Gauss-Newton method for convex composite optimizations on Riemannian manifolds
type: Item
instance of: scholarly article
title: Gauss-Newton method for convex composite optimizations on Riemannian manifolds
zbMATH Open document ID: 1254.65076
author: Jen-Chih Yao
author: J. H. Wang
author: C. Li
published in: Journal of Global Optimization
publication date: 2012-09-20T00:00:00Z
review text: The notions of quasi-regularity for \(p_0 \in M\) (a Riemannian manifold), quasi-regular radius \(r_{p_0}\) and quasi-regular bound function \(\beta{p_0}\) attached to a quasi-regular point \(p_0\) are extended to Riemannian manifolds. The authors show that if the initial point \(p_0\) is a quasi-regular point with \((r_{p_0}, \beta_{p_0})\) and if \(DF\) (\(F\) is a nonlinear Fréchet differentiable mapping from \(\mathbb R^m\) to \(\mathbb R^n\), \(DF(p)\) denotes the derivative of \(F\) at \(p\)) satisfies a Lipschitz type condition, then the Gauss-Newton sequence \(\{p_n\}\) generated by the algorithm \(R (\eta, \Delta, p_0)\) converges at a quadratic rate to some \(p^*\) with \(F(p^*) \in C\) (in particular, \(p^*\) solves: \(\min_{p\in M} f(p):=h(F(p))\), where \(h\) is a real-valued convex function on \(\mathbb R\)). Furthermore, two applications to special cases, one is for the case of regularities of Riemannian manifolds and the other is for the case when \(C\) is a cone and \((DF (p_0)(.)-C)\) is surjective, are proposed. Main result: The authors extend the Gauss-Newton method to a Riemannian manifold to solve the convex composite optimization on the Riemannian manifold \(M\)which is formulated as follows: \[ \min_{p \in M} f(p):=h(F(p)).\tag{1} \] The majorizing function technique to establish the semi-local convergence of sequences generated by the Gauss-Newton method for the convex composite function \(h_0F\) on the Riemannian manifold is used. The extended Gauss-Newton for the convex composite optimization problem on Riemannian manifold (1) is defined. The authors extend the notion of the quasi-regular point to Riemannian manifolds and specialize own results in two important cases: the classical Lipschitz condition and the \(\gamma\)-condition (implies the \(L\)-average Lipschitz condition) which have been used extensively in the study of convergence of Newton's method both in Banach spaces and Riemannian manifolds. If \(F\) satisfies the \(\gamma\)-con…
reviewed by: Jan Lovíšek
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 90C48
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 6084586
zbMATH Keywords: Gauss-Newton method
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: \(L\)-average Lipschitz condition
zbMATH Keywords: quasi-regularity
zbMATH Keywords: convergence criterion
zbMATH Keywords: algorithm
zbMATH Keywords: regular point
zbMATH Keywords: convex process
zbMATH Keywords: sequence
zbMATH Keywords: convex composite optimization
zbMATH Keywords: semi-local convergence
MaRDI profile type: Publication
full work available at URL: s10898-010-9638-1
OpenAlex ID: W2085379846
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DOI: 10.1007/S10898-010-9638-1
description: scientific article; zbMATH DE number 6084586