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方法。它通过偶尔接受劣质解来避免陷入局部最优。为了更严谨地理解这一能力，我们在 CLIFF 基准函数上对 MA 进行了数学运行时分析。除一个局部最优解外，悬崖函数单调递增至全局最优解。因此，优化悬崖函数时，MA 仅需接受一次劣质解。尽管这看似是 MA 发挥其主要工作原理的理想基准，但我们的数学运行时分析表明，这一期望并未实现。即使采用最优温度（MA 的唯一参数），MA 在优化大多数悬崖函数时的效率也低于简单的精英进化算法（EA），后者只能通过生成可能远距离的优质解来脱离局部最优。这一结果表明，我们对于 MA 在实践中常获成功的理解尚不完整。我们的工作还建议为 MA 配备全局变异算子，初步实验支持了这一构想。