Approximate invariant subspaces and quasi-Newton optimization methods

Serge Gratton, P.L. Toint

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New approximate secant equations are shown to result from the knowledge of (problem dependent) invariant subspace information, which in turn suggests improvements in quasi-Newton methods for unconstrained minimization. A new limited-memory Broyden-Fletcher-Goldfarb-Shanno using approximate secant equations is then derived and its encouraging behaviour illustrated on a small collection of multilevel optimization examples. The smoothing properties of this algorithm are considered next, and automatic generation of approximate eigenvalue information demonstrated. The use of this information for improving algorithmic performance is finally investigated on the same multilevel examples. © 2010 Taylor & Francis.
Original languageEnglish
Pages (from-to)507-529
Number of pages23
JournalOptimization Methods and Software
Issue number4
Publication statusPublished - 1 Aug 2010


  • secant equation
  • multigrid techniques
  • quasi-Newton methods
  • Unconstrained optimization


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