A Global Index Theorem for Degenerate Variational Inequalities

We derive a global index theorem for degenerate variational inequality problems defined by a continuously differentiable function F over a convex set M represented by a finite number of inequality constraints. Our index theorem can be applied when the solutions are non-singular and possibly degenerate, as long as they also satisfy the injective normal map property, which is implied by strong stability. Using this index theorem, we derive local matrix theoretic sufficiency conditions for the global uniqueness of solutions to the variational inequality problem. These uniqueness conditions generalize earlier results which required either the function F to satisfy global conditions or each solution to be non-degenerate. We apply our main results to derive uniqueness theorems for mixed and nonlinear complementarity problems. ∗E-mail: [email protected] †E-mail: [email protected] ‡E-mail: [email protected]