Multigrid preconditioning of linear systems for semismooth Newton methods applied to optimization problems constrained by smoothing operators
For researchers solving large-scale optimization problems with box constraints, this provides an efficient preconditioning strategy, though it is incremental as it extends known multigrid techniques to constrained settings.
This paper develops multigrid preconditioners for linear systems in semismooth Newton methods applied to constrained optimization problems with smoothing operators. The preconditioner quality improves with mesh refinement, yielding iteration counts that decrease with mesh size, though the spectral distance is suboptimal.
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.