Conditionning, halting criteria and choosing λ

We show the convergence of 1+λ-ES with standard step-size update-rules on a large family of fitness functions without any convexity assumption or quasi-convexity assumptions ([5, 6]). The result provides a rule for choosing λ and shows the consistency of halting criteria based on thresholds on the step-size. The family of functions under work is defined through a conditionnumber that generalizes usual condition-numbers in a manner that only depends on level-sets. We consider that the definition of this conditionnumber is the relevant one for evolutionary algorithms; in particular, global convergence results without convexity or quasi-convexity assumptions are proved when this condition-number is finite.