Optimization of lipschitz continuous functions

Abstract

This paper contains basic results that are useful for building algorithms for the optimization of Lipschitz continuous functionsf on compact subsets of En. In this settingf is differentiable a.e. The theory involves a set-valued mappingxźźźf(x) whose range is the convex hull of existing values of źf and limits of źf on a closedź-ball,B(x, ź). As an application, simple descent algorithms are formulated that generate sequence {xk} whose distance from some stationary set (see Section 2) is 0, and where {f(xk)} decreases monotonously. This is done with the aid of anyone of the following three hypotheses: Forź arbitrarily small, a point is available that in arbitrarily close to:(1)the minimizer off onB(x, ź),(2)the closest point inźźf(x) to the origin,(3)ź(h) ź źźf(x), where [ź(h), h] = max {[ź, h]: ź ź źźf(x)}. Observe that these three problems are simplified iff has a tractable local approximation. The minimax problem is taken as an example, and algorithms for it are sketched. For this example, all three hypotheses may be satisfied. A class of functions called uniformly-locally-convex is introduced that is also tractable.