Existence of solutions and star-shapedness in Minty variational inequalities

Crespi Giovanni P., Ginchev Ivan () and Rocca Matteo ()
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Crespi Giovanni P.: Department of Economics, University of Insubria, Italy
Ginchev Ivan: Department of Mathematics Varna, Bulgaria
Rocca Matteo: Department of Economics, University of Insubria, Italy

Economics and Quantitative Methods from Department of Economics, University of Insubria

Abstract: Minty Variational Inequalities (for short, MVI) have proved to characterize a kind of equilibrium more qualified than Stampacchia Variational Inequalities (for short, SVI). This conclusion leads to argue that, when a MVI admits a solution and the operator F admits a primitive minimization problem (that is the function f to minimize is such that F = f0), then f has some regularity property, e.g. convexity or generalized convexity. In this paper we put in terms of the lower Dini directional derivative a problem, referred to as GMV I(f0 -,K), which can be considered a nonlinear extension of the MVI with F = f0 (K denotes a subset of Rn). We investigate, in the case K star-shaped, the existence of a solution of GMV I(f0 -,K) and the property of f to increase along rays starting at x (for short, f 2 IAR(K, x)). We prove that GMV I(f0 -,K) with a radially lower semicontinuous function f has a solution x 2 kerK if and only if f 2 IAR(K, x). Furthermore we investigate, with regard to optimization problems, some properties of functions increasing along rays, which can be considered as extensions of analogous properties holding for convex functions. In particular we show that functions belonging to the class IAR(K, x) enjoy some well-posebness properties. Key words: Minty variational inequality, generalized convexity, star-shaped sets, existence of solutions, well-posedness

Pages: 17 pages
Date: 2004-09
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