Sard theorems for Lipschitz functions and applications in optimization

Luc Barbet (1) , Marc Dambrine (1) , Aris Daniilidis (2) , Ludovic Rifford (3, 4)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 UAB - Departament de Matemàtiques [Barcelona]
3 LJAD - Laboratoire Jean Alexandre Dieudonné
4 McTAO - Mathematics for Control, Transport and Applications

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.

Mots clés: Sard theorem; Lipschitz function; Clarke generalized Jacobian; semi-infinite optimization

Domaines: Mathématiques [math]

Luc Barbet, Marc Dambrine, Aris Daniilidis, Ludovic Rifford. Sard theorems for Lipschitz functions and applications in optimization. Israel Journal of Mathematics, 2016, 212 (2), pp.757-790. ⟨10.1007/s11856-016-1308-7⟩. ⟨hal-01336328⟩