Sard theorems for Lipschitz functions and applications in optimization

Abstract

We establish a "preparatory Sard theorem" for smooth functions with a partially affine structure. By means of this result, we improve a previous result of Rifford [17, 19] concerning the generalized (Clarke) critical values of Lipschitz functions defined as minima of smooth functions. We also establish a nonsmooth Sard theorem for the class of Lipschitz functions from R (d) to R (p) that can be expressed as finite selections of C (k) functions (more generally, continuous selections over a compact countable set). This recovers readily the classical Sard theorem and extends a previous result of Barbet-Daniilidis-Dambrine [1] to the case p > 1. Applications in semi-infinite and Pareto optimization are given.