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

摘要

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