Generalized Poincare-Hopf theorem for compact nonsmooth regions

• 成果类型: Article
• 署名作者: Simsek, Alp; Ozdaglar, Asuman; Acemoglu, Daron
• 署名单位: Massachusetts Institute of Technology (MIT); Massachusetts Institute of Technology (MIT); Massachusetts Institute of Technology (MIT)
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.1060.0235
• 发表日期: 2007
• 页码: 193-214
• 关键词: equilibria; number

摘要

This paper presents an extension of the Poincare-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 -> R, 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 Poincare-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.