A geometric representation of the Morse fan
type: Item
instance of: scholarly article
title: A geometric representation of the Morse fan
zbMATH Open document ID: 1105.49018
published in: Journal of Global Optimization
publication date: 2005-05-17T00:00:00Z
review text: This article bases on the result by \textit{H. Th. Jongen} and \textit{D. Pallaschke} [Optimization 19, No. 3, 343--353 (1988; Zbl 0671.90099)] that around a nondegenerate critical point a piecewise smooth Morse function is topologically equivalent to the sum of a constant, a normalized nondegenerate quadratic form and the term \(\ell(y_1,...,y_k)\) where \(\ell\) is a continuous selection of the functions \(y_1,...,y_k\) and \(-\sum_{i=1}^k y_i\). The present paper studies a collection of cones in \(\mathbb R^k\), on which the latter continuous selections are linear. The construction of these cones goes back to \textit{S. G. Bartels, L. Kuntz} and \textit{S. Scholtes [Nonlinear Anal., Theory Methods Appl. 24, No. 3, 385--407 (1995; Zbl 0846.49005)]. The considered collection of cones form a complete polyhedral fan and are here briefly called Morse fan. The main result of the article is that the Morse fan is a refinement of the normal fan of the polytope which results from taking the Minkowski sum of a pyramid \(P\) and of \(-P\), where \(P\) is the convex hull of the unit vectors \(e_1,\dots,e_k\) and their negative sum \(-\sum_{i=1}^k e_i\). Furthermore, several details about the structure of the Morse fan are mentioned.}
reviewed by: Oliver Stein
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 90C46
Mathematics Subject Classification ID: 90C47
zbMATH DE Number: 2168244
zbMATH Keywords: combinatorial convexity
zbMATH Keywords: nonsmooth Morse theory
zbMATH Keywords: nonsmooth optimization
author: J. Grzybowski
author: D. Pallaschke
author: Ryszard Urbański
MaRDI profile type: Publication
full work available at URL: s10898-004-8275-y
OpenAlex ID: W1973743958
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DOI: 10.1007/S10898-004-8275-Y
description: scientific article; zbMATH DE number 2168244