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Trust-region and other regularisations of linear least-squares problems

Coralia Cartis (coralia.cartis***at***ed.ac.uk)
Nicholas Gould (nick.gould***at***comlab.ox.ac.uk)
Philippe Toint (philippe.toint***at***fundp.ac.be)

Abstract: We consider methods for regularising the least-squares solution of the linear system Ax = b. In particular, we propose iterative methods for solving large problems in which a trust-region bound ||x|| <= Delta is imposed on the size of the solution, and in which the least value of linear combinations of ||Ax-b||_2^q and a regularisation term ||x||_2^p for various p and q =1,2 is sought. In each case, one of more ``secular'' equations are derived, and fast Newton-like solution procedures are suggested. The resulting algorithms are available as part of the GALAHAD optimization library.

Keywords: linear least-squares, regularisation, trust-region, secular equation

Category 1: Nonlinear Optimization (Nonlinear Systems and Least-Squares )

Citation: @article{CartGoulToin09a, author = {C. Cartis and N. I. M. Gould and Ph. L. Toint}, title = {Trust-region and other regularisation of linear least-squares problems}, journal = {BIT}, volume = 49, number = 1, pages = {21--53}, year = 2009}

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Entry Submitted: 02/03/2008
Entry Accepted: 02/03/2008
Entry Last Modified: 04/14/2011

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