A class of metaheuristic techniques called estimation-of-distribution algorithms (EDAs) are employed in optimization as more sophisticated substitutes for traditional strategies like evolutionary algorithms. EDAs generally drive the search for the optimum by creating explicit probabilistic models of potential candidate solutions through repeated sampling and selection from the underlying search space. Most theoretical research on EDAs has focused on pseudo-Boolean optimization. Jedidia et al. (GECCO 2023) proposed the first EDAs for optimizing problems involving multi-valued decision variables. By building a framework, they have analyzed the runtime of a multi-valued UMDA on the r-valued LeadingOnes function. Using their framework, here we focus on the multi-valued compact genetic algorithm (r-cGA) and provide a first runtime analysis of a generalized OneMax function. To prove our results, we investigate the effect of genetic drift and progress of the probabilistic model towards the optimum. After finding the right algorithm parameters, we prove that the r-cGA solves this r-valued OneMax problem efficiently. We show that with high probability, the runtime bound is O(r2 n log2 r log3 n). At the end of experiments, we state one conjecture related to the expected runtime of another variant of multi-valued OneMax function.

翻译：一类称为估计分布算法（EDAs）的元启发式技术被用于优化领域，作为进化算法等传统策略的更复杂替代方案。EDAs通常通过从底层搜索空间中重复采样和选择，为潜在候选解创建显式概率模型，从而驱动寻优过程。目前关于EDAs的理论研究大多集中在伪布尔优化上。Jedidia等人（GECCO 2023）提出了首个用于优化涉及多值决策变量问题的EDAs。通过构建一个框架，他们分析了多值UMDA在r值LeadingOnes函数上的运行时。利用该框架，我们聚焦于多值紧凑遗传算法（r-cGA），并首次对广义OneMax函数进行了运行时分析。为证明我们的结论，我们研究了遗传漂变的影响以及概率模型向最优解演进的进程。在找到正确的算法参数后，我们证明r-cGA能够高效求解该r值OneMax问题。我们表明，在大概率下，运行时上界为O(r² n log² r log³ n)。实验最后，我们针对多值OneMax函数另一变体的期望运行时提出了一个猜想。