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

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

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Article ID: iaor2013168
Volume: 49
Issue: 3
Start Page Number: 365
End Page Number: 379
Publication Date: Mar 2011
Journal: Journal of Global Optimization
Authors: , , ,
Keywords: quasi-Newton method, Hilbert space
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.

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