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Local Indices for Degenerate Variational Inequalities

Department of Economics, Massachusetts Institute of Technology, Office: E52-303, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Office: 32D-630, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Department of Economics, Massachusetts Institute of Technology, Office: E52-380B, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
alpstein{at}mit.edu, http://web.mit.edu/alpstein/www
asuman{at}mit.edu, http://web.mit.edu/asuman/www
daron{at}mit.edu, http://econ-www.mit.edu/faculty/?prof_id=acemoglu

We provide an index formula for solutions of 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 formula can be applied when the solutions are nonsingular and possibly degenerate, as long as they also satisfy the injective normal map (INM) property, which is implied by strong stability. We show that when the INM property holds, the degeneracy in a solution can be removed by perturbing the function F slightly, i.e., the index of a degenerate solution is equal to the index of a nondegenerate solution of a slightly perturbed variational inequality problem. We further show that our definition of the index is equivalent to the topological index of the normal map at the zero corresponding to the solution. As an application of our index formula, we provide a global index theorem for variational inequalities which holds even when the solutions are degenerate.

Key Words: variational inequality; index theory; complementarity problem; uniqueness
History: Received: July 24, 2006; revision received: April 14, 2007;