We establish a "preparatory Sard theorem" for smooth functions with a partial affine structure. By means of this result, we improve a previous result of Rifford [14,16] 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.

Artículo de publicación ISIThe Morse–Sard theorem states that the set of critical values of a Ck smooth function defined on a Euclidean space Rd has Lebesgue measure zero, provided k ≥ d. This result is hereby extended for (generalized) critical values of continuous selections over a compactly indexed countable family of Ck functions: it is shown that these functions are Lipschitz continuous and the set of their Clarke critical values is null

Abstract. Given a C 1,1 -function f : U → R (where U ⊂ R n open) we deal with the question of whether or not at a given point x 0 ∈ U there exists a local minorant ϕ of f of class C 2 that satisfies ϕ(. This question is motivated by the second-order viscosity theory of the PDE, since for nonsmooth functions, an analogous result between subgradients and first-order viscosity subjets is known to hold in every separable Asplund space. In this work we show that the aforementioned second-order result holds true whenever Hf (x 0 ) has a minimum with respect to the semidefinite cone (thus in particular, in one dimension), but it fails in two dimensions even for piecewise polynomial functions. We extend this result by introducing a new notion of directional minimum of Hf (x 0 ).