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Characterizations of the Solution Sets of Convex Programs and Variational Inequality Problems

Author

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  • Z. L. Wu

    (National Cheng Kung University)

  • S. Y. Wu

    (National Cheng Kung University)

Abstract

For a convex program in a normed vector space with the objective function admitting the Gâteaux derivative at an optimal solution, we show that the solution set consists of the feasible points lying in the hyperplane whose normal vector equals the Gâteaux derivative. For a general continuous convex program, a feasible point is an optimal solution iff it lies in a hyperplane with a normal vector belonging to the subdifferential of the objective function at this point. In several cases, the solution set of a variational inequality problem is shown to coincide with the solution set of a convex program with its dual gap function as objective function, while the mapping involved can be used to express the above normal vectors.

Suggested Citation

  • Z. L. Wu & S. Y. Wu, 2006. "Characterizations of the Solution Sets of Convex Programs and Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 130(2), pages 341-360, August.
    • Handle: RePEc:spr:joptap:v:130:y:2006:i:2:d:10.1007_s10957-006-9108-6
    DOI: 10.1007/s10957-006-9108-6
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    Citations

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    Cited by:

    1. X. M. Yang, 2009. "On Characterizing the Solution Sets of Pseudoinvex Extremum Problems," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 537-542, March.
    2. Satoshi Suzuki, 2019. "Optimality Conditions and Constraint Qualifications for Quasiconvex Programming," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 963-976, December.
    3. V. Jeyakumar & G. M. Lee & G. Li, 2015. "Characterizing Robust Solution Sets of Convex Programs under Data Uncertainty," Journal of Optimization Theory and Applications, Springer, vol. 164(2), pages 407-435, February.
    4. Vsevolod I. Ivanov, 2013. "Optimality Conditions and Characterizations of the Solution Sets in Generalized Convex Problems and Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 65-84, July.
    5. Satoshi Suzuki & Daishi Kuroiwa, 2015. "Characterizations of the solution set for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential," Journal of Global Optimization, Springer, vol. 62(3), pages 431-441, July.

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