A multigrid approach to discretized optimization problems

Abstract

Many large optimization problems represent a family of models of varying size, corresponding to different discretizations. An example is optimal control problems where the solution is a function that is approximated by its values at finitely many points. We discuss optimization techniques suitable for nonlinear programs of this type, with an emphasis on algorithms that guarantee global convergence. The goal is to exploit the similar structure among the subproblems, using the solutions of smaller subproblems to accelerate the solution of larger, more refined subproblems.

On the convergence of the MG/OPT method

“…For example the truncated Newton scheme used in [5]. Given an initial approximation x 0 k to the solution of (1), the application of…”

“…This claim appears to be true as far as a line search along the coarse-grid correction is performed. Also in [5] it is reported that MG/OPT without line search diverges in some cases. Therefore line search appears to be necessary for convergence.…”

“…In some recent papers, Nash [5] and Lewis and Nash [4], propose a multigrid approach to optimization problems, called MG/OPT, which closely resembles the well known full approximation storage (FAS) scheme [1] and is similar to the nonlinear multigrid (NMGM) methods discussed in [2]. One novelty of the MG/OPT approach is the extension of the multigrid strategy to optimization problems.…”

“…In [4,5] it is emphasized that under appropriate assumptions, the multigrid coarse-grid correction provides a descent direction and, therefore, combining this fact with a line search procedure and a minimizing 'smoothing' iteration, a globally convergent algorithm is obtained. Numerical experiments, e.g.…”

“…Numerical experiments, e.g. [5], demonstrate that MG/OPT greatly improves the efficiency of the underlying optimization scheme used as 'smoother', suggesting that the MG/OPT scheme may be beneficial in combination with well known optimization algorithms. This claim appears to be true as far as a line search along the coarse-grid correction is performed.…”