  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.
