### Abstract

This paper studies the convergence properties of algorithms belonging to the class of self-scaling (SS) quasi-Newton methods for unconstrained optimization. This class depends on two parameters, say θ_{k} and τ_{k}, for which the choice τ_{k} = 1 gives the Broyden family of unscaled methods, where θ_{k} = 1 corresponds to the well known DFP method. We propose simple conditions on these parameters that give rise to global convergence with inexact line searches, for convex objective functions. The q-superlinear convergence is achieved if further restrictions on the scaling parameter are introduced. These convergence results are an extension of the known results for the unscaled methods. Because the scaling parameter is heavily restricted, we consider a subclass of SS methods which satisfies the required conditions. Although convergence for the unscaled methods with θ_{k}≥1 is still an open question, we show that the global and superlinear convergence for SS methods is possible and present, in particular, a new SS-DFP method.

Original language | English |
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Pages (from-to) | 191-203 |

Number of pages | 13 |

Journal | Computational Optimization and Applications |

Volume | 9 |

Issue number | 2 |

Publication status | Published - Feb 1998 |

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### ASJC Scopus subject areas

- Management Science and Operations Research
- Applied Mathematics
- Computational Mathematics
- Control and Optimization

### Cite this

**Global and superlinear convergence of a restricted class of self-scaling methods with inexact line searches, for convex functions.** / Al-Baali, M.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Global and superlinear convergence of a restricted class of self-scaling methods with inexact line searches, for convex functions

AU - Al-Baali, M.

PY - 1998/2

Y1 - 1998/2

N2 - This paper studies the convergence properties of algorithms belonging to the class of self-scaling (SS) quasi-Newton methods for unconstrained optimization. This class depends on two parameters, say θk and τk, for which the choice τk = 1 gives the Broyden family of unscaled methods, where θk = 1 corresponds to the well known DFP method. We propose simple conditions on these parameters that give rise to global convergence with inexact line searches, for convex objective functions. The q-superlinear convergence is achieved if further restrictions on the scaling parameter are introduced. These convergence results are an extension of the known results for the unscaled methods. Because the scaling parameter is heavily restricted, we consider a subclass of SS methods which satisfies the required conditions. Although convergence for the unscaled methods with θk≥1 is still an open question, we show that the global and superlinear convergence for SS methods is possible and present, in particular, a new SS-DFP method.

AB - This paper studies the convergence properties of algorithms belonging to the class of self-scaling (SS) quasi-Newton methods for unconstrained optimization. This class depends on two parameters, say θk and τk, for which the choice τk = 1 gives the Broyden family of unscaled methods, where θk = 1 corresponds to the well known DFP method. We propose simple conditions on these parameters that give rise to global convergence with inexact line searches, for convex objective functions. The q-superlinear convergence is achieved if further restrictions on the scaling parameter are introduced. These convergence results are an extension of the known results for the unscaled methods. Because the scaling parameter is heavily restricted, we consider a subclass of SS methods which satisfies the required conditions. Although convergence for the unscaled methods with θk≥1 is still an open question, we show that the global and superlinear convergence for SS methods is possible and present, in particular, a new SS-DFP method.

UR - http://www.scopus.com/inward/record.url?scp=0031999308&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031999308&partnerID=8YFLogxK

M3 - Article

VL - 9

SP - 191

EP - 203

JO - Computational Optimization and Applications

JF - Computational Optimization and Applications

SN - 0926-6003

IS - 2

ER -