Journal of Convex Analysis 28 (2021), No. 1, 157--178
Copyright Heldermann Verlag 2021
Versions of the Sard Theorem for Essentially Smooth Lipschitz Maps and Applications in Optimization and Nonsmooth Equations
Truong Xuan Duc Ha
Institute for Mathematics and Applied Sciences, Thang Long University, Hanoi, Vietnam
The classical Sard theorem (in a special case) states that the set of critical values of a C1-map from an open set of Rn to Rn 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.
Keywords: Sard theorem, essentially smooth Lipschitz map, critical points, optimization, nonsmooth equation.
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