An SQP-based multiple shooting algorithm for large-scale PDE-constrained optimal control problems

Abstract

Multiple shooting methods for solving optimal control problems have been developed rapidly in the past decades and are widely considered a promising direction to speed up the optimization process. Here we propose and analyze a new multiple shooting algorithm based on a sequential quadratic programming (SQP) method that is suitable for optimal control problems governed by large-scale time-dependent partial-differential equations (PDEs). We investigate the structure of the KKT matrix and solve the large-scale KKT system by a preconditioned conjugate gradient algorithm. A simplified block Schur complement preconditioner is proposed, that allows for the parallelization of the method in the time domain. The proposed algorithm is first validated for an optimal control problem constrained by the Nagumo equation. The results indicate that considerable accelerations can be achieved for multiple shooting approaches with appropriate starting guesses and scaling of the matching conditions. We further apply the proposed algorithm to a two-dimensional velocity tracking problem governed by the Navier–Stokes equations. We find algorithmic speed-ups of up to 12 versus single shooting on up to 50 shooting windows. We also compare results with earlier work that uses an augmented Lagrangian algorithm instead of SQP, showing better performance of the SQP method for most of the cases.