STABILITY ANALYSIS OF VARIATIONAL-INEQUALITIES AND NONLINEAR COMPLEMENTARITY-PROBLEMS, VIA THE MIXED LINEAR COMPLEMENTARITY-PROBLEM AND DEGREE THEORY

This paper is concerned with the mixed linear complementarity problem and the role it and its variants play in the stability analysis of the nonlinear complementarity problem and the Karush-Kuhn-Tucker system o f a variational inequality problem. Under a nonsingular assumption, th e mixed linear complementarity problem can be converted to The standar d problem; in this case, the rich theory of the latter can be directly applied to the former. In this work, we employ degree theory to deriv e some sufficient conditions for the existence of a solution to the mi xed linear complementarity problem in the absence of the nonsingularit y property. Next, we extend this existence theory to the mixed nonline ar complementarity problem and establish a main stability result under a certain degree-theoretic assumption concerning the linearized probl em. We then specialize this stability result and its consequences to t he parametric variational inequality problem under the assumption of a unique set of multipliers. Finally, we consider the latter problem wi th the uniqueness assumption of the multipliers replaced by a convexit y assumption and obtain stability results under some weak second-order conditions. In addition to the new existence results for the mixed li near complementarity problem, the main contributions of this paper in the stability category are the following: a resolution to a conjecture concerning the local solvability of a parametric Variational inequali ty, the use of the generalized linear complementarity problem as a too l to broaden the second-order conditions, the characterization of the solution stability of the linear complementarity problem and the affin e variational inequality problem in terms of the solution isolatedness under some weak hypotheses, and Various stability theorems under some weak second-order conditions.