We present a line search multigrid method for solving discretized versions of
general unconstrained infinite dimensional optimization problems. Introducing a
new condition to a backtracking line search procedure, the step generated from
the coarser levels is guaranteed to be a descent direction. This method is
globally convergent under fairly minimal requirements on the
minimization method used at all grid levels. In particular, our
method does not require that these minimizations, or
so-called ``smoothing'' steps, be taken at each grid level in
contrast with multigrid
algorithms for PDEs, which fail to converge without such steps.
Preliminary numerical experiments show that our method is promising.