A Parallel Algorithm for Long Horizon Optimal Control Problems Using the Mixed Coordination Method

Abstract

In this paper, we present a new approach for solving long horizon, discrete-time optimal control problems by using the mixed coordination method. The idea is to decompose a long horizon problem into subproblems along the time axis. The requirement that the initial state of a subproblem equals the terminal state of the preceding subproblem is relaxed by using Lagrange multipliers. The Lagrange multipliers and the initial state of each subproblem are then selected as high level variables. The equivalence of the two-level formulation and the original problem is proved for both the convex and nonconvex cases. Under the two-level formulation, the low level subproblems are optimal control problems with a shorter time horizon, and are solved in parallel by using the extended Differential Dynamic Programming (DDP). An efficient way for finding the gradient and Hessian of a low level objective function with respect to high level variables is developed. The high level problem, on the other hand, is solved by using the Modified Newton's Method. An effective procedure is developed to select initial values of multipliers based on the given initial trajectory. Since both the DDP and the Modified Newton's Method have fast convergence rate and are compatible with each other, the method is very efficient. Furthermore, because of the specific way in selecting high level variables, the method can convexify the high level problem while maintain the separability of an originally nonconvex problem.