An Augmented Lagrangian Method for State Constrained Linear Parabolic Optimal Control Problems

Hailing Wang (), Changjun Yu () and Yongcun Song ()
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Hailing Wang: Shanghai University
Changjun Yu: Shanghai University
Yongcun Song: Friedrich-Alexander-Universität Erlangen-Nürnberg

Journal of Optimization Theory and Applications, 2024, vol. 203, issue 1, No 9, 196-226

Abstract: Abstract In this paper, we consider a class of state constrained linear parabolic optimal control problems. Instead of treating the inequality state constraints directly, we reformulate the problem as an equality-constrained optimization problem, and then apply the augmented Lagrangian method (ALM) to solve it. We prove the convergence of the ALM without any existence or regularity assumptions on the corresponding Lagrange multipliers, which is an essential complement to the classical theoretical results for the ALM because restrictive regularity assumptions are usually required to guarantee the existence of the Lagrange multipliers associated with the state constraints. In addition, under an appropriate choice of penalty parameter sequence, we can obtain a super-linear non-ergodic convergence rate for the ALM. Computationally, we apply a semi-smooth Newton (SSN) method to solve the ALM subproblems and design an efficient preconditioned conjugate gradient method for solving the Newton systems. Some numerical results are given to illustrate the effectiveness and efficiency of our algorithm.

Keywords: Optimal control problem; State constraints; Augmented Lagrangian method; Convergence analysis; Convergence rate; 49M41; 35Q90; 35Q93; 65K05; 90C25 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02494-3

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