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Evolutionary Computation

Spring 2020, Vol. 28, No. 1, Pages 27-53
(doi: 10.1162/evco_a_00248)
© 2019 Massachusetts Institute of Technology
Global Convergence of the (1 + 1) Evolution Strategy to a Critical Point
Article PDF (745.2 KB)
Abstract
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.