椭圆最优控制问题的修改交替方向乘子计算方法
A Modified Alternating Direction Method of Multipliers Algorithm for Elliptic Optimal Control Problem
摘要: 本文研究了具有混合控制–状态约束的椭圆最优控制问题。我们采用有限元法离散化连续型最优化问题，并使用由正定矩阵诱导的范数作为非精项，近似计算经典交替方向乘子算法中的子问题，由此提出了修改的交替方向乘子(mADMM)算法。此外我们证明了mADMM算法的全局收敛性和收敛速率。最后，利用数值例子来验证提出的理论结果。
Abstract: In this paper, an elliptic optimal control problem with mixed control-state constraint is considered. The finite element method is used to discretize the continuous optimization problem. And the norm induced by a positive definite matrix is used as the inexact term to approximate the subproblem in the classical alternating direction method of the multipliers algorithm. And a modified alternating direction method of multipliers (mADMM) algorithm is proposed. In addition, the global conver-gence and convergence rate of mADMM algorithm are proven. Finally, a numerical example demon-strates the theoretical results.
文章引用：林继桐. 椭圆最优控制问题的修改交替方向乘子计算方法[J]. 应用数学进展, 2022, 11(11): 7936-7945. https://doi.org/10.12677/AAM.2022.1111840

参考文献

[1] Abdulla, U.G., Bukshtynov, V. and Seif, S. (2021) Cancer Detection through Electrical Impedance Tomography and Optimal Control Theory: Theoretical and Computational Analysis. Mathematical Biosciences and Engineering, 18, 4834-4859. [Google Scholar] [CrossRef] [PubMed]
[2] Grass, D., Uecker, H. and Upmann, T. (2019) Optimal Fishery with Coastal Catch. Natural Resource Modeling, 32, e12235. [Google Scholar] [CrossRef] 
[3] Ciambella, J. and Tomassetti, G. (2020) A Form-Finding Strategy for Magneto-Elastic Actuators. International Journal of Non-Linear Mechanics, 119, Article ID: 103297. [Google Scholar] [CrossRef] 
[4] Casas, E. (1985) L^2 Estimates for the Finite Element Method for the Dirichlet Problem with Singular Data. Numerische Mathematik, 47, 627-632. [Google Scholar] [CrossRef] 
[5] Hinze, M. and Meyer, C. (2010) Variational Discretization of Lavrentiev-Regularized State Constrained Elliptic Optimal Control Problems. Computational Optimization and Applications, 46, 487-510. [Google Scholar] [CrossRef] 
[6] Ciarlet, P.G. (1978) The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam.
[7] Zhang, K., Li, J.S., Song, Y.C. and Wang, X.S. (2017) An Alternating Direction Method of Multipliers for Elliptic Equation Constrained Optimization Problem. Science China Mathematics, 60, 361-378. [Google Scholar] [CrossRef] 
[8] Song, X.L., Yu, B., Wang, Y.Y. and Zhang, X.P. (2018) An FE-Inexact Heterogeneous ADMM for Elliptic Optimal Control Problems with L^1-Control Cost. Journal of Systems Science and Complexity, 31, 1659-1697. [Google Scholar] [CrossRef] 
[9] Song, X.L. and Yu, B. (2017) A Two-Phase Strategy for Control Constrained Elliptic Optimal Control Problems. Numerical Linear Algebra with Applications, 25, e2138. [Google Scholar] [CrossRef] 
[10] Chen, Z.X., Song, X.L., Zhang, X.P. and Yu, B. (2019) A FE-ADMM Algorithm for Lavrentiev-Regularized State-Con- strained Elliptic Control Problem. ESAIM-Control Optimisation and Calculus of Varuations, 25, Article No. 5. [Google Scholar] [CrossRef] 
[11] Chen, X.T., Song, X.L., Chen, Z.X. and Yu, B. (2020) A Multi-Level ADMM Algorithm for Elliptic PDE-Constrained Optimization Problems. Computational and Applied Mathematics, 39, Article No. 331. [Google Scholar] [CrossRef] 
[12] He, B.S. and Yuan, X.M. (2015) On Non-Ergodic Convergence Rate of Douglas-Rachford Alternating Direction Method of Multipliers. Numerische Mathematik, 130,567-577. [Google Scholar] [CrossRef] 
[13] Chen, L., Sun, D.F. and Toh, K.-C. (2017) An Efficient Inexact Symmetric Gauss-Seidel Based Majorized ADMM for High-Dimensional Convex Composite Conic Programming. Mathematical Programming, 161, 237-270. [Google Scholar] [CrossRef] 

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