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

C. Cartis, N. I. M. Gould, P. L. Toint

Research output: Contribution to journalArticlepeer-review

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||(q)(2) and a regularisation term ||x||(2)(p) 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 GALAHAD optimization library.

Original languageEnglish
Pages (from-to)21-53
Number of pages33
JournalBit numerical mathematics
Volume49
Issue number1
DOIs
Publication statusPublished - Mar 2009

Keywords

  • Linear least-squares
  • Regularisation
  • Trust-region
  • Secular equation
  • CONJUGATE-GRADIENT METHOD
  • GLOBAL PERFORMANCE
  • REGULARIZATION
  • OPTIMIZATION
  • ALGORITHM
  • EQUATIONS
  • LSQR

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