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

Authors:
Boubakeur Benahmed;Hocine Mokhtar-Kharroubi;Bruno Malafosse;Adnan Yassine
Affiliations:
Département de Mathématiques et Informatique, ENSET d'Oran, El m'naouer, Algérie;Département de Mathématiques, Faculté des sciences, Université d'Oran, El m'naouer, Algérie;Laboratoire de Mathématiques Appliquées du Havre, Université du Havre, Le Havre Cedex, France 76058;Laboratoire de Mathématiques Appliquées du Havre, Université du Havre, Le Havre Cedex, France 76058
Venue:
Journal of Global Optimization
Year:
2011

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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.