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Acta Mathematica Vietnamica

CODERIVATIVE CALCULATION RELATED TO A PARAMETRIC AFFINE VARIATIONAL INEQUALITY PART 1: BASIC CALCULATIONS

J.-C. YAO, N. D. YEN

Abstract

Consider a parametric affine variational inequality 0Mx+q+N(x;Δ(A;b)); denoted by AVI(M,q,A,b); for which the pair (q,b)Rn×Rm describes the linear perturbations. Here the matrices MRn×n and ARm×n are the given data, Δ(A,b)={xRn:Axb} is a polyhedral convex constraint set, and N(x;Δ(A,b)) denotes the normal cone to Δ(A,b) at x. We study the normal coderivative of the normal-cone operator (x,b)N(x;Δ(A;b)). In the second part of this paper [20], combining the obtained results with some theorems from Mordukhovich [11], Levy and Mordukhovich [10], Yen and Yao [21], we get sufficient conditions for the Aubin property (the Lipschitz-like property) and the local metric regularity in Robinson’s sense of the solution map (q,b)S(q,b) of the problem AVI(M,q,A,b) and of the solution map (w,b)S(w,b) of the problem 0f(x,w)+N(x;Δ(A,b)) where f:Rn×RsRn is a given C1 vector function. Our investigation complements the well-known work of Dontchev and Rockafellar [3] where the Aubin property of the solution maps qS(q,b) and wS(w,b) (b is fixed) was established via a critical face condition.