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Multi-Secant Equations, Approximate Invariant Subspaces and Multigrid Optimization

Serge Gratton (gratton***at***cerfacs.fr)
Philippe L. Toint (philippe.toint***at***fundp.ac.be)

Abstract: 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. It is also shown that this type of information may often be extracted from the multigrid structure of discretized infinite dimensional problems. A new limited-memory BFGS 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.

Keywords: Unconstrained optimization, structure, multigrid, quasi-Newton methods

Category 1: Nonlinear Optimization (Unconstrained Optimization )

Category 2: Applications -- Science and Engineering (Optimization of Systems modeled by PDEs )

Citation: Report 07/11, Department of Mathematics, University of Namur -FUNDP, Namur, Belgium

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Entry Submitted: 11/13/2007
Entry Accepted: 11/13/2007
Entry Last Modified: 11/20/2007

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