Integral equations: theory and numerical treatment
Integral equations: theory and numerical treatment
SIAM Journal on Control and Optimization
Semismooth Newton Methods for Operator Equations in Function Spaces
SIAM Journal on Optimization
The Primal-Dual Active Set Strategy as a Semismooth Newton Method
SIAM Journal on Optimization
Object-oriented software for quadratic programming
ACM Transactions on Mathematical Software (TOMS)
V-cycle convergence of some multigrid methods for ill-posed problems
Mathematics of Computation
A Multigrid Scheme for Elliptic Constrained Optimal Control Problems
Computational Optimization and Applications
Interior Point Methods in Function Space
SIAM Journal on Control and Optimization
SC '05 Proceedings of the 2005 ACM/IEEE conference on Supercomputing
Primal-dual interior-point methods for PDE-constrained optimization
Mathematical Programming: Series A and B
Multigrid Methods for PDE Optimization
SIAM Review
Multilevel Algorithms for Large-Scale Interior Point Methods
SIAM Journal on Scientific Computing
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In this article 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 $\lambda0$ 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 Drăgănescu 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.