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

Coralia Cartis, Nick Gould, Philippe Toint

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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 || and a regularisation term ||x|| 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 G ALAHAD optimization library. © 2009 Springer Science + Business Media B.V.
Original languageEnglish
Pages (from-to)21-53
Number of pages33
JournalBIT Numerical Mathematics
Issue number1
Publication statusPublished - 1 Mar 2009


  • algorithms
  • least-squares
  • Regularisation


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