Using Inexact Gradients in a Multilevel Optimization Algorithm

Many optimization algorithms require gradients of the model functions, but computing accurate gradients can be computationally expensive. We study the implications of using inexact gradients in the context of the multilevel optimization algorithm MGOpt. MGOpt recursively uses (typically cheaper) coarse models to obtain search directions for finer-level models. However, MGOpt requires the gradient on the … Read more

Convergence and Descent Properties for a Class of Multilevel Optimization Algorithms

I present a multilevel optimization approach (termed MG/Opt) for the solution of constrained optimization problems. The approach assumes that one has a hierarchy of models, ordered from fine to coarse, of an underlying optimization problem, and that one is interested in finding solutions at the finest level of detail. In this hierarchy of models calculations … Read more

Automatically Assessing the Performance of an Optimization-Based Multigrid Method

Many large nonlinear optimization problems are based upon discretizations of underlying function spaces. Optimization-based multigrid methods—that is, multigrid methods based on solving coarser versions of an optimization problem—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 … Read more

Model Problems for the Multigrid Optimization of Systems Governed by Differential Equations

We present a multigrid approach to the optimization of systems governed by differential equations. Such optimization problems have many applications, and are a broader class of problems than systems of equations. Using several model problems we give evidence (both theoretical and numerical) that a multigrid approach can often be successful in the setting of optimization. … Read more