A Fast Temporal Decomposition Procedure for Long-horizon Nonlinear Dynamic Programming

TL;DR

This paper introduces a rapid temporal decomposition method for long-horizon nonlinear dynamic programming, leveraging SQP with an augmented Lagrangian, achieving global convergence and efficient solutions with numerical validation.

Contribution

It develops a novel fast decomposition approach using a single Newton step within SQP, improving efficiency over existing Schwarz schemes for long-horizon problems.

Findings

Proven global convergence of the method.

Achieved local linear convergence rate matching recent schemes.

Numerical experiments demonstrate superior performance.

Abstract

We propose a fast temporal decomposition procedure for solving long-horizon nonlinear dynamic programs. The core of the procedure is sequential quadratic programming (SQP) that utilizes a differentiable exact augmented Lagrangian as the merit function. Within each SQP iteration, we approximately solve the Newton system using an overlapping temporal decomposition strategy. We show that the approximate search direction is still a descent direction of the augmented Lagrangian, provided the overlap size and penalty parameters are suitably chosen, which allows us to establish the global convergence. Moreover, we show that a unit stepsize is accepted locally for the approximate search direction, and further establish a uniform, local linear convergence over stages. This local convergence rate matches the rate of the recent Schwarz scheme by Na et al., 2022. However, the Schwarz scheme has to solve nonlinear subproblems to optimality in each iteration, while we only perform a single Newton step instead. Numerical experiments validate our theories and demonstrate the superiority of our method.