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New Set-Valued Directional Derivatives: Calculus and Optimality Conditions

Author

Listed:
  • Nguyen Minh Tung

    (Banking University of Ho Chi Minh City)

  • Nguyen Xuan Duy Bao

    (University of Science
    Vietnam National University)

Abstract

In this paper, we propose a new notion called radial directional derivative and derive its existence as well as main calculus rules. Then we employ them to investigate optimality conditions for a nonsmooth vector optimization problem subjected to an inclusion constraint in Banach spaces. With a directional (Hölder) metric subregularity assumption and a constraint qualification, necessary optimality conditions for both local weak and strict solutions are given in types of Karush–Kuhn–Tucker multiplier rules. The sufficient conditions are also established for local strict solutions whenever the decision space is finite-dimensional without any convexity assumption. Examples are provided to show advantages of the presented results over recent existing ones.

Suggested Citation

  • Nguyen Minh Tung & Nguyen Xuan Duy Bao, 2023. "New Set-Valued Directional Derivatives: Calculus and Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 411-437, May.
    • Handle: RePEc:spr:joptap:v:197:y:2023:i:2:d:10.1007_s10957-023-02185-5
    DOI: 10.1007/s10957-023-02185-5
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    References listed on IDEAS

    as
    1. Phan Quoc Khanh & Nguyen Minh Tung, 2016. "Second-Order Conditions for Open-Cone Minimizers and Firm Minimizers in Set-Valued Optimization Subject to Mixed Constraints," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 45-69, October.
    2. P. Q. Khanh & N. D. Tuan, 2007. "Optimality Conditions for Nonsmooth Multiobjective Optimization Using Hadamard Directional Derivatives," Journal of Optimization Theory and Applications, Springer, vol. 133(3), pages 341-357, June.
    3. A. Guerraggio & D. T. Luc, 2001. "Optimality Conditions for C1,1 Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 109(3), pages 615-629, June.
    4. P. Q. Khanh & N. M. Tung, 2015. "Second-Order Optimality Conditions with the Envelope-Like Effect for Set-Valued Optimization," Journal of Optimization Theory and Applications, Springer, vol. 167(1), pages 68-90, October.
    5. Stephen M. Robinson, 1976. "Regularity and Stability for Convex Multivalued Functions," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 130-143, May.
    6. A. Guerraggio & D.T. Luc, 2003. "Optimality Conditions for C 1,1 Constrained Multiobjective Problems," Journal of Optimization Theory and Applications, Springer, vol. 116(1), pages 117-129, January.
    7. S. J. Li & S. K. Zhu & X. B. Li, 2012. "Second-Order Optimality Conditions for Strict Efficiency of Constrained Set-Valued Optimization," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 534-557, November.
    8. C. Gutiérrez & B. Jiménez & V. Novo, 2009. "New Second-Order Directional Derivative and Optimality Conditions in Scalar and Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 85-106, July.
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