The Metropolis algorithm (MA) is a classic stochastic local search heuristic. It avoids getting stuck in local optima by occasionally accepting inferior solutions. To better and in a rigorous manner understand this ability, we conduct a mathematical runtime analysis of the MA on the CLIFF benchmark. Apart from one local optimum, cliff functions are monotonically increasing towards the global optimum. Consequently, to optimize a cliff function, the MA only once needs to accept an inferior solution. Despite seemingly being an ideal benchmark for the MA to profit from its main working principle, our mathematical runtime analysis shows that this hope does not come true. Even with the optimal temperature (the only parameter of the MA), the MA optimizes most cliff functions less efficiently than simple elitist evolutionary algorithms (EAs), which can only leave the local optimum by generating a superior solution possibly far away. This result suggests that our understanding of why the MA is often very successful in practice is not yet complete. Our work also suggests to equip the MA with global mutation operators, an idea supported by our preliminary experiments.

翻译：Metropolis算法（MA）是一种经典的随机局部搜索启发式算法。它通过偶尔接受劣质解来避免陷入局部最优。为更深入、严谨地理解这一特性，我们在CLIFF基准函数上对其进行了数学运行时间分析。除一个局部最优点外，悬崖函数单调递增至全局最优。因此，优化悬崖函数时，MA只需一次接受劣质解。尽管这看似是MA发挥核心工作原理的理想基准，但我们的数学运行时间分析表明，这一期望并未实现。即使在最优温度（MA的唯一参数）下，MA对多数悬崖函数的优化效率仍低于简单精英进化算法（EA）——后者仅能通过生成可能相距甚远的优越解来脱离局部最优。这一结果提示，我们对于MA在实践中常获成功的原因尚未完全理解。此外，我们的工作建议为MA配备全局变异算子，初步实验已支持这一观点。