Gradient Evaluation for Feasible Path Approach

When only the control is discretized as described in section 5.7, integration of the model equations is required to evaluate the performance index and to obtain the gradients. The evaluation of such gradients will consume a significant part of the total computational time needed to solve the optimisation problem. 

A number of methods can be found in the literature to evaluate the gradients that are required for the NLP problems  

The adjoint system approach requires integration of the model equations forward in time before integration of the adjoint system equations backward. 

On the other hand, the trajectory sensitivity equations method requires simultaneous integration of a greater number of equations than the adjoint system approach. However, it is more stable than the adjoint system approach due to the requirement of forward integration only. It is usually preferred in the area of parameter estimation and sensitivity (Kalogerakis and Luus, 1983 Caracotsios and 

Stewart, 1985). Advent of high-speed computers has improved significantly the performance of the finite difference and the trajectory sensitivity approaches. 

---