Randomized search heuristics have been applied successfully to a plethora of problems. This success is complemented by a large body of theoretical results. Unfortunately, the vast majority of these results regard problems with binary or continuous decision variables -- the theoretical analysis of randomized search heuristics for unbounded integer domains is almost nonexistent. To resolve this shortcoming, we start the runtime analysis of multi-objective evolutionary algorithms, which are among the most successful randomized search heuristics, for unbounded integer search spaces. We analyze single- and full-dimensional mutation operators with three different mutation strengths, namely changes by plus/minus one (unit strength), random changes following a law with exponential tails, and random changes following a power-law. The performance guarantees we prove on a recently proposed natural benchmark problem suggest that unit mutation strengths can be slow when the initial solutions are far from the Pareto front. When setting the expected change right (depending on the benchmark parameter and the distance of the initial solutions), the mutation strength with exponential tails yields the best runtime guarantees in our results -- however, with a wrong choice of this expectation, the performance guarantees quickly become highly uninteresting. With power-law mutation, which is an essentially parameter-less mutation operator, we obtain good results uniformly over all problem parameters and starting points. We complement our mathematical findings with experimental results that suggest that our bounds are not always tight. Most prominently, our experiments indicate that power-law mutation outperforms the one with exponential tails even when the latter uses a near-optimal parametrization. Hence, we suggest to favor power-law mutation for unknown problems in integer spaces.

翻译：随机搜索启发式算法已在众多问题中成功应用，这一成功得到了大量理论成果的补充。然而，这些成果绝大多数关注的是具有二进制或连续决策变量的问题——针对无界整数域的随机搜索启发式算法的理论分析几乎不存在。为弥补这一不足，我们首次对无界整数搜索空间中的多目标进化算法（作为最成功的随机搜索启发式算法之一）展开运行时分析。我们分析了具有三种不同变异强度的单维和全维变异算子：加减一的单位强度变异、遵循指数尾分布律的随机变异，以及遵循幂律分布的随机变异。我们在近期提出的自然基准问题上证明的性能保证表明，当初始解远离帕累托前沿时，单位变异强度可能导致收敛缓慢。若恰当设置期望变化量（取决于基准参数和初始解的距离），指数尾变异强度在我们的结果中获得了最佳的运行时保证——然而，若该期望值选择不当，性能保证会迅速变得毫无意义。对于本质上无需参数调整的幂律变异，我们在所有问题参数和起始点上均获得了良好的统一结果。我们通过实验数据补充了数学分析，这些实验表明我们的理论边界并非总是紧致的。最显著的是，实验显示即使指数尾变异采用接近最优的参数配置，幂律变异仍表现更优。因此，我们建议对未知整数空间问题优先采用幂律变异算子。