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.
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}