Multigrid preconditioning of linear systems for semismooth Newton methods applied to optimization problems constrained by smoothing operators

TL;DR

This paper develops multigrid preconditioners for linear systems in semismooth Newton methods applied to large-scale constrained optimization problems with smoothing operators, showing improved efficiency as mesh size decreases.

Contribution

It introduces a multigrid preconditioner tailored for constrained optimization problems, analyzing its spectral properties and demonstrating its effectiveness in reducing iteration counts.

Findings

Preconditioner quality improves as mesh size decreases.

Number of iterations decreases with mesh refinement.

Spectral distance is suboptimal but still effective.

Abstract

This article is concerned with the question of constructing effcient multigrid preconditioners for the linear systems arising when applying semismooth Newton methods to large-scale linear-quadratic optimization problems constrained by smoothing operators with box-constraints on the controls. It is shown that, for certain discretizations of the optimization problem, the linear systems to be solved at each semismooth Newton iteration reduce to inverting principal minors of the Hessian of the associated unconstrained problem. As in the case when box-constraints on the controls are absent, the multigrid preconditioner introduced here is shown to increase in quality as the mesh-size decreases, resulting in a number of iterations that decreases with mesh-size. However, unlike the unconstrained case, the spectral distance between the preconditioners and the Hessian is shown to be of suboptimal order in general.