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 boundx ≤ Δ is imposed on the size of the solution, and in which the least value of linear combinations of Ax-b₂^q and a regularisation term x₂^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.

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