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Nonsmooth Vector Optimization Problems and Minty Vector Variational Inequalities

Author

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  • Q. H. Ansari

    (Aligarh Muslim University)

  • G. M. Lee

    (Pukyong National University)

Abstract

The vector optimization problem may have a nonsmooth objective function. Therefore, we introduce the Minty vector variational inequality (Minty VVI) and the Stampacchia vector variational inequality (Stampacchia VVI) defined by means of upper Dini derivative. By using the Minty VVI, we provide a necessary and sufficient condition for a vector minimal point (v.m.p.) of a vector optimization problem for pseudoconvex functions involving Dini derivatives. We establish the relationship between the Minty VVI and the Stampacchia VVI under upper sign continuity. Some relationships among v.m.p., weak v.m.p., solutions of the Stampacchia VVI and solutions of the Minty VVI are discussed. We present also an existence result for the solutions of the weak Minty VVI and the weak Stampacchia VVI.

Suggested Citation

  • Q. H. Ansari & G. M. Lee, 2010. "Nonsmooth Vector Optimization Problems and Minty Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 145(1), pages 1-16, April.
    • Handle: RePEc:spr:joptap:v:145:y:2010:i:1:d:10.1007_s10957-009-9638-9
    DOI: 10.1007/s10957-009-9638-9
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    References listed on IDEAS

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    1. Q. H. Ansari & J> C> Yao, 2000. "On Nondifferentiable and Nonconvex Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 106(3), pages 475-488, September.
    2. G. P. Crespi & I. Ginchev & M. Rocca, 2004. "Minty Variational Inequalities, Increase-Along-Rays Property and Optimization1," Journal of Optimization Theory and Applications, Springer, vol. 123(3), pages 479-496, December.
    3. S. Al-Homidan & Q. H. Ansari, 2010. "Generalized Minty Vector Variational-Like Inequalities and Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 144(1), pages 1-11, January.
    4. X. M. Yang & X. Q. Yang & K. L. Teo, 2004. "Some Remarks on the Minty Vector Variational Inequality," Journal of Optimization Theory and Applications, Springer, vol. 121(1), pages 193-201, April.
    5. Giorgio Giorgi & Sándor Komlósi, 1995. "Dini derivatives in optimization — Part III," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 18(1), pages 47-63, March.
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    Cited by:

    1. M. Golestani & H. Sadeghi & Y. Tavan, 2018. "Nonsmooth Multiobjective Problems and Generalized Vector Variational Inequalities Using Quasi-Efficiency," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 896-916, December.
    2. Giovanni P. Crespi & Matteo Rocca & Carola Schrage, 2015. "Variational Inequalities Characterizing Weak Minimality in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 804-824, September.
    3. M. Oveisiha & J. Zafarani, 2012. "Vector optimization problem and generalized convexity," Journal of Global Optimization, Springer, vol. 52(1), pages 29-43, January.
    4. M. Oveisiha & J. Zafarani, 2014. "On Characterization of Solution Sets of Set-Valued Pseudoinvex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 387-398, November.
    5. Qamrul Ansari & Mahboubeh Rezaie & Jafar Zafarani, 2012. "Generalized vector variational-like inequalities and vector optimization," Journal of Global Optimization, Springer, vol. 53(2), pages 271-284, June.
    6. Giovanni P. Crespi & Carola Schrage, 2021. "Applying set optimization to weak efficiency," Annals of Operations Research, Springer, vol. 296(1), pages 779-801, January.
    7. Shashi Kant Mishra & Vivek Laha & Mohd Hassan, 2024. "On Quasiconvex Multiobjective Optimization and Variational Inequalities Using Greenberg–Pierskalla Based Generalized Subdifferentials," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 1169-1186, September.
    8. Syed Shakaib Irfan & Mijanur Rahaman & Iqbal Ahmad & Rais Ahmad & Saddam Husain, 2019. "Generalized Nonsmooth Exponential-Type Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problems in Asplund Spaces," Mathematics, MDPI, vol. 7(4), pages 1-11, April.

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