Conjugate direction methods for optimal control

This correspondence extends two algorithms for unconstrained minimization in R
ⁿ
, Davidon's method and a projected gradient algorithm, to optimal control problems. Both require only the value and gradient of the functional being minimized; both find the current search direction by operating on the negative gradient with a dyadic operator; and both generate conjugate directions when applied to a quadratic functional. To compute the direction of search at iteration
i
, the Davidon algorithm requires that
2i + 2
functions, generated in past and current cycles, be stored. The projected gradient method requires only
i + 2
. Both decrease the value of the functional being minimized at each step. The storage demands will require that both methods be restarted periodically. However, recent computational results indicate that this may improve the rate of convergence.