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Optimality and Duality in Nonsmooth Conic Vector Optimization

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  • Thai Doan Chuong

    (Ton Duc Thang University
    Ton Duc Thang University)

Abstract

This article is concerned with a nonsmooth vector optimization problem involving conic constraints. We employ some advanced tools of variational analysis and generalized differentiation to establish necessary conditions for (weakly) efficient solutions of the conic vector optimization problem, where the fuzzy necessary condition and sequential necessary condition are expressed in terms of the Fréchet subdifferential and the exact necessary condition is in terms of the limiting/Mordukhovich subdifferential of the related functions. Sufficient conditions for (weakly) efficient solutions of the underlying problem are also provided by means of introducing the concepts of (strictly) generalized convex vector functions with respect to a cone. In addition, we propose a dual problem to the conic vector optimization problem and explore weak, strong, and converse duality relations between these two problems.

Suggested Citation

  • Thai Doan Chuong, 2019. "Optimality and Duality in Nonsmooth Conic Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 471-489, November.
    • Handle: RePEc:spr:joptap:v:183:y:2019:i:2:d:10.1007_s10957-019-01577-w
    DOI: 10.1007/s10957-019-01577-w
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    References listed on IDEAS

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    1. I. Ginchev & A. Guerraggio & M. Rocca, 2009. "Dini Set-Valued Directional Derivative in Locally Lipschitz Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 143(1), pages 87-105, October.
    2. Thai Doan Chuong & Do Sang Kim, 2014. "Nonsmooth Semi-infinite Multiobjective Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 748-762, March.
    3. Jia Chen & Yeol Cho & Jong Kim & Jun Li, 2011. "Multiobjective optimization problems with modified objective functions and cone constraints and applications," Journal of Global Optimization, Springer, vol. 49(1), pages 137-147, January.
    4. Thai Chuong & Do Kim, 2014. "Optimality conditions and duality in nonsmooth multiobjective optimization problems," Annals of Operations Research, Springer, vol. 217(1), pages 117-136, June.
    5. A. J. V. Brandão & M. A. Rojas-Medar & G. N. Silva, 1999. "Optimally Conditions for Pareto Nonsmooth Nonconvex Programming in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 103(1), pages 65-73, October.
    6. Boris S. Mordukhovich & T. T. A. Nghia, 2014. "Nonsmooth Cone-Constrained Optimization with Applications to Semi-Infinite Programming," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 301-324, May.
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    Cited by:

    1. Thai Doan Chuong, 2022. "Approximate solutions in nonsmooth and nonconvex cone constrained vector optimization," Annals of Operations Research, Springer, vol. 311(2), pages 997-1015, April.
    2. Thai Doan Chuong, 2021. "Optimality and duality in nonsmooth composite vector optimization and applications," Annals of Operations Research, Springer, vol. 296(1), pages 755-777, January.

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