2 (|a|_1) \cdot \left(\log (\log (|a|_1))\right)^{1 + \epsilon})$. Furthermore, we show that RLS with step size adaptation achieves an optimization time of $\Theta(n \cdot \log(|a|_1))$. We conclude with an empirical analysis, comparing the above algorithms also with a variant of CMA-ES for discrete search spaces.

翻译：虽然离散随机搜索启发式算法的多数理论运行时间分析集中于有限搜索空间，本文考虑搜索空间$\mathbb{Z}^n$。这是多值决策变量$\{0,\ldots,r-1\}^n$搜索空间的进一步推广。我们采用基于$L_1$度量到（唯一）非零最优值$a$的距离作为适应度函数，并采用通过在每个确定变异的分量上应用步长算子进行变异的\ooea算法。对于$\pm 1$的步长变化，我们证明期望优化时间为$\Theta(n \cdot (|a|_{\infty} + \log(|a|_H)))$。特别地，该时间与最优值$a$的最大值呈线性关系。通过采用另一种步长算子，该算子从期望值为无穷大的重尾分布中选择步长，我们得到优化时间为$O(n \cdot \log^2 (|a|_1) \cdot \left(\log (\log (|a|_1))\right)^{1 + \epsilon})$。此外，我们证明具有步长自适应的RLS算法能够达到$\Theta(n \cdot \log(|a|_1))$的优化时间。最后通过实证分析，将上述算法同时与一种适用于离散搜索空间的CMA-ES变体进行比较。