Global convergence of algorithms with nonmonotone line search strategy in unconstrained optimization
type: Item
instance of: scholarly article
title: Global convergence of algorithms with nonmonotone line search strategy in unconstrained optimization
zbMATH Open document ID: 1033.65041
DOI: 10.1007/BF03322774
author: Björn Hüther
published in: Results in Mathematics
publication date: 2003-10-29T00:00:00Z
review text: Let \(f: \mathbb{R}^{n}\rightarrow \mathbb{R}\) be a continuously differentiable function. For the unconstrained optimization problem \[ \min f(x), x \in \mathbb{R}^{n}, \] the author studies some methods generating an approximation sequence \((x_{k})\) by the iteration process \[ x_{k+1} =x_{k} + \alpha_{k}d_{k}, k=0,1\dots, \] with some given starting points \(x_{0}\in \mathbb{R}^{n}\). If the sequence \((f(x_{k}))\) is monotone, then the line search strategy is called monotone. The focus of this paper are nonmonotone line search strategies; the hypotheses do not force the sequence \((f(x_{k}))\) to be monotone. The author proves that the nonmonotone Wolfe line search strategy is \(M\)-efficient and that the Armijo line search strategy is semi-\(M\)-efficient. Using these results he shows the global convergence of the Broyden-Fletcher-Goldfarb-Shannon method with nonmonotone line search strategy.
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 90C52
zbMATH DE Number: 1997875
zbMATH Keywords: nonmonotone line search strategy
zbMATH Keywords: BFGS method
zbMATH Keywords: global convergence
zbMATH Keywords: unconstrained optimization
zbMATH Keywords: Wolfe line search strategy
zbMATH Keywords: Armijo line search strategy
zbMATH Keywords: Broyden-Fletcher-Goldfarb-Shannon method
reviewed by: Dorel I. Duca
MaRDI profile type: Publication
cites work: A Nonmonotone Line Search Technique for Newton’s Method
cites work: Minimization of functions having Lipschitz continuous first partial derivatives
cites work: Convergence of the nonmonotone Perry and Shanno method for optimization
cites work: A class of nonmonotone conjugate gradient methods for unconstrained optimization
cites work: Q4312995
cites work: Descent Property and Global Convergence of the Fletcher—Reeves Method with Inexact Line Search
cites work: Convergence Conditions for Ascent Methods
cites work: Global convergence analysis of a new nonmonotone BFGS algorithm on convex objective functions
cites work: Global convergece of the bfgs algorithm with nonmonotone linesearch∗∗this work is supported by national natural science foundation$ef:
cites work: A class on nonmonotone stabilization methods in unconstrained optimization
cites work: An Assessment of Nonmonotone Linesearch Techniques for Unconstrained Optimization
cites work: Effiziente Schrittweitenfunktionen bei unrestringierten Optimierungsaufgaben
cites work: Über die globale Konvergenz von Variable-Metrik-Verfahren mit nicht- exakter Schrittweitenbestimmung
cites work: Q4103338
cites work: A new approach to variable metric algorithms
cites work: Variable metric methods of minimisation
cites work: On the Convergence of the Variable Metric Algorithm
description: scientific article; zbMATH DE number 1997875