Optimal algorithms and the BFGS updating techniques for solving unconstrained nonlinear minimization problems
type: Item
instance of: scholarly article
title: Optimal algorithms and the BFGS updating techniques for solving unconstrained nonlinear minimization problems
zbMATH Open document ID: 1406.65042
DOI: 10.1155/2014/324181
author: Chein-Shan Liu
published in: Journal of Applied Mathematics
publication date: 2019-02-01T00:00:00Z
review text: Summary: To solve an unconstrained nonlinear minimization problem, we propose an optimal algorithm (OA) as well as a globally optimal algorithm (GOA), by deflecting the gradient direction to the best descent direction at each iteration step, and with an optimal parameter being derived explicitly. An invariant manifold defined for the model problem in terms of a locally quadratic function is used to derive a purely iterative algorithm and the convergence is proven. Then, the rank-two updating techniques of BFGS are employed, which result in several novel algorithms as being faster than the steepest descent method (SDM) and the variable metric method (DFP). Six numerical examples are examined and compared with exact solutions, revealing that the new algorithms of OA, GOA, and the updated ones have superior computational efficiency and accuracy.
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 90C53
zbMATH DE Number: 7010598
Wikidata QID: Q59050906
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/324181
OpenAlex ID: W2020357931
description: scientific article; zbMATH DE number 7010598