Second-order constrained dynamic optimization

This paper provides an overview, analysis, and comparison of second-order dynamic optimization algorithms, that is, constrained differential dynamic programming (DDP) and sequential quadratic programming (SQP). Although a variety of these algorithms have been proposed and used successfully, there exists a gap in understanding the key differences and advantages, which we aim to provide in this work. For constrained DDP, we choose methods that incorporate nonlinear programming techniques to handle state and control constraints, including augmented Lagrangian (AL), interior point, primal-dual augmented Lagrangian (PDAL), and alternating direction method of multipliers (ADMM). Both DDP and SQP are provided in single- and multiple-shooting formulations, where constraints that arise from dynamics are encoded implicitly and explicitly, respectively. As a byproduct of the review, we propose a single-shooting PDAL DDP that has more favorable properties than the standard AL variant, such as the robustness to the growth of penalty parameters. We perform extensive numerical experiments on a variety of systems with increasing complexity to investigate the quality of the solutions, the levels of constraint violation, and the sensitivity of final solutions with respect to initialization, as well as targets. The results show that single-shooting PDAL DDP and multiple-shooting SQP are the most robust methods. For multiple-shooting formulation, both DDP and SQP can enjoy informed initial guesses, while the latter appears to be more advantageous in complex systems. It is also worth highlighting that DDP provides favorable computational complexity and feedback gains as a byproduct of optimization as is.