Morse theory for some lower-\(C^ 2\) functions in finite dimension
type: Item
instance of: scholarly article
title: Morse theory for some lower-\(C^ 2\) functions in finite dimension
zbMATH Open document ID: 0646.49029
DOI: 10.1007/BF01580761
author: M. L. Bougeard
published in: Mathematical Programming. Series A. Series B
publication date: 1988-01-01T00:00:00Z
review text: The author proves the Morse inequalities for quadratically minorized (i.e., \(f(x)\geq\) const. + const.\(| x|^ 2)\) real-valued lower- \(C^ 2\) functions (i.e. f is representable as a maximum of a family of \(C^ 2\)-smooth functions) defined on open subsets of a finite- dimensional Hilbert space under the assumption that the critical points of f are strongly nondegenerate (a difficult concept not explained here) and \(f^{-1}([a,b])\) are compact subsets for all \(a,b\in {\mathbb{R}}\). The method is based on the Moreau-Yosida approximations \(f_{\epsilon}(x)=\inf_{w}(f(w)+| x-w|^ 2/2\epsilon)\) which are \(C^ 2\)-smooth and preserve the critical points together with certain generalized Morse index (a new concept for the lower-\(C^ 2\) functions), thus reducing the problem to the classical case.
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 90C30
zbMATH DE Number: 4054434
zbMATH Keywords: Morse inequalities
zbMATH Keywords: lower-\(C^ 2\) functions
zbMATH Keywords: critical points
zbMATH Keywords: Moreau- Yosida approximations
zbMATH Keywords: generalized Morse index
MaRDI profile type: Publication
cites work: Convexification procedures and decomposition methods for nonconvex optimization problems
cites work: Approximation and decomposition properties of some classes of locally d.c. functions
cites work: Generalized Gradients and Applications
cites work: On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface
cites work: A Note on Differentiability of Global Optimal Values
cites work: Q3042868
cites work: Q3707765
cites work: Q3205869
cites work: Q4160120
cites work: Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming
cites work: Q4750055
cites work: Strong and Weak Convexity of Sets and Functions
cites work: On Functions Representable as a Supremum of a Family of Smooth Functions
description: scientific article; zbMATH DE number 4054434