Generalized Poincaré-Hopf Theorem for Compact Nonsmooth Regions
Tóm tắt
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 ↦ ℝn, 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.
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Tài liệu tham khảo
Berge C., 1963, Topological Spaces
Bertsekas D., 1995, Nonlinear Programming
Bertsekas D., 2003, Convex Analysis and Optimization
Clarke F. H., 2003, Optimization and Nonsmooth Analysis
Clarke F. H., 1995, J. Convex Anal., 2, 117
Cornet B., 2000, Comm. Appl. Nonlinear Anal., 7, 21
Eraslan H., McLennan A. Uniqueness of stationary equilibrium payoffs in coalitional bargaining (2005) Working paper, University of Pennsylvania, Philadelphia, PA. http://finance.wharton.upenn.edu/∼eraslan/HulyaEraslanWebPage/Papers/bar_latex060107.pdf
Facchinei F., 2003, Finite-Dimensional Variational Inequalities and Complementarity Problems, 1
Gottlieb D. H., 1995, New York J. Math., 1, 130
Guillemin V., 1974, Differential Topology
Hildenbrand W., 1988, Equilibrium Analysis and Variations on Themes by Edgeworth and Walras
Jongen Th. H., 2000, Nonlinear Optimization in Finite Dimensions, Nonconvex Optimization and Its Applications, 47
Milnor J. W., 1997, Topology from the Differential Viewpoint
Ortega J. M., 1970, Iterative Solution of Nonlinear Equations in Several Variables
Varian H. R., 1975, Econometrica, 43, 985