  We establish global convergence of the (1+1) evolution strategy, i.e.,
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.
