Nonlinear optimization in finite dimensions. Morse theory, Chebyshev approximation, transversality, flows, parametric aspects
type: Item
instance of: scholarly article
title: Nonlinear optimization in finite dimensions. Morse theory, Chebyshev approximation, transversality, flows, parametric aspects
zbMATH Open document ID: 0985.90083
author: Hubertus T. Jongen
author: Peter Jonker
author: Frank Twilt
published in: Nonconvex Optimization and Its Applications
publication date: 2000-11-15T00:00:00Z
review text: This book presents methods and ideas from differential topology for readers interested in finite-dimensional nonlinear optimization theory. After an introduction into some basic concepts and notions from critical point theory in chapter 1, the chapters 2, 3 and 5 are devoted to Morse theory. The classical results on the homotopy of sublevel sets are proved in chapter 2 for functions \(f:\mathbb R^n\rightarrow\mathbb R\) which satisfy the Palais-Smale condition. These results are extended in chapter 3 to functions defined on ''manifolds with generalized boundary''. These are subsets of \(\mathbb R^n\) which locally look like \(\mathbb R^k\times(\mathbb R_+)^p\). In chapter 5 singular homology is introduced and the Morse relations are proved. This presentation of Morse theory is interrupted in chapter 4 for an application of Morse theory to Chebyshev approximation where one wants to approximate a function \(f\in{\mathcal{C}}^2(\mathbb R^n)\) by an element from a family \(\{F(.,y)\in{\mathcal{C}}^2(\mathbb R^N):y\in M\subset\mathbb R^m\}\) of functions parametrized by a compact manifold \(M\) with generalized boundary. The remaining chapters cover a variety of topics. In chapter 6 the stability of nondegenerate critical points under \({\mathcal{C}}^2\)-perturbations of \(f\) is proved. Chapter 7 contains an introduction to transversality in order to treat Morse-Smale gradient flows in chapter 8. Chapter 9 deals with Newton flows, in particular with gradient Newton flows. If \(f:\mathbb R^n\rightarrow\mathbb R\) is smooth and \(F=\text{grad} f:\mathbb R^n\rightarrow\mathbb R^n\), then the (gradient) Newton flow is the flow associated to the differential equation \(\dot{x}=-DF(x)^{-1}F(x)\) defined on the set \(\{x\in\mathbb R^n:\det DF(x)\not=0\}\). The Euler discretization of this equation yields a Newton-Raphson iteration formula for finding critical points, in particular local minima of \(f\). The last chapter 10 treats parameter dependent problems like one-parameter…
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 90C31
Mathematics Subject Classification ID: 90C52
Mathematics Subject Classification ID: 90C53
zbMATH DE Number: 1530909
zbMATH Keywords: critical point theory
zbMATH Keywords: Morse theory
zbMATH Keywords: Chebyshev approximation
zbMATH Keywords: transversality
zbMATH Keywords: gradient flows
zbMATH Keywords: Newton flows
zbMATH Keywords: optimization problems
zbMATH Keywords: stability of critical points
reviewed by: Thomas J. Bartsch
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1530909