Numerical solutions for optimal power generation in a pumping power cycle for Airborne Wind Energy is presented. The problem is essentially an infinite horizon optimal control problem which is then attempted to solve by three methods, finite time horizon approximation, search for the optimal periodic limit cycle and time transformation of the infinite horizon to a finite half-open space. We find the optimal periodic limit cycle formulation to be the most computationally efficient (least time consuming) of the three formulations. While the finite time approximation yields a solution close to the optimal limit cycle, it suffers due to nonlinearities in the system failing to give a consistent limit cycle and also requires large computational time. The time transformation method fails to solve the problem at all as it requires problems to have solutions, exponentially converging to a steady state which is not the case for our system in the time domain.