Sard theorems for Lipschitz functions and applications in optimization

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Abstract

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 Rd to Rp that can be expressed as finite selections of Ck 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.

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Barbet, L., Dambrine, M., Daniilidis, A., & Rifford, L. (2016). Sard theorems for Lipschitz functions and applications in optimization. Israel Journal of Mathematics, 212(2), 757–790. https://doi.org/10.1007/s11856-016-1308-7

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