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Generalized Poincaré-Hopf Theorem for Compact Nonsmooth Regions

Department of Economics, Massachusetts Institute of Technology, Office: 32D-740, 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

This paper presents an extension of the Poincaré-Hopf theorem to generalized critical points of a function on a compact region with nonsmooth boundary, M, defined by a finite number of smooth inequality constraints. Given a function F: M ⁿ, we define the generalized critical points of F over M, define the index for the critical point, and show that the sum of the indices of the critical points is equal to the Euler characteristic of M. We use the generalized Poincaré-Hopf theorem to present sufficient (local) conditions for the uniqueness of solutions to finite-dimensional variational inequalities and the uniqueness of stationary points in nonconvex optimization problems.

Key Words: Poincaré-Hopf theorem; index theory; variational inequality; Euclidean projection; generalized equilibrium; nonlinear optimization
History: Received: July 25, 2005; revision received: September 30, 2006;