Damped techniques for enforcing convergence of quasi-Newton methods

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

This paper extends the technique used in the damped BFGS method of Powell [Algorithms for nonlinear constraints that use Lagrange functions, Math. Program. 14 (1978), 224-248] to the Broyden family of quasi-Newton methods with applications to unconstrained optimization problems. Appropriate conditions on the damped technique are proposed to enforce safely the positive definiteness property for all Broyden's updates. It is shown that this technique maintains the q-superlinear convergence property of the restricted Broyden family of methods for uniformly convex functions. It also extends the global convergence property to all members of the family. Preliminary numerical results are described which show that appropriate ways for employing the proposed technique improve the performance of all members of the Broyden family of methods substantially and significantly in certain cases. They also enforce convergence of divergent quasi-Newton methods.

Original languageEnglish
Pages (from-to)919-936
Number of pages18
JournalOptimization Methods and Software
Volume29
Issue number5
DOIs
Publication statusPublished - Sep 3 2014

Fingerprint

Quasi-Newton Method
Newton-Raphson method
Damped
Convergence Properties
BFGS Method
Positive Definiteness
Uniformly Convex
Nonlinear Constraints
Superlinear Convergence
Unconstrained Optimization
Global Convergence
Lagrange
Convex function
Update
Optimization Problem
Numerical Results
Family

Keywords

  • damped techniques
  • line search framework
  • quasi-Newton methods
  • unconstrained optimization

ASJC Scopus subject areas

  • Control and Optimization
  • Software
  • Applied Mathematics

Cite this

Damped techniques for enforcing convergence of quasi-Newton methods. / Al-Baali, Mehiddin.

In: Optimization Methods and Software, Vol. 29, No. 5, 03.09.2014, p. 919-936.

Research output: Contribution to journalArticle

@article{2ee4e5afec3f4b8e96d642d9147e39e0,
title = "Damped techniques for enforcing convergence of quasi-Newton methods",
abstract = "This paper extends the technique used in the damped BFGS method of Powell [Algorithms for nonlinear constraints that use Lagrange functions, Math. Program. 14 (1978), 224-248] to the Broyden family of quasi-Newton methods with applications to unconstrained optimization problems. Appropriate conditions on the damped technique are proposed to enforce safely the positive definiteness property for all Broyden's updates. It is shown that this technique maintains the q-superlinear convergence property of the restricted Broyden family of methods for uniformly convex functions. It also extends the global convergence property to all members of the family. Preliminary numerical results are described which show that appropriate ways for employing the proposed technique improve the performance of all members of the Broyden family of methods substantially and significantly in certain cases. They also enforce convergence of divergent quasi-Newton methods.",
keywords = "damped techniques, line search framework, quasi-Newton methods, unconstrained optimization",
author = "Mehiddin Al-Baali",
year = "2014",
month = "9",
day = "3",
doi = "10.1080/10556788.2014.891030",
language = "English",
volume = "29",
pages = "919--936",
journal = "Optimization Methods and Software",
issn = "1055-6788",
publisher = "Taylor and Francis Ltd.",
number = "5",

}

TY - JOUR

T1 - Damped techniques for enforcing convergence of quasi-Newton methods

AU - Al-Baali, Mehiddin

PY - 2014/9/3

Y1 - 2014/9/3

N2 - This paper extends the technique used in the damped BFGS method of Powell [Algorithms for nonlinear constraints that use Lagrange functions, Math. Program. 14 (1978), 224-248] to the Broyden family of quasi-Newton methods with applications to unconstrained optimization problems. Appropriate conditions on the damped technique are proposed to enforce safely the positive definiteness property for all Broyden's updates. It is shown that this technique maintains the q-superlinear convergence property of the restricted Broyden family of methods for uniformly convex functions. It also extends the global convergence property to all members of the family. Preliminary numerical results are described which show that appropriate ways for employing the proposed technique improve the performance of all members of the Broyden family of methods substantially and significantly in certain cases. They also enforce convergence of divergent quasi-Newton methods.

AB - This paper extends the technique used in the damped BFGS method of Powell [Algorithms for nonlinear constraints that use Lagrange functions, Math. Program. 14 (1978), 224-248] to the Broyden family of quasi-Newton methods with applications to unconstrained optimization problems. Appropriate conditions on the damped technique are proposed to enforce safely the positive definiteness property for all Broyden's updates. It is shown that this technique maintains the q-superlinear convergence property of the restricted Broyden family of methods for uniformly convex functions. It also extends the global convergence property to all members of the family. Preliminary numerical results are described which show that appropriate ways for employing the proposed technique improve the performance of all members of the Broyden family of methods substantially and significantly in certain cases. They also enforce convergence of divergent quasi-Newton methods.

KW - damped techniques

KW - line search framework

KW - quasi-Newton methods

KW - unconstrained optimization

UR - http://www.scopus.com/inward/record.url?scp=84903541767&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84903541767&partnerID=8YFLogxK

U2 - 10.1080/10556788.2014.891030

DO - 10.1080/10556788.2014.891030

M3 - Article

VL - 29

SP - 919

EP - 936

JO - Optimization Methods and Software

JF - Optimization Methods and Software

SN - 1055-6788

IS - 5

ER -