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| Application of the Alternating Direction Method of Multipliers to Control Constrained Parabolic Optimal Control Problems and Beyond |
| Roland Glowinski1,Yongcun Song,Xiaoming Yuan,Hangrui Yue |
| (Department of Mathematics, University of Houston, 44800 Calhoun Road,
Houston, TX 77204, USA; Department of Mathematics, Hong Kong Baptist University, Kowloon,
Hong Kong, China;Chair for Dynamics, Control and Numerics, Alexander von
Humboldt-Professorship, Department of Data Science,
Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, 91058 Erlangen,
Germany;Department of Mathematics, The University of Hong Kong, Hong Kong,
China;School of Mathematical Sciences, Nankai University, Tianjin 300071,
China) |
| DOI： |
| Abstract: |
| Control constrained parabolic optimal control problems are gener ally challenging, from either theoretical analysis or algorithmic design perspec tives. 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 individu ally 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 itera tions, 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 em pirically 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 guar anteed rigorously. Efficiency of this ADMM implementation is promisingly val idated 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. |
| Key words: Parabolic optimal control problem, control constraint, alternating direction
method of multipliers, inexactness criterion, nested iteration, convergence analysis. |