An Inexact Alternating Direction Method of Multipliers for Constrained Parabolic Optimal Distributed Control Problems
Haiming Song, Jinda Yang, Yuran Yang, Jianhua Yuan
Abstract
Solving parabolic optimal control problems can be inherently challenging in the field of science and engineering, especially with constraints on the nonsmooth distributed control. Motivated by the extensive applicability of the alternating direction method of multipliers, in this paper we develop a novel inexact algorithmic framework for parabolic optimal distributed control problems with control constraints. By decoupling the control constraint and possible nonsmooth objective from the optimal control problem, our aim is to efficiently solve the subproblem constrained by the parabolic state equation, for which computing a sufficiently accurate numerical solution can be prohibitively expensive. Given this high computational cost, we consider that it may not always be justifiable to compute a highly accurate solution of the subproblem at every iteration. Hence, we propose an inexact strategy for solving the parabolic equation constrained subproblem. Under mild and flexible conditions on the parameters, we prove global convergence and a linear convergence rate for the resulting algorithmic framework. In practice, our inexact algorithmic framework is easily implementable with applicable nested iterations. Numerical experiments are performed on different cases, and the results demonstrate the validity and trustworthy performance of the proposed methods.