Cause of poor convergence of first-order IPOPT on the coupled Rosenbrock problem

Determine the underlying cause of the unexpectedly poor convergence rate of IPOPT when configured without exact second derivatives (first-order IPOPT) on the coupled multidimensional Rosenbrock optimization problem whose objective sums adjacent Rosenbrock terms; specifically, explain why this behavior occurs even for low-dimensional instances.

Background

In the scalability paper on multidimensional Rosenbrock problems, the authors evaluated multiple optimizers, including IPOPT in both first- and second-order configurations. The coupled variant is non-separable and typically more challenging, with steep curvatures and a local minimum that can trap some methods.

While second-order IPOPT consistently converged and showed robustness, the authors observed that first-order IPOPT (i.e., IPOPT run without exact Hessians) exhibited an unexpectedly poor convergence rate even for low-dimensional instances of the coupled Rosenbrock problem, and they noted that the cause of this behavior is unclear.