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Second-Order Optimality Conditions for Strict Efficiency of Constrained Set-Valued Optimization

Author

Listed:
  • S. J. Li

    (Chongqing University
    Chongqing University)

  • S. K. Zhu

    (Chongqing University)

  • X. B. Li

    (Chongqing University)

Abstract

In this paper, we propose several second-order derivatives for set-valued maps and discuss their properties. By using these derivatives, we obtain second-order necessary optimality conditions for strict efficiency of a set-valued optimization problem with inclusion constraints in real normed spaces. We also establish second-order sufficient optimality conditions for strict efficiency of the set-valued optimization problem in finite-dimensional normed spaces. As applications, we investigate second-order sufficient and necessary optimality conditions for a strict local efficient solution of order two of a nonsmooth vector optimization problem with an abstract set and a functional constraint.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:155:y:2012:i:2:d:10.1007_s10957-012-0076-8
    DOI: 10.1007/s10957-012-0076-8
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    References listed on IDEAS

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    1. J. Jahn & A. A. Khan & P. Zeilinger, 2005. "Second-Order Optimality Conditions in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 125(2), pages 331-347, May.
    2. 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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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. X. L. Guo & S. J. Li, 2014. "Optimality Conditions for Vector Optimization Problems with Difference of Convex Maps," Journal of Optimization Theory and Applications, Springer, vol. 162(3), pages 821-844, September.
    5. Nguyen Minh Tung, 2020. "New Higher-Order Strong Karush–Kuhn–Tucker Conditions for Proper Solutions in Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 448-475, May.

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