On second-order properties of the moreau-yosida regularization for constrained nonsmooth convex programs

Abstract

In this paper, we attempt to investigate a class of constrained nonsmooth convex optimization problems, that is, piecewise C2 convex objectives with smooth convex inequality constraints. By using the Moreau-Yosida regularization, we convert these problems into unconstrained smooth convex programs. Then, we investigate the second-order properties of the Moreau-Yosida regularization η. By introducing the (GAIPCQ) qualification, we show that the gradient of the regularized function n is piecewise smooth, thereby, semismooth.