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

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|| ≤ Δ 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.