The Application of Variational Inequalities to Complementarity Problems and Existence Theorems

Abstract

If $F : C \rightarrow R^{n}$ is a continuous (nonlinear) mapping on a closed, convex subset $C$ of $R^{n}$, it is shown that very weak coercivity conditins on $F$ guarantee the existence of a solution $x^{*} in $C$ to the variational inequality $(x-x^{*}, Fx^{*}) \geq 0$ for each $x$ in $C$. By restricting the shape of $C$, it is shown that $x^{*}$ solves different problems, and in each case we are able to obtain new existence results. If $C$ is a cone $K$, the $x^{*}$ is a solution to the complementarity problem: Find an $x^{*}$ in $K$ such that $Fx^{*}$ belongs to the polar of $K$ and $(x^{*}, Fx^{*})=0$. In this case it is possible to generalize some of the feasibility results available in the linear theory and to give an iterative scheme for finding $x^{*}. If $C$ is similar to a simplex then $x^{*}$ turns out to be a solution of nonlinear inequalities in the preorder induced by a cone, while if $C$ or $K$ is $R^{n}$, then $Fx^{*} =0$.