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Second-Order Optimality Conditions with the Envelope-Like Effect for Set-Valued Optimization

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  • P. Q. Khanh

    (International University, Vietnam National University Ho Chi Minh City)

  • N. M. Tung

    (University of Science, Vietnam National University Ho Chi Minh City)

Abstract

We consider Karush–Kuhn–Tucker second-order optimality conditions for nonsmooth set-valued optimization with attention to the envelope-like effect. To analyse the critical feasible directions, which produce this phenomenon, we use the contingent derivatives, the adjacent derivatives and the corresponding asymptotic derivatives, since directions are explicitly involved in these kinds of derivatives. To pursue strong multiplier rules, we impose cone-Aubin conditions to deal with the objective and constraint maps separately. In this way, we can invoke constraint qualifications of the Kurcyusz–Robinson–Zowe type. To our knowledge, some of the results are new; they will be indicated explicitly. The paper also discusses improvements or extensions of known results.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:167:y:2015:i:1:d:10.1007_s10957-015-0728-6
    DOI: 10.1007/s10957-015-0728-6
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    References listed on IDEAS

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    1. 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.
    2. P. Q. Khanh & N. D. Tuan, 2008. "Variational Sets of Multivalued Mappings and a Unified Study of Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 139(1), pages 47-65, October.
    3. 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.
    4. 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.
    5. Stephen M. Robinson, 1976. "Regularity and Stability for Convex Multivalued Functions," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 130-143, May.
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    Cited by:

    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. Nguyen Xuan Duy Bao & Phan Quoc Khanh & Nguyen Minh Tung, 2022. "Quasi-contingent derivatives and studies of higher-orders in nonsmooth optimization," Journal of Global Optimization, Springer, vol. 84(1), pages 205-228, September.
    3. 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.
    4. 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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