Using approximate secant equations in limited memory methods
          for multilevel unconstrained optimization

       Serge Gratton, Vincent Malmedy, Philippe L. Toint
                  Report TR09-18, November 2009

The properties of multilevel optimization problems defined on a
hierarchy of discretization grids can be used to define approximate secant
equations, which describe the second-order behaviour of the objective
function.  Following earlier work by Gratton and Toint (2009), we introduce a
quasi-Newton method (with a linesearch) and a nonlinear conjugate  gradient
method that both take advantage of this new second-order information.  We then
present numerical experiments with these methods and formulate recommendations
for their practical use.

Keywords: nonlinear optimization, multilevel problems,
quasi-Newton methods, nonlinear conjugate gradient methods,
limited-memory algorithms.