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Parabolic optimal control problems with control constraints are generallychallenging, from either theoretical analysis or algorithmic designperspectives. Conceptually, the well-known alternating direction method ofmultipliers (ADMM) can be directly applied to such a problem. An attractiveadvantage of this direct ADMM application is that the control constraint can beuntied from the parabolic PDE constraint; these two inherently differentconstraints thus can be treated individually in iterations. At each iterationof the ADMM, the main computation is for solving an unconstrained parabolicoptimal control problem. Because of its high dimensionality afterdiscretization, the unconstrained parabolic optimal control problem at eachiteration can be solved only inexactly by implementing certain numerical schemeinternally and thus a two-layer nested iterative scheme is required. It thenbecomes important to find an easily implementable and efficient inexactnesscriterion to execute the internal iterations, and to prove the overallconvergence rigorously for the resulting two-layer nested iterative scheme. Toimplement the ADMM efficiently, we propose an inexactness criterion that isindependent of the mesh size of the involved discretization, and it can beexecuted automatically with no need to set empirically perceived constantaccuracy a prior. The inexactness criterion turns out to allow us to solve theresulting unconstrained optimal control problems to medium or even low accuracyand thus saves computation significantly, yet convergence of the overalltwo-layer nested iterative scheme can be still guaranteed rigorously.Efficiency of this ADMM implementation is promisingly validated by preliminarynumerical results. Our methodology can also be extended to a range of optimalcontrol problems constrained by other linear PDEs such as elliptic equationsand hyperbolic equations.

Implementation of the ADMM to Parabolic Optimal Control Problems with Control Constraints and Beyond