An Inexact Alternating Direction Method of Multipliers for Constrained Parabolic Optimal Distributed Control Problems

TL;DR

This paper introduces an inexact alternating direction method of multipliers tailored for constrained parabolic optimal control problems, effectively balancing computational efficiency and convergence guarantees.

Contribution

It develops a novel inexact algorithmic framework that decouples control constraints, enabling efficient solutions with proven convergence for complex parabolic control problems.

Findings

Proven global convergence of the inexact method.

Achieved linear convergence rate under mild conditions.

Numerical experiments validate the method's effectiveness.

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.