报告题目：Implementation of the ADMM to Parabolic Optimal Control Problems with Control Constraints and Beyond

报 告 人：宋永存博士 香港大学

报告时间：2021年4月16日 14:00-15:00 （北京时间）

报告地点：腾讯会议 ID：702 109 768

会议密码：271828

点击入会或添加至会议列表：https://meeting.tencent.com/s/Nur3urkUTtaF

校内联系人：宋海明

报告摘要：Optimal control problems subject to both parabolic partial differential equation (PDE) constraints and additional constraints on the control variables 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 a problem. An attractive advantage of this direct ADMM application is that the additional constraint on the control variable can be untied from the parabolic PDE constraint; these two inherently different constraints thus can be treated individually in iterations. At each iteration of the ADMM, the main computation is for solving an optimal control problem with a parabolic PDE constraint while it is not interacted with the constraint on the control variable. Because of its inevitably high dimensionality after the space-time discretization, the parabolic optimal control problem at each iteration can be solved only inexactly by implementing certain numerical scheme internally and thus a two-layer nested iterative scheme is required. It then becomes important to find an easily implementable and efficient inexactness criterion to execute the internal iterations, and to prove the overall convergence rigorously for the resulting two-layer nested iterative scheme. To implement the ADMM efficiently, we propose an inexactness criterion that is independent of the mesh size of the involved discretization, and it can be executed automatically with no need to set empirically perceived constant accuracy a prior. The inexactness criterion turns out to allow us to solve the resulting optimal control problems with the only parabolic PDE constraints to medium or even low accuracy and thus saves computation significantly, yet convergence of the overall two-layer nested iterative scheme can be still guaranteed rigorously. Efficiency of this ADMM implementation is promisingly validated by preliminary numerical results. Our methodology can also be extended to a range of optimal control problems constrained by other linear PDEs such as elliptic equations, hyperbolic equations, convection-diffusion equations and fractional parabolic equations.

报告人简介：Yongcun Song is now a PhD candidate at the Department of Mathematics, The University of Hong Kong, after he graduated from Jilin University in 2016. His research area includes numerical optimization, operator splitting algorithms, and optimal control problems. He won the best paper prize in the 5th Graduate Forum of the Mathematical Programming Branch of Operational Research Society of China.