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 C 2 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 η is piecewise smooth, thereby, semismooth.

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