### Technical Report ERGO 08-001

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

*Coralia Cartis, Nicholas I. M. Gould and Philippe L. Toint*

##### 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 problems, Tikhonov regularisation, ridge regression, trust-region method, Newton's method, secular equation

##### Download:

ERGO-08-001.pdf

##### History:

Written: 3 February 2008

##### Status:

Published in *BIT*.