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Variational Inequalities Characterizing Weak Minimality in Set Optimization

Author

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  • Giovanni P. Crespi

    (Loc. Grand Chemin 73-75)

  • Matteo Rocca

    (Università degli Studi dell’Insubria)

  • Carola Schrage

    (Loc. Grand Chemin 73-75)

Abstract

We introduce the notion of weak minimizer in set optimization. Necessary and sufficient conditions in terms of scalarized variational inequalities of Stampacchia and Minty type, respectively, are proved. As an application, we obtain necessary and sufficient optimality conditions for weak efficiency of vector optimization in infinite-dimensional spaces. A Minty variational principle in this framework is proved as a corollary of our main result.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:166:y:2015:i:3:d:10.1007_s10957-014-0679-3
    DOI: 10.1007/s10957-014-0679-3
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    References listed on IDEAS

    as
    1. J. Li & G. Mastroeni, 2010. "Vector Variational Inequalities Involving Set-valued Mappings via Scalarization with Applications to Error Bounds for Gap Functions," Journal of Optimization Theory and Applications, Springer, vol. 145(2), pages 355-372, May.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
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    Cited by:

    1. Giovanni P. Crespi & Carola Schrage, 2021. "Applying set optimization to weak efficiency," Annals of Operations Research, Springer, vol. 296(1), pages 779-801, January.

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