Computing proximal points of nonconvex functions
type: Item
instance of: scholarly article
title: Computing proximal points of nonconvex functions
zbMATH Open document ID: 1168.90010
author: B. E. Eshmatov
published in: Mathematical Programming. Series A. Series B
publication date: 2008-12-16T00:00:00Z
review text: From authors' abstract: The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mappings was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with the critical points of the original function. This suggests that the many uses of proximal points, and their corresponding proximal envelopes (Moreau envelopes), will have a natural extension from convex optimization to nonconvex optimization. For example, the inexact proximal point methods for convex optimization might be redesigned to work for nonconvex functions. In order to begin the practical implementation of proximal points in a nonconvex setting, a first crucial step would be to design efficient methods of approximating nonconvex proximal points. This would provide a solid foundation on which future design and analysis for nonconvex proximal point methods could flourish. In this paper we present a methodology based on the computation of proximal points of piecewise affine models of the nonconvex function. These models can be built with only the knowledge obtained from a black box providing, for each point, the function value and one subgradient. Convergence of the method is proved for the class of nonconvex functions that are prox-bounded and lower-\(C^2\) and encouraging preliminary numerical testing is reported.
Mathematics Subject Classification ID: 90C26
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 49J53
Mathematics Subject Classification ID: 49M05
zbMATH DE Number: 5382675
zbMATH Keywords: nonconvex optimization
zbMATH Keywords: nonsmooth optimization
zbMATH Keywords: proximal point
zbMATH Keywords: prox-regular
zbMATH Keywords: lower-\({\mathcal{C}}^2\)
reviewed by: Fabián Flores-Bazan
describes a project that uses: PNEW
MaRDI profile type: Publication
full work available at URL: s10107-007-0124-6
OpenAlex ID: W2007277085
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DOI: 10.1007/S10107-007-0124-6
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description: scientific article; zbMATH DE number 5382675