We argue that proven exponential upper bounds on runtimes, an established area in classic algorithms, are interesting also in heuristic search and we prove several such results. We show that any of the algorithms randomized local search, Metropolis algorithm, simulated annealing, and (1+1) evolutionary algorithm can optimize any pseudo-Boolean weakly monotonic function under a large set of noise assumptions in a runtime that is at most exponential in the problem dimension n. This drastically extends a previous such result, limited to the (1+1) EA, the LeadingOnes function, and one-bit or bit-wise prior noise with noise probability at most 1/2, and at the same time simplifies its proof. With the same general argument, among others, we also derive a sub-exponential upper bound for the runtime of the $(1,\lambda )$ evolutionary algorithm on the OneMax problem when the offspring population size λ is logarithmic, but below the efficiency threshold. To show that our approach can also deal with non-trivial parent population sizes, we prove an exponential upper bound for the runtime of the mutation-based version of the simple genetic algorithm on the OneMax benchmark, matching a known exponential lower bound.

我们认为，在经典算法中已建立的运行时，经过证明的指数上限在启发式搜索中也很有趣，我们证明了一些这样的结果。我们证明了随机算法中的任何算法，Metropolis算法，模拟退火算法和（1 + 1）进化算法都可以在运行时噪声指数最大的情况下优化任何伪布尔弱单调函数，而该假设在运行时最多为指数。问题维度n。这极大地扩展了先前的此类结果，仅限于（1 + 1）EA，LeadingOnes函数以及具有最多1/2噪声概率的按位或逐位先验噪声，同时简化了其证明。使用相同的一般参数，我们还可以得出运行时的次指数上限$（\mathrm{1个}，\lambda ）$后代种群大小λ为对数但低于效率阈值的OneMax问题的进化算法。为了证明我们的方法还可以处理非平凡的父母群体大小，我们在OneMax基准上证明了基于突变版本的简单遗传算法的运行时的指数上限，并与已知的指数下限匹配。