Factors Affecting the Performance of Optimization-based Multigrid Methods

Summary

Many large nonlinear optimization problems are based upon discretizations of underlying continuous functions. Optimization-based multigrid methods are designed to solve such discretized problems efficiently by taking explicit advantage of the family of discretizations. The methods are generalizations of more traditional multigrid methods for solving partial differential equations. The goal of this paper is to clarify the factors that affect the performance of an optimization-based multigrid method. There are five main factors involved: (1) global convergence, (2) local convergence, (3) role of the underlying optimization method, (4) role of the multigrid recursion, and (5) properties of the optimization model. We discuss all five of these issues, and illustrate our analysis with computational examples. Optimization-based multigrid methods are an intermediate tool between general-purpose optimization software and customized software. Because discretized optimization problems arise in so many practical settings we think that they could become a valuable tool for engineering design.

This research was supported by the National Aeronautics and Space Administration under NASA Grant NCC-1-02029, and by the National Science Foundation under grant DMS-0215444.