Second-order convergence properties of trust-region methods using 
                    incomplete curvature information,
              with an application to multigrid optimization

              by S. Gratton, A. Sartenaer and Ph. L. Toint
                    Report 05/08  November 2005

Convergence properties of trust-region methods for unconstrained nonconvex
optimization is considered in the case where information on the objective
function's local curvature is incomplete, in the sense that it may be
restricted to a fixed set of ``test directions'' and may not be available at
every iteration.  It is shown that convergence to local ``weak'' minimizers
can still be obtained under some additional but algorithmically realistic
conditions. These theoretical results are then applied to recursive multigrid
trust-region methods, which suggests a new class of algorithms with guaranteed
second-order convergence properties.