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Application of the Alternating Direction Method of Multipliers to Control Constrained Parabolic Optimal Control Problems and Beyond

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Annals of Applied Mathematics 2022年第2期38卷 115-158页

作者： Roland Glowinski Yongcun Song Xiaoming Yuan Hangrui Yue Department of Mathematics University of Houston44800 Calhoun RoadHoustonTX 77204USA Department of Mathematics Hong Kong Baptist UniversityKowloonHong KongChina Chair for Dynamics Control and NumericsAlerander uon Humboldt-ProfessorshipDepartment of Data ScienceFriedrich-Alexander-Universitat Erlangen-Nurnberg91058 ErlangenGermany Department of Mathematics The University of Hong KongHong KongChina School of Mathematical Sciences Nankai UniversityTianjin 300071China

Control constrained parabolic optimal control problems are generally challenging,from either theoretical analysis or algorithmic design ***,the well-known alternating direction method of multipliers(ADMM)can be directly applied to such *** 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 *** each iteration of the ADMM,the main computation is for solving an unconstrained parabolic optimal control *** of its inevitably high dimensionality after space-time discretization,the parabolicoptimal 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 *** 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 *** 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 *** 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 *** of this ADMM implementation is promisingly validated by some numerical *** 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.

关键词： Parabolic optimal control problem control constraint alternating direction method of multipliers inexactness criterion nested iteration convergence analysis

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