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.