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Optimization and Variational Inequalities with Pseudoconvex Functions

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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.

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Authors and Affiliations

  1. Department of Mathematics, Technical University of Varna, 9010, Varna, Bulgaria

    V. I. Ivanov

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  1. V. I. Ivanov
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Correspondence to V. I. Ivanov.

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Communicated by G. Di Pillo.

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Ivanov, V.I. Optimization and Variational Inequalities with Pseudoconvex Functions. J Optim Theory Appl 146, 602–616 (2010). https://doi.org/10.1007/s10957-010-9682-5

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