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Optimization and Variational Inequalities with Pseudoconvex Functions

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  • V. I. Ivanov

    (Technical University of Varna)

Abstract

In the present paper we consider a pseudoconvex (in an extended sense) function f using higher order Dini directional derivatives. A Variational Inequality, which is a refinement of the Stampacchia Variational Inequality, is defined. We prove that the solution set of this problem coincides with the set of global minimizers of f if and only if f is pseudoconvex. We introduce a notion of pseudomonotone Dini directional derivatives (in an extended sense). It is applied to prove that the solution sets of the Stampacchia Variational Inequality and Minty Variational Inequality coincide if and only if the function is pseudoconvex. At last, we obtain several characterizations of the solution set of a program with a pseudoconvex objective function.

Suggested Citation

  • V. I. Ivanov, 2010. "Optimization and Variational Inequalities with Pseudoconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 602-616, September.
    • Handle: RePEc:spr:joptap:v:146:y:2010:i:3:d:10.1007_s10957-010-9682-5
    DOI: 10.1007/s10957-010-9682-5
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    References listed on IDEAS

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    1. 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.
    2. Ivan Ginchev & Vsevolod I. Ivanov, 2007. "Higher-order Pseudoconvex Functions," Lecture Notes in Economics and Mathematical Systems, in: Generalized Convexity and Related Topics, pages 247-264, Springer.
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    Cited by:

    1. Vsevolod Ivanov, 2013. "Characterizations of pseudoconvex functions and semistrictly quasiconvex ones," Journal of Global Optimization, Springer, vol. 57(3), pages 677-693, November.
    2. 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.
    3. Vsevolod I. Ivanov, 2019. "Characterizations of Solution Sets of Differentiable Quasiconvex Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 144-162, April.

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