Journal of Global Optimization

, Volume 49, Issue 3, pp 365–379 | Cite as

Quasi-Newton methods in infinite-dimensional spaces and application to matrix equations

  • Boubakeur Benahmed
  • Hocine Mokhtar-Kharroubi
  • Bruno de Malafosse
  • Adnan Yassine
Article

Abstract

In the first part of this paper, we give a survey on convergence rates analysis of quasi-Newton methods in infinite Hilbert spaces for nonlinear equations. Then, in the second part we apply quasi-Newton methods in their Hilbert formulation to solve matrix equations. So, we prove, under natural assumptions, that quasi-Newton methods converge locally and superlinearly; the global convergence is also studied. For numerical calculations, we propose new formulations of these methods based on the matrix representation of the dyadic operator and the vectorization of matrices. Finally, we apply our results to algebraic Riccati equations.

Keywords

Nonlinear equations Optimization problems Quasi-Newton methods Rate of convergence Linear convergence Superlinear convergence Hilbert space Matrix equations Algebraic Riccati equation 

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Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  • Boubakeur Benahmed
    • 1
  • Hocine Mokhtar-Kharroubi
    • 2
  • Bruno de Malafosse
    • 3
  • Adnan Yassine
    • 3
  1. 1.Département de Mathématiques et InformatiqueENSET d’OranEl m’naouerAlgérie
  2. 2.Département de Mathématiques, Faculté des sciencesUniversité d’OranEl m’naouerAlgérie
  3. 3.Laboratoire de Mathématiques Appliquées du HavreUniversité du HavreLe Havre CedexFrance

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