Multi-Secant Equations, Approximate Invariant
Subspaces and Multigrid Optimization
S. Gratton and Ph. L. Toint
Report 07/11, Dept of Mathematics, Univerisity of Namur
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.