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Outer Limit of Subdifferentials and Calmness Moduli in Linear and Nonlinear Programming

Author

Listed:
  • M. J. Cánovas

    (Miguel Hernández University of Elche)

  • R. Henrion

    (Weierstrass Institute for Applied Analysis and Stochastics)

  • M. A. López

    (University of Alicante)

  • J. Parra

    (Miguel Hernández University of Elche)

Abstract

With a common background and motivation, the main contributions of this paper are developed in two different directions. Firstly, we are concerned with functions, which are the maximum of a finite amount of continuously differentiable functions of n real variables, paying special attention to the case of polyhedral functions. For these max-functions, we obtain some results about outer limits of subdifferentials, which are applied to derive an upper bound for the calmness modulus of nonlinear systems. When confined to the convex case, in addition, a lower bound on this modulus is also obtained. Secondly, by means of a Karush–Kuhn–Tucker index set approach, we are also able to provide a point-based formula for the calmness modulus of the argmin mapping of linear programming problems, without any uniqueness assumption on the optimal set. This formula still provides a lower bound in linear semi-infinite programming. Illustrative examples are given.

Suggested Citation

  • M. J. Cánovas & R. Henrion & M. A. López & J. Parra, 2016. "Outer Limit of Subdifferentials and Calmness Moduli in Linear and Nonlinear Programming," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 925-952, June.
    • Handle: RePEc:spr:joptap:v:169:y:2016:i:3:d:10.1007_s10957-015-0793-x
    DOI: 10.1007/s10957-015-0793-x
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    References listed on IDEAS

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    1. Diethard Klatte & Bernd Kummer, 2015. "On Calmness of the Argmin Mapping in Parametric Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 708-719, June.
    2. Stephen M. Robinson, 1977. "A Characterization of Stability in Linear Programming," Operations Research, INFORMS, vol. 25(3), pages 435-447, June.
    3. David G. Luenberger & Yinyu Ye, 2008. "Linear and Nonlinear Programming," International Series in Operations Research and Management Science, Springer, edition 0, number 978-0-387-74503-9, March.
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    Cited by:

    1. Zhou Wei & Christiane Tammer & Jen-Chih Yao, 2021. "Characterizations for Strong Abadie Constraint Qualification and Applications to Calmness," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 1-18, April.
    2. M. A. Goberna & M. A. López, 2018. "Recent contributions to linear semi-infinite optimization: an update," Annals of Operations Research, Springer, vol. 271(1), pages 237-278, December.

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