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

C. Cartis; N. I. M. Gould; P. L. Toint

doi:10.1007/s10543-008-0206-8

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

C. Cartis, N. I. M. Gould, P. L. Toint

School of Mathematics

Research output: Contribution to journal › Article › peer-review

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||(q)(2) and a regularisation term ||x||(2)(p) for various p and q = 1, 2 is sought. In each case, one or 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.

Original language	English
Pages (from-to)	21-53
Number of pages	33
Journal	Bit numerical mathematics
Volume	49
Issue number	1
DOIs	https://doi.org/10.1007/s10543-008-0206-8
Publication status	Published - Mar 2009

Keywords / Materials (for Non-textual outputs)

Linear least-squares
Regularisation
Trust-region
Secular equation
CONJUGATE-GRADIENT METHOD
GLOBAL PERFORMANCE
REGULARIZATION
OPTIMIZATION
ALGORITHM
EQUATIONS
LSQR

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10.1007/s10543-008-0206-8

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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||(q)(2) and a regularisation term ||x||(2)(p) for various p and q = 1, 2 is sought. In each case, one or 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.",

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T1 - Trust-region and other regularisations of linear least-squares problems

AU - Cartis, C.

AU - Gould, N. I. M.

AU - Toint, P. L.

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N2 - 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||(q)(2) and a regularisation term ||x||(2)(p) for various p and q = 1, 2 is sought. In each case, one or 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.

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KW - Linear least-squares

KW - Regularisation

KW - Trust-region

KW - Secular equation

KW - CONJUGATE-GRADIENT METHOD

KW - GLOBAL PERFORMANCE

KW - REGULARIZATION

KW - OPTIMIZATION

KW - ALGORITHM

KW - EQUATIONS

KW - LSQR

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DO - 10.1007/s10543-008-0206-8

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EP - 53

JO - Bit numerical mathematics

JF - Bit numerical mathematics

IS - 1

ER -

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

Abstract

Keywords / Materials (for Non-textual outputs)

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