Multigrid Preconditioning of Linear Systems for Interior Point Methods Applied to a Class of Box-constrained Optimal Control Problems
We construct and analyze multigrid preconditioners for discretizations of operators of the form ${\mathcal D}_{\lambda}+{\mathcal K}^*{\mathcal K}$, where $D_{\lambda}$ is the multiplication with a relatively smooth function $\lambda>0$ and ${\mathcal K}$ is a compact linear operator. These systems arise when applying interior point methods to the minimization problem $\min_{u} \frac{1}{2}(|\!|{\mathcal K} u-f|\!|^2 +\beta|\!|u|\!|^2)$ with box-constraints $\underline{u}\leqslant u\leqslant\overline{u}$ on the controls. The presented preconditioning technique is closely related to the one developed by Draganescu and Dupont [Math. Comp., 77 (2008), pp. 2001–2038] for the associated unconstrained problem and is intended for large-scale problems. As in that work, the quality of the resulting preconditioners is shown to increase as $h\downarrow 0$, but it decreases as the smoothness of $\lambda$ declines. We test this algorithm on a Tikhonov-regularized backward parabolic equation with box-constraints on the control and 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 finest-scale matrix-vector multiplications, is decreasing with increasing resolution, thus showing the method to be potentially very efficient for truly large-scale problems.