Local Convergence Behaviour of Generalized Gauss-Newton Multiple Shooting, Single Shooting and Differential Dynamic Programming

TL;DR

This paper analyzes the local convergence rates of three classical optimal control methods—multiple shooting, single shooting, and differential dynamic programming—showing they all converge linearly under Gauss-Newton or generalized Gauss-Newton Hessian approximations.

Contribution

It demonstrates that these methods share the same linear convergence rate when using GGN Hessian approximations, clarifying their theoretical behavior in widely used implementations.

Findings

All three methods converge linearly with GGN approximations.

Convergence rates are identical for multiple shooting, single shooting, and DDP.

Results apply to practical implementations like iLQR.

Abstract

We revisit three classical numerical methods for solving unconstrained optimal control problems - multiple shooting, single shooting, and differential dynamic programming - and examine their local convergence behaviour. In particular, we show that all three methods converge with the same linear rate if a Gauss-Newton (GN), or more general a Generalized Gauss-Newton (GGN), Hessian approximation is used, which is the case in widely used implementations such as iLQR.