Convergence analysis of the Gauss-Newton method for convex inclusion and convex-composite optimization problems
type: Item
instance of: scholarly article
title: Convergence analysis of the Gauss-Newton method for convex inclusion and convex-composite optimization problems
zbMATH Open document ID: 1239.65036
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-02-27T00:00:00Z
review text: \textit{S. M. Robinson} [Numer. Math. 19, 341--347 (1972; Zbl 0227.65040)] proposed the so-called extended Newton method for solving the convex inclusion problem defined by a nonempty closed convex cone \(C\) in a Banach space \(Y\), and a smooth function \(F\) from a reflexive Banach space \(X\) into \(Y\). Under two important assumptions, he established the existence of solution to the convex inclusion problem, and provided the convergence criterion of this method for the convex inclusion problem. In this paper, the authors consider the convergence analysis of the Gauss-Newton method for the convex inclusion problem by using the convex process theory. Under some mild assumptions (which are weaker than Robinson's assumptions), they establish the convergence criterion of the Gauss-Newton method for solving convex-composite optimization problem. Particularly, they also obtain the convergence criterion for the extended Newton method for solving the convex inclusion problem presented by Robinson. Even in the special case when \(X,Y\) are finite-dimensional and \(T_{x_0}\) is surjective, their result is sharper than Robinson's result. Furthermore, the convergence criterions given in this paper are affine-invariant.
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 90C48
zbMATH DE Number: 6009674
zbMATH Keywords: Gauss-Newton method
zbMATH Keywords: convex process
zbMATH Keywords: convex-composite optimization
zbMATH Keywords: weak-Robinson condition
zbMATH Keywords: majorizing function
zbMATH Keywords: convergence criterion
zbMATH Keywords: Banach space
zbMATH Keywords: convex inclusion problem
author: C. Li
author: K. F. Ng
reviewed by: Guoqiang Wang
describes a project that uses: NewtonLib
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.11.062
OpenAlex ID: W2064882318
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DOI: 10.1016/J.JMAA.2011.11.062
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description: scientific article; zbMATH DE number 6009674