Global Convergence of a Line-Search Filter Differential Dynamic Programming Method

2026-05-31

AI Summary

This work addresses the lack of global convergence guarantees in existing Differential Dynamic Programming (DDP) algorithms for optimal control problems with nonlinear state and control constraints. We propose FilterDDP, a novel algorithm that integrates a filter-based line search mechanism into the DDP framework. Instead of conventional damped Newton steps, FilterDDP generates search directions via backward recursion and trial points through forward simulation, thereby satisfying both dynamics and nonlinear constraints while ensuring iterative convergence. We provide the first rigorous proof of global convergence for this backward–forward procedure on a class of constrained optimal control problems and establish its theoretical equivalence to filter methods. This result offers a new solution paradigm that combines theoretical rigor with practical effectiveness for constrained optimal control.

Abstract

In this article, we establish the global convergence properties of the FilterDDP algorithm, which extends the discrete-time differential dynamic programming (DDP) algorithm of Mayne and Jacobson [\emph{International Journal of Control}, 3, (1966), pp. 85-95] to handle nonlinear constraints over states and controls, in addition to the dynamics. FilterDDP adopts a line-search filter procedure for step acceptance. However, instead of a damped Newton step applied in the general nonlinear programming setting, the computation of a trial point involves applying a backward recursion and a forward simulation. We establish the global convergence of FilterDDP by showing that for a subset of constrained optimal control problems, the this backward-forward procedure satisfies the same properties as a Newton step for the purpose of establishing global convergence of a line-search filter method, following the analysis of Wächter and Biegler [\emph{SIAM Journal on Optimization}, 16 (2005), pp. 1-31].

Problem

• differential dynamic programming
• global convergence
• nonlinear constraints
• optimal control
• line-search filter

Innovation

• Differential Dynamic Programming
• Filter Method
• Global Convergence
• Nonlinear Constraints
• Line-Search