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Optimality conditions in optimization problems with convex feasible set using convexificators

Author

Listed:
  • Alireza Kabgani

    (University of Tehran)

  • Majid Soleimani-damaneh

    (University of Tehran
    Institute for Research in Fundamental Sciences (IPM))

  • Moslem Zamani

    (University of Tehran
    Institute for Research in Fundamental Sciences (IPM))

Abstract

In this paper, we consider a nonsmooth optimization problem with a convex feasible set described by constraint functions which are neither convex nor differentiable nor locally Lipschitz necessarily. Utilizing upper regular convexificators, we characterize the normal cone of the feasible set and derive KKT type necessary and sufficient optimality conditions. Under some assumptions, we show that the set of KKT multipliers is bounded. We also characterize the set of optimal solutions and introduce a linear approximation corresponding to the original problem which is useful in checking optimality. The obtained outcomes extend various results existing in the literature to a more general setting.

Suggested Citation

  • Alireza Kabgani & Majid Soleimani-damaneh & Moslem Zamani, 2017. "Optimality conditions in optimization problems with convex feasible set using convexificators," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(1), pages 103-121, August.
    • Handle: RePEc:spr:mathme:v:86:y:2017:i:1:d:10.1007_s00186-017-0584-2
    DOI: 10.1007/s00186-017-0584-2
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    References listed on IDEAS

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    1. X. F. Li & J. Z. Zhang, 2010. "Existence and Boundedness of the Kuhn-Tucker Multipliers in Nonsmooth Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 145(2), pages 373-386, May.
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    8. Do Luu, 2014. "Necessary and Sufficient Conditions for Efficiency Via Convexificators," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 510-526, February.
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    Cited by:

    1. E. Allevi & J. E. Martínez-Legaz & R. Riccardi, 2020. "Optimality conditions for convex problems on intersections of non necessarily convex sets," Journal of Global Optimization, Springer, vol. 77(1), pages 143-155, May.
    2. Felipe Lara & Alireza Kabgani, 2021. "On global subdifferentials with applications in nonsmooth optimization," Journal of Global Optimization, Springer, vol. 81(4), pages 881-900, December.
    3. Felipe Serrano & Robert Schwarz & Ambros Gleixner, 2020. "On the relation between the extended supporting hyperplane algorithm and Kelley’s cutting plane algorithm," Journal of Global Optimization, Springer, vol. 78(1), pages 161-179, September.
    4. A. Kabgani & F. Lara, 2022. "Strong subdifferentials: theory and applications in nonconvex optimization," Journal of Global Optimization, Springer, vol. 84(2), pages 349-368, October.
    5. Kabgani, Alireza & Soleimani-damaneh, Majid, 2022. "Semi-quasidifferentiability in nonsmooth nonconvex multiobjective optimization," European Journal of Operational Research, Elsevier, vol. 299(1), pages 35-45.

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