Inner Moreau Envelope of Nonsmooth Conic Chance-Constrained Optimization Problems

Abstract

Optimization problems with uncertainty in the constraints occur in many applications. Particularly, probability functions present a natural form to deal with this situation. Nevertheless, in some cases, the resulting probability functions are nonsmooth, which motivates us to propose a regularization employing the Moreau envelope of a scalar representation of the vector inequality. More precisely, we consider a probability function that covers most of the general classes of probabilistic constraints: [Formula: see text]where [Formula: see text] is a convex cone of a Banach space. The conic inclusion [Formula: see text] represents an abstract system of inequalities, and ξ is a random vector. We propose a regularization by applying the Moreau envelope to the scalarization of the function [Formula: see text]. In this paper, we demonstrate, under mild assumptions, the smoothness of such a regularization and that it satisfies a type of variational convergence to the original probability function. Consequently, when considering an appropriately structured problem involving probabilistic constraints, we can, thus, entail the convergence of the minimizers of the regularized approximate problems to the minimizers of the original problem. Finally, we illustrate our results with examples and applications in the field of (nonsmooth) joint, semidefinite, and probust chance-constrained optimization problems. Funding: P. Pérez-Aros was supported by Centro de Modelamiento Matemático [Grants ACE210010 and FB210005] and BASAL funds for center of excellence and ANID-Chile grant Fondecyt Regular [Grants 1200283 and 1190110] and Fondecyt Exploración [Grant 13220097]. C. Soto was supported by the National Agency for Research and Development (ANID)/Scholarship Program/Doctorado Nacional Chile [Grant 2017-21170428]. E. Vilches was supported by Centro de Modelamiento Matemático [Grants ACE210010 and FB210005] and BASAL funds for center of excellence and Fondecyt Regular [Grant 1200283] and Fondecyt Exploración [Grant 13220097] from ANID-Chile.