Problems defined on binary decision spaces have been intensively studied in the theory of multi-objective evolutionary algorithms (MOEAs). In contrast, no mathematical runtime analyses exist so far for MOEAs dealing with decision variables that take a finite number $r > 2$ of values, despite the prevalence of such problems in practice. In this work, we begin to fill this research gap. We analyze how the classic SEMO algorithm with unit-strength local mutation computes the Pareto front of an $r$-valued counterpart of the classic \oneminmax benchmark. For the expected number of function evaluations until the Pareto front is covered by the population of this MOEA, we prove an upper bound of $O(n^2 r^2 \log n)$ and a near-tight lower bound of $Ω(n^2 r (r + \log n))$. We can close the small remaining gap between these two bounds by considering a variant of the algorithm that accepts only strictly better solutions; for this variant, we show an upper bound of $O(n^2 r (r + \log n))$, matching our lower bound (which also holds for this variant). Our results suggest that classic MOEAs encounter no significant additional difficulties when dealing with multi-valued decision variables. However, significantly more advanced tools may be required to obtain tight bounds for algorithms with more complex population dynamics.

翻译：针对二进制决策空间定义的问题已在多目标进化算法（MOEA）理论中得到深入研究。然而，尽管实践普遍存在决策变量取值有限个数（$r > 2$）的问题，但目前尚无针对此类MOEA的数学运行时间分析。本文首次填补这一研究空白。我们分析了采用单位强度局部变异的标准SEMO算法在计算经典\oneminmax基准的$r$值对应版本的帕累托前沿时的表现。针对MOEA种群覆盖帕累托前沿所需的预期函数评估次数，我们证明了上界$O(n^2 r^2 \log n)$和近乎紧的下界$\Omega(n^2 r (r + \log n))$。通过考虑仅接受严格更优解的算法变体，我们可缩小两者间的微小差距；对于该变体，我们证明了上界$O(n^2 r (r + \log n))$，与下界（同样适用于该变体）匹配。结果表明，经典MOEA在处理多值决策变量时并未面临显著额外困难，但针对种群动态更复杂的算法获得紧界可能需要更先进的工具。