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.

Key Words:

• Moreau–Yosida regularization
• Piecewise C ^k functions
• Semi-smooth functions

AMS Subject Classification (1991):

• 90C25
• 65K10
• 52A41

Acknowledgment

The research of the first author was supported by Grant R-146-000-036-592 of IHPC-CIM, National University of Singapore.