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

In this contribution, we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, for example, 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 structured quadratic programs (QPs) 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 QPs with many controls. This algorithm is based on a hybrid null-space range-space approach to exploit the block structure of the QPs 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 behaviour of both methods for a vehicle control problem in which the integer gear decision is convexified.