Multigrid Preconditioning of Linear Systems for Interior Point Methods Applied to a Class of Box-constrained Optimal Control Problems

In this article we construct and analyze multigrid preconditioners for discretizations of operators of the form Dλ + K K, where Dλ is the multiplication with a relatively smooth function λ > 0 and K is a compact linear operator. These systems arise when applying interior point methods to the minimization problem minu 1 2 (||Ku − f || + β||u||) with box-constraints u 6 u 6 u on the controls. The presented preconditioning technique is closely related to the one developed by Drăgănescu and Dupont in [13] for the associated unconstrained problem, and is intended for large-scale problems. As in [13], the quality of the resulting preconditioners is shown to increase as h ↓ 0, but decreases as the smoothness of λ declines. We test this algorithm first on a Tikhonovregularized backward parabolic equation with box-constraints on the control, and then on a standard elliptic-constrained optimization problem. In both cases it is shown that the number of linear iterations per optimization step, as well as the total number of fine-scale matrix-vector multiplications is decreasing with increasing resolution, thus showing the method to be potentially very efficient for truly large-scale problems.