We establish global convergence of the (1 + 1) evolution strategy, that is, convergence to a critical point independent of the initial state. More precisely, we show the existence of a critical limit point, using a suitable extension of the notion of a critical point to measurable functions. At its core, the analysis is based on a novel progress guarantee for elitist, rank-based evolutionary algorithms. By applying it to the (1 + 1) evolution strategy we are able to provide an accurate characterization of whether global convergence is guaranteed with full probability, or whether premature convergence is possible. We illustrate our results on a number of example applications ranging from smooth (non-convex) cases over different types of saddle points and ridge functions to discontinuous and extremely rugged problems.

中文翻译：

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(1+1) 进化策略全局收敛到临界点

我们建立了（1+1）进化策略的全局收敛，即收敛到一个独立于初始状态的临界点。更准确地说，我们使用临界点概念对可测量函数的适当扩展，证明了临界极限点的存在。该分析的核心是基于精英的、基于等级的进化算法的新进展保证。通过将其应用于 (1 + 1) 演化策略，我们能够准确表征是否全概率保证全局收敛，或者是否可能过早收敛。我们在许多示例应用程序中说明了我们的结果，从不同类型的鞍点和脊函数上的平滑（非凸）情况到不连续和极其崎岖的问题。