# Comparison of Derivative Estimation Methods in Solving Optimal Control Problems Using Direct Collocation

## TL;DR

This paper compares four derivative estimation methods for solving large sparse nonlinear programming problems from optimal control, finding hyper-dual methods to be most efficient and easiest to implement.

## Contribution

It provides a comprehensive comparison of finite-difference, bicomplex-step, hyper-dual, and automatic differentiation methods in the context of direct collocation for optimal control.

## Key findings

- Hyper-dual method has lower total computation time.
- Bicomplex-step and hyper-dual are easier to implement.
- Automatic differentiation is less efficient than Taylor series-based methods.

## Abstract

A study is conducted to evaluate four derivative estimation methods when solving a large sparse nonlinear programming problem that arises from the approximation of an optimal control problem using a direct collocation method. In particular, the Taylor series-based finite-difference, bicomplex-step, and hyper-dual derivative estimation methods are evaluated and compared alongside a well known automatic differentiation method. The performance of each derivative estimation method is assessed based on the number of iterations, the computation time per iteration, and the total computation time required to solve the nonlinear programming problem. The efficiency of each of the four derivative estimation methods is compared by solving three benchmark optimal control problems. It is found that while central finite-differencing is typically more efficient per iteration than either the hyper-dual or bicomplex-step, the latter two methods have significantly lower overall computation times due to the fact that fewer iterations are required by the nonlinear programming problem when compared with central finite-differencing. Furthermore, while the bicomplex-step and hyper-dual methods are similar in performance, the hyper-dual method is significantly easier to implement. Moreover, the automatic differentiation method is found to be substantially less computationally efficient than any of the three Taylor series-based methods. The results of this study show that the hyper-dual method offers several benefits over the other three methods both in terms of computational efficiency and ease of implementation.