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Preconditioned Conjugate Gradient Method for Optimal Control Problems with Control and State Constraints

Optimality systems and their linearizations arising in optimal control of partial differential equations with pointwise control and (regularized) state constraints are considered. The preconditioned conjugate gradient (PCG) method in a nonstandard inner product is employed for their efficient solution. Preconditioned condition numbers are estimated for problems with pointwise control constraints, mixed control-state constraints, and of Moreau-Yosida penalty type. Numerical results for elliptic problems demonstrate the performance of the PCG iteration. Regularized state-constrained problems in three dimensions with more than 750,000 variables are solved.