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Control constrained parabolic optimal control problems are generally challenging, from either theoretical analysis or algorithmic design perspectives. Conceptually, the well-known alternating direction method of multipliers (ADMM) can be directly applied to such problems. An attractive advantage of this direct ADMM application is that the control constraints can be untied from the parabolic optimal control problem and thus can be treated individually in the iterations. At each iteration of the ADMM, the main computation is for solving an unconstrained parabolic optimal control subproblem. Because of its inevitably high dimensionality after space-time discretization, the parabolic optimal control subproblem at each iteration can be solved only inexactly by implementing certain numerical scheme internally and thus a two-layer nested iterative algorithm is required. It then becomes important to find an easily implementable and efficient inexactness criterion to perform the internal iterations, and to prove the overall convergence rigorously for the resulting two-layer nested iterative algorithm. To implement the ADMM efficiently, we propose an inexactness criterion that is independent of the mesh size of the involved discretization, and that can be performed automatically with no need to set empirically perceived constant accuracy a priori. The inexactness criterion turns out to allow us to solve the resulting parabolic optimal control subproblems to medium or even low accuracy and thus save computation significantly, yet convergence of the overall two-layer nested iterative algorithm can be still guaranteed rigorously. Efficiency of this ADMM implementation is promisingly validated by some numerical results. Our methodology can also be extended to a range of optimal control problems modeled by other linear PDEs such as elliptic equations, hyperbolic equations, convection-diffusion equations, and fractional parabolic equations.

}, issn = {}, doi = {https://doi.org/10.4208/aam.OA-2022-0004}, url = {http://global-sci.org/intro/article_detail/aam/20452.html} }Control constrained parabolic optimal control problems are generally challenging, from either theoretical analysis or algorithmic design perspectives. Conceptually, the well-known alternating direction method of multipliers (ADMM) can be directly applied to such problems. An attractive advantage of this direct ADMM application is that the control constraints can be untied from the parabolic optimal control problem and thus can be treated individually in the iterations. At each iteration of the ADMM, the main computation is for solving an unconstrained parabolic optimal control subproblem. Because of its inevitably high dimensionality after space-time discretization, the parabolic optimal control subproblem at each iteration can be solved only inexactly by implementing certain numerical scheme internally and thus a two-layer nested iterative algorithm is required. It then becomes important to find an easily implementable and efficient inexactness criterion to perform the internal iterations, and to prove the overall convergence rigorously for the resulting two-layer nested iterative algorithm. To implement the ADMM efficiently, we propose an inexactness criterion that is independent of the mesh size of the involved discretization, and that can be performed automatically with no need to set empirically perceived constant accuracy a priori. The inexactness criterion turns out to allow us to solve the resulting parabolic optimal control subproblems to medium or even low accuracy and thus save computation significantly, yet convergence of the overall two-layer nested iterative algorithm can be still guaranteed rigorously. Efficiency of this ADMM implementation is promisingly validated by some numerical results. Our methodology can also be extended to a range of optimal control problems modeled by other linear PDEs such as elliptic equations, hyperbolic equations, convection-diffusion equations, and fractional parabolic equations.

*Annals of Applied Mathematics*.

*38*(2). 115-158. doi:10.4208/aam.OA-2022-0004