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Second-Order Strong Karush/Kuhn–Tucker Conditions for Proper Efficiencies in Multiobjective Optimization

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  • Min Feng

    (Chongqing University)

  • Shengjie Li

    (Chongqing University)

Abstract

In this paper, strong Karush/Kuhn–Tucker conditions are studied for smooth multiobjective optimization with inequality constraints. We introduce a new second-order regularity condition of Abadie type in terms of the second-order directional derivatives and then obtain a second-order strong Karush/Kuhn–Tucker necessary condition at a Borwein-properly efficient solution. Simultaneously, we also use an example to show that, if the Abadie type regularity condition is weakened to the Guignard type one, the second-order strong Karush/Kuhn–Tucker necessary condition may not hold. Finally, then we also apply the second-order strong Karush/Kuhn–Tucker conditions to derive a sufficient result for local Geoffrion-proper efficiency.

Suggested Citation

  • Min Feng & Shengjie Li, 2019. "Second-Order Strong Karush/Kuhn–Tucker Conditions for Proper Efficiencies in Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 766-786, June.
    • Handle: RePEc:spr:joptap:v:181:y:2019:i:3:d:10.1007_s10957-019-01484-0
    DOI: 10.1007/s10957-019-01484-0
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    References listed on IDEAS

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    1. Vsevolod I. Ivanov, 2015. "Second-Order Optimality Conditions for Vector Problems with Continuously Fréchet Differentiable Data and Second-Order Constraint Qualifications," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 777-790, September.
    2. R. Andreani & C. E. Echagüe & M. L. Schuverdt, 2010. "Constant-Rank Condition and Second-Order Constraint Qualification," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 255-266, August.
    3. Gulati, T. R. & Islam, M. A., 1990. "Efficiency and proper efficiency in nonlinear vector maximum problems," European Journal of Operational Research, Elsevier, vol. 44(3), pages 373-382, February.
    4. T. Maeda, 2004. "Second-Order Conditions for Efficiency in Nonsmooth Multiobjective Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 122(3), pages 521-538, September.
    5. X. F. Li & J. Z. Zhang, 2005. "Stronger Kuhn-Tucker Type Conditions in Nonsmooth Multiobjective Optimization: Locally Lipschitz Case," Journal of Optimization Theory and Applications, Springer, vol. 127(2), pages 367-388, November.
    6. Regina S. Burachik & M. M. Rizvi, 2012. "On Weak and Strong Kuhn–Tucker Conditions for Smooth Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 477-491, November.
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    Cited by:

    1. Min Feng & Shengjie Li & Jie Wang, 2022. "On Tucker-Type Alternative Theorems and Necessary Optimality Conditions for Nonsmooth Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 480-503, November.

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