This paper presents a class of efficient Newton-type algorithms for solving the nonlinear programs (NLPs) arising from applying a direct collocation approach to continuous time optimal control. The idea is based on an implicit lifting technique including a condensing and expansion step, such that the structure of each subproblem corresponds to that of the multiple shooting method for direct optimal control. We establish the mathematical equivalence between the Newton iteration based on direct collocation and the proposed approach, and we discuss the computational advantages of a lifted collocation integrator. In addition, we investigate different inexact versions of the proposed scheme and study their convergence and computational properties. The presented algorithms are implemented as part of the open-source ACADO code generation software for embedded optimization. Their performance is illustrated on a benchmark case study of the optimal control for a chain of masses. Based on these results, the use of lifted collocation within direct multiple shooting allows for a computational speedup factor of about 10 compared to a standard collocation integrator and a factor in the range of 10 − 50 compared to direct collocation using a general-purpose sparse NLP solver.
published in Mathematical Programming Computation (May 2017)