Generalized Poincaré-Hopf Theorem for Compact Nonsmooth Regions

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 → , 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.