Global Convergence of a Line-Search Filter Differential Dynamic Programming Method
This provides theoretical guarantees for a practical algorithm in constrained optimal control, addressing a gap in convergence theory for DDP-based methods.
The paper establishes global convergence properties for the FilterDDP algorithm, which extends differential dynamic programming to handle nonlinear constraints. It proves that for a subset of constrained optimal control problems, the algorithm's backward-forward recursion satisfies properties akin to a Newton step, ensuring global convergence via a line-search filter method.
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].