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Epi-convergence: The Moreau Envelope and Generalized Linear-Quadratic Functions

Author

Listed:
  • Chayne Planiden

    (University of Wollongong)

  • Xianfu Wang

    (University of British Columbia Okanagan)

Abstract

This work explores the class of generalized linear-quadratic functions, constructed using maximally monotone symmetric linear relations. Calculus rules and properties of the Moreau envelope for this class of functions are developed. In finite dimensions, on a metric space defined by Moreau envelopes, we consider the epigraphical limit of a sequence of quadratic functions and categorize the results. We examine the question of when a quadratic function is a Moreau envelope of a generalized linear-quadratic function; characterizations involving nonexpansiveness and Lipschitz continuity are established. This work generalizes some results by Hiriart-Urruty and by Rockafellar and Wets.

Suggested Citation

  • Chayne Planiden & Xianfu Wang, 2018. "Epi-convergence: The Moreau Envelope and Generalized Linear-Quadratic Functions," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 21-63, April.
    • Handle: RePEc:spr:joptap:v:177:y:2018:i:1:d:10.1007_s10957-018-1254-0
    DOI: 10.1007/s10957-018-1254-0
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    Cited by:

    1. Lixin Shen & Bruce W. Suter & Erin E. Tripp, 2019. "Structured Sparsity Promoting Functions," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 386-421, November.

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