Block-structured quadratic programming for the direct multiple shooting method for optimal control

Abstract

Abstract. In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, e.g., from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved using sequential quadratic programming methods. We review the classical condensing algorithm that preprocesses the large but sparse quadratic programs to obtain small but dense ones. We show that this approach leaves room for improvement when applied in conjunction with outer convexification. To this end, we present a new complementary condensing algorithm for quadratic programs with many controls. This algorithm is based on a hybrid null-space range-space approach to exploit the block sparse structure of the quadratic programs that is due to direct multiple shooting. An assessment of the theoretical run time complexity reveals significant advantages of the proposed algorithm. We give a detailed account on the required number of floating point operations, depending on the process dimensions. Finally we demonstrate the merit of the new complementary condensing approach by comparing the behavior of both methods for a vehicle control problem in which the integer gear decision is convexified.

A factorization with update procedures for a KKT matrix arising in direct optimal control

“…We employed this method to solve a partially convexified and relaxed mixed-integer optimal control problem. The results reported in [34] indicate a significant reduction of the computational effort compared to the classical condensing method. The described linear algebra techniques however lack exploitation of simple bounds and matrix update procedures are not provided.…”

“…We propose for the first time update procedures to this factorization for all kinds of active set changes that may occur when solving a QP with the presented block structure. These updates improve the run time complexity to O(mn 2 ) as compared to O(mn 3 ) in [34]. They are based on well established results by Gill and Golub [23] for Cholesky and QR factorizations, and extend the techniques commonly used in dense null-space active set QP solvers, cf.…”

“…Improving over [34] we show how to efficiently exploit simple bounds on the QP's unknowns in this factorization. We propose for the first time update procedures to this factorization for all kinds of active set changes that may occur when solving a QP with the presented block structure.…”

“…[36]. We extend the run time comparisons of [34] to include a number of generic sparse or structure linear algebra codes, and study the performance gains obtained by the new matrix updates. Computational evidence obtained is promising such that the presented factorization and update procedures are intended to lay the foundation for a fast block structured primal and primal-dual active set QP code to be used in optimal control and model-predictive control.…”

“…This motivates our investigation of structured active set techniques. In [34] we examined a block structured factorization due to [40] and its applicability to a KKT matrix arising in an active set QP method for optimal control and model-predictive control. We employed this method to solve a partially convexified and relaxed mixed-integer optimal control problem.…”