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Minty variational inequalities, increase-along-rays property and optimization

  • Crespi Giovanni P.


    (Facoltà di Scienze Economiche Aosta, Italy)

  • Ginchev Ivan


    (Department of Mathematics Varna, Bulgaria)

  • Rocca Matteo


    (Department of Economics, University of Insubria, Italy)

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    Let E be a linear space, K E and f : K ? R. We put in terms of the lower Dini directional derivative a problem, referred to as GMV I(f ,K), which can be considered as a generalization of the Minty variational inequality of differential type (for short, MV I(f ,K)). We investigate, in the case of K star-shaped (for short, st-sh), the existence of a solution x of GMV I(f ,K) and the property of f to increase-along-rays starting at x (for short, f IAR(K, x )). We prove that GMV I(f ,K) with radially l.s.c. function f has a solution x ker K if and only if f IAR(K, x ). Further, we prove, that the solution set of GMV I(f ,K) is a convex and radially closed subset of kerK. We show also that, if GMV I(f ,K) has a solution x K, then x is a global minimizer of the problem f(x) ? min, x K. Moreover, we observe that the set of the global minimizers of the related optimization problem, its kernel, and the solution set of the variational inequality can be different. Finally, we prove, that in case of a quasi-convex function f, these sets coincide. Key words: Minty variational inequality, Generalized variational inequality, Existence of solutions, Increase along rays, Quasi-convex functions.

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    Paper provided by Department of Economics, University of Insubria in its series Economics and Quantitative Methods with number qf04019.

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    Length: 21 pages
    Date of creation: Oct 2004
    Date of revision:
    Handle: RePEc:ins:quaeco:qf04019
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