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A multigrid method for constrained optimal control problems

: Engel, M.; Griebel, M.


Journal of computational and applied mathematics 235 (2011), Nr.15, S.4368-4388
ISSN: 0377-0427
ISSN: 0771-050X
Fraunhofer SCAI ()

We consider the fast and efficient numerical solution of linear-quadratic optimal control problems with additional constraints on the control. Discretization of the first-order conditions leads to an indefinite linear system of saddle point type with additional complementarity conditions due to the control constraints. The complementarity conditions are treated by a primal-dual active set strategy that serves as outer iteration. At each iteration step, a KKT system has to be solved. Here, we develop amultigrid method for its fast solution. To this end, we use a smoother which is based on an inexact constraint preconditioner.

We present numerical results which show that the proposed multigrid method possesses convergence rates of the same order as for the underlying (elliptic) PDE problem. Furthermore, when combined with a nested iteration, the solver is of optimal complexity and achieves the solution of the optimization problem at only a small multiple of the cost for the PDE solution.