A Line Search Multigrid Method for Large-Scale Nonlinear Optimization

Abstract. We present a line search multigrid method for solving discretized versions of general unconstrained infinite dimensional optimization problems. At each iteration on each level, the algorithm computes either a “direct search” direction on the current level or a “recursive search” direction from coarser level models. Introducing a new condition that must be satisfied by a backtracking line search procedure, the “recursive search” direction is guaranteed to be a descent direction. Global convergence is proved under fairly minimal requirements on the minimization method used at all grid levels. Using a limited memory BFGS quasi-Newton method to produce the “direct search” direction, preliminary numerical experiments show that our line search multigrid approach is promising.