Second-Order Constrained Dynamic Optimization

This paper provides an overview, analysis, and comparison of second-order dynamic optimization algorithms, i.e., constrained Differential Dynamic Programming (DDP) and Sequential Quadratic Programming (SQP). Although a variety of these algorithms has 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 nolinear 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. Both DDP and SQP are provided in single- and multiple-shooting formulations, where constraints that arise from dynamics are encoded implicitly and explicitly, respectively. In addition to reviewing these methods, we propose a single-shooting PDAL DDP. As a byproduct of the review, we also propose a single-shooting PDAL DDP which is robust to the growth of penalty parameters and performs better than the normal AL variant. We perform extensive numerical experiments on a variety of systems with increasing complexity towards investigating the quality of the solutions, the levels of constraint violation, iterations for convergence, and the sensitivity of final solutions with respect to initialization. The results show that DDP often has the advantage of finding better local minima, while SQP tends to achieve better constraint satisfaction. 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.