Versions of the Sard Theorem for Essentially Smooth Lipschitz Maps and Applications in Optimization and Nonsmooth Equations

The classical Sard theorem (in a special case) states that the set of critical values of a $C^1$-map from an open set of $\mathbb{R}^n$ to $\mathbb{R}^n$ has Lebesgue measure zero. Motivated by a recent work of Barbet, Dambrine, Daniilidis and Rifford [Sard theorems for Lipschitz functions and applications in optimization, Israel J. Math. 212 (2016) 757-–790], we obtain in this paper versions of this theorem for a finite family of essentially smooth Lipschitz maps and for a locally Lipschitz continuous selection of this family. Here, a locally Lipschitz map is essentially smooth if its Clarke's subdifferential reduces to a singleton almost everywhere. As applications, we establish the genericity of Karush-Kuhn-Tucker type necessary condition for scalar/vector parametrized constrained optimization problems, Lebesgue zero measure of the set of Pareto optimal values of a map and the genericity of the finiteness of the solution set for a nonsmooth equation.