Semismoothness of solutions to generalized equations and the Moreau-Yosida regularization

Authors:
Fanwen Meng;Defeng Sun;Gongyun Zhao
Affiliations:
School of Mathematics, University of Southampton, SO17 1BJ, Highfield, Southampton, United Kingdom;Department of Mathematics, National University of Singapore, 117543, Singapore, Southampton, Republic of Singapore;Department of Mathematics, National University of Singapore, 117543, Singapore, Southampton, Republic of Singapore
Venue:
Mathematical Programming: Series A and B
Year:
2005

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Abstract

We show that a locally Lipschitz homeomorphism function is semismooth at a given point if and only if its inverse function is semismooth at its image point. We present a sufficient condition for the semismoothness of solutions to generalized equations over cone reducible (nonpolyhedral) convex sets. We prove that the semismoothness of solutions to the Moreau-Yosida regularization of a lower semicontinuous proper convex function is implied by the semismoothness of the metric projector over the epigraph of the convex function.