The classical Sard theorem states that the set of critical values of a $C^{k}$-map from an open set of $\R^n$ to $\R^p$ ($n\geq p$) has Lebesgue measure zero provided $k\geq n-p+1$. In the recent paper by Barbet, Dambrine, Daniilidis and Rifford, the so called ``preparatory Sard theorem" for a compact countable set $I$ of $C^k$ maps from $\R^n$ to $\R^p$ and a Sard theorem for a locally Lipschitz continuous selection of this family have been established under the assumption that $k\geq n-p+1$. Here, we show that, in the special case $n=p$ and $I$ is finite, the $C^1$ smoothness assumption in these results can be relaxed to ``essentially smooth locally Lipschitz". Then we apply the obtained results to study Karush-Kuhn-Tucker type necessary condition for scalar/vector parametrized constrained optimization problems and the set of Pareto optimal values of a continuous selection of a finite family of essentially smooth locally Lipschitz maps.
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View The Sard theorem for essentially smooth locally Lipschitz maps and applications in optimization