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Applying augmented weak subdifferentials and normal cones for nonconvex mathematical programming problems

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  • Tran Van Su

    (The University of Danang - University of Science and Education)

  • Chu Van Tiep

    (The University of Danang - University of Science and Education)

Abstract

The paper is devoted to the study of some necessary and sufficient optimality conditions for a nonconvex extended-real-valued function having a global minimum/and a vector-valued mapping having a weakly efficient solution at a given point in terms of the augmented weak subdifferentials and the augmented normal cones in reflexive Banach spaces without any convexity assumption. By applying a special separation theorem for the nonconvex sets in reflexive Banach spaces, some necessary optimality conditions for a nonconvex (scalar/vector) function having a global minimum/and a (weakly) efficient solution concern the existence of a weakly subgradient pair are derived. Under some suitable assumptions, one of such conditions becomes the sufficient optimality condition respectively. An application of the obtained result for the nonsmooth nonconvex multiobjective mathematical programming problem having set, inequality and equality constraints is presented accordingly.

Suggested Citation

  • Tran Van Su & Chu Van Tiep, 2025. "Applying augmented weak subdifferentials and normal cones for nonconvex mathematical programming problems," Journal of Global Optimization, Springer, vol. 91(4), pages 765-786, April.
    • Handle: RePEc:spr:jglopt:v:91:y:2025:i:4:d:10.1007_s10898-025-01474-9
    DOI: 10.1007/s10898-025-01474-9
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    References listed on IDEAS

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    1. Frank H. Clarke, 1976. "A New Approach to Lagrange Multipliers," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 165-174, May.
    2. Gulcin Dinc Yalcin & Refail Kasimbeyli, 2020. "On weak conjugacy, augmented Lagrangians and duality in nonconvex optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(1), pages 199-228, August.
    3. R. T. Rockafellar, 1981. "Proximal Subgradients, Marginal Values, and Augmented Lagrangians in Nonconvex Optimization," Mathematics of Operations Research, INFORMS, vol. 6(3), pages 424-436, August.
    4. Elena Constantin, 2021. "Necessary conditions for weak minima and for strict minima of order two in nonsmooth constrained multiobjective optimization," Journal of Global Optimization, Springer, vol. 80(1), pages 177-193, May.
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