A class of superlinearly convergent projection algorithms with relaxed stepsizes
type: Item
instance of: scholarly article
title: A class of superlinearly convergent projection algorithms with relaxed stepsizes
zbMATH Open document ID: 0563.65041
DOI: 10.1007/BF01449032
published in: Applied Mathematics and Optimization
publication date: 1984-01-01T00:00:00Z
review text: Let \(f: {\mathbb{R}}^ n\to {\mathbb{R}}\) and \(g: {\mathbb{R}}^ n\to {\mathbb{R}}^ m\) be differentiable functions and let \(D=\{x\in {\mathbb{R}}^ n| g(x)\geq 0\}\) be a convex set. Consider the problem: min\(\{\) f(x)\(| x\in D\}\). This typical constrained optimization problem is considered by the author, who investigates, in this work, a quasi-Newton extension of the Goldstein- Levitin-Polyak projected gradient algorithm. This extension projects an unconstrained descent step on to D, the feasible region and is closed to the work of \textit{J. C. Allwright} [J. Optimization Theory Appl. 30, 1-18 (1980; Zbl 0393.90069)]. In the algorithm presented here, the determination of the stepsize is divided into two stages. The first one determines the lengths of the unconstrained step and the second stage is the determination of the stepsize from the range [0,1] that shortens the projected step, if the projection on D does not reduce the objective function f. In the first stage, the objective function f decreases and is bounded by a conventional linear functional, while the second stage uses a quadratic functional as a bound. We remark that the well known Goldstein-Levitin-Polyak algorithm is relaxed here and so the quadratic subproblem involved, becomes simple.
reviewed by: A.Donescu
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 90C30
zbMATH DE Number: 3898137
zbMATH Keywords: relaxed stepsizes
zbMATH Keywords: superlinear convergence
zbMATH Keywords: constrained optimization
zbMATH Keywords: quasi-Newton
zbMATH Keywords: Goldstein-Levitin-Polyak projected gradient algorithm
zbMATH Keywords: quadratic subproblem
author: Berç Rustem
MaRDI profile type: Publication
cites work: A feasible direction algorithm for convex optimization: Global convergence rates
cites work: On the Goldstein-Levitin-Polyak gradient projection method
cites work: Projected Newton Methods for Optimization Problems with Simple Constraints
cites work: On the Local Convergence of Quasi-Newton Methods for Constrained Optimization
cites work: Quasi-Newton Methods, Motivation and Theory
cites work: Rates of Convergence for Conditional Gradient Algorithms Near Singular and Nonsingular Extremals
cites work: Newton’s Method and the Goldstein Step-Length Rule for Constrained Minimization Problems
cites work: Global and Asymptotic Convergence Rate Estimates for a Class of Projected Gradient Processes
cites work: Q5588268
cites work: Convex programming in Hilbert space
cites work: Superlinearly convergent variable metric algorithms for general nonlinear programming problems
cites work: Q5566063
cites work: The Gradient Projection Method under Mild Differentiability Conditions
cites work: Q5672476
cites work: Projection methods in constrained optimisation and applications to optimal policy decisions
description: scientific article; zbMATH DE number 3898137