Sard theorems for Lipschitz functions and applications in optimization
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We establish a “preparatory Sard theorem” for smooth functions with a partially affine structure. By means of this result, we improve a previous result of Rifford [17, 19] 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  to the case p > 1. Applications in semi-infinite and Pareto optimization are given.
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- A. Jofre, Image of critical points set for nonsmooth mappings and applications, in Contributions à l’analyse non-diffrentiable avec applications à l’optimisation et à l’analyse non-linéaire, Thèse 1989, Université de Pau et des Pays de l’Adour, France.Google Scholar