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

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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.

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Authors and Affiliations

  1. School of Mathematics, University of Southampton, Highfield, Southampton, SO17 1BJ, United Kingdom

    Fanwen Meng

  2. Department of Mathematics, National University of Singapore, Singapore, 117543, Republic of Singapore

    Defeng Sun

  3. Department of Mathematics, National University of Singapore, Singapore, 117543, Republic of Singapore

    Gongyun Zhao

Authors
  1. Fanwen Meng
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  2. Defeng Sun
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  3. Gongyun Zhao
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Correspondence to Defeng Sun.

Additional information

This paper is dedicated to Terry Rockafellar on the occasion of his seventieth birthday

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Meng, F., Sun, D. & Zhao, G. Semismoothness of solutions to generalized equations and the Moreau-Yosida regularization. Math. Program. 104, 561–581 (2005). https://doi.org/10.1007/s10107-005-0629-9

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  • DOI: https://doi.org/10.1007/s10107-005-0629-9

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