An ADMM numerical approach to linear parabolic state constrained optimal control problems

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.