Geodesic
Versions of the Sard Theorem for Essentially Smooth Lipschitz Maps and Applications in Optimization and Nonsmooth Equations
Journal of convex analysis, Tome 28 (2021) no. 1, pp. 157-178
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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.
Classification : 49J40
Mots-clés : Sard theorem, essentially smooth Lipschitz map, critical points, optimization, nonsmooth equation 
@article{JCA_2021_28_1_JCA_2021_28_1_a11,
     author = {T. X. D. Ha},
     title = {Versions of the {Sard} {Theorem} for {Essentially} {Smooth} {Lipschitz} {Maps} and {Applications} in {Optimization} and {Nonsmooth} {Equations}},
     journal = {Journal of convex analysis},
     pages = {157--178},
     year = {2021},
     volume = {28},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JCA_2021_28_1_JCA_2021_28_1_a11/}
}
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T. X. D. Ha. Versions of the Sard Theorem for Essentially Smooth Lipschitz Maps and Applications in Optimization and Nonsmooth Equations. Journal of convex analysis, Tome 28 (2021) no. 1, pp. 157-178. http://geodesic.mathdoc.fr/item/JCA_2021_28_1_JCA_2021_28_1_a11/