Optimal control problems arising from systems modeled by linear parabolic equations may be difficult for both theoretical analysis and algorithmic design. For the case where there are additional constraints on the state variables, restrictive regularity assumptions are usually required to guarantee the existence of the associated Lagrange multiplier and thus some regularization type methods such as the Moreau–Yosida and Lavrentiev methods have been discussed in the literature. In this article, we study the application of the alternating direction method of multipliers (ADMM) to linear parabolic state constrained optimal control problems, and propose an ADMM numerical approach. We prove the convergence of the ADMM algorithm without any existence or regularity assumption on the Lagrange multiplier, and estimate its worst-case convergence rate in both the ergodic and nonergodic senses. An important feature of the ADMM approach is that it decouples the state constraints and the parabolic optimal control problems inside each iteration. We show the efficiency of the ADMM approach by testing some control problems in two space dimensions.

中文翻译：

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线性抛物线状态约束最优控制问题的 ADMM 数值方法

由线性抛物线方程建模的系统产生的最优控制问题对于理论分析和算法设计来说可能是困难的。对于状态变量有额外约束的情况，通常需要限制性正则假设来保证相关拉格朗日乘子的存在，因此文献中已经讨论了一些正则化类型的方法，例如 Moreau-Yosida 和 Lavrentiev 方法。在本文中，我们研究了乘法器交替方向法 (ADMM) 在线性抛物线状态约束最优控制问题中的应用，并提出了一种 ADMM 数值方法。我们证明了 ADMM 算法的收敛性，在拉格朗日乘子上没有任何存在或规律性假设，并估计其在遍历和非遍历意义上的最坏情况收敛率。ADMM 方法的一个重要特征是它在每次迭代中解耦了状态约束和抛物线最优控制问题。我们通过测试二维空间中的一些控制问题来展示 ADMM 方法的效率。