By R. Karp
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I t can be seen that there are significant differences between the two trajectories although the final states are similar. T h e first performance index J1 leads to a far more economical fuel consumption although the majority of the trajectory is a long way off the nominal. T h e second, J 2 , leads to a trajectory that follows the nominal far more closely although much more control effort is needed. Of course, the terminal state may not be the only consideration when a trajectory is chosen. On-board experiments may dictate other factors.
This example was suggested by Rosenbrock (1960) to illustrate an optimization technique that avoids the explicit evaluation of the gradient L, . Rosenbrock, H. H. (1960). , Vol. 3, p. 175. 8. Over the last few years several papers have been published that deal exclusively with conjugate direction methods. In particular, the paper by Kelley and Myers (1967) effects a comparison of several versions of the procedure. Beckman, F. S. (1960). “The Solution of Linear Equations by the Conjugate Gradient Method,” Muthematical Methods for Digital Computers (A.
J. and Myers, G. E. (1967). ” Presented at the 18th Congress of the International Astronautical Federation, Belgrade, Yugoslavia. Myers, G. E. (1968). “Properties of the Conjugate Gradient and Davidon Methods,” J . , Vol. 2, No. 4, p. 209. Wilde and Beightler (1967) provide a wealth of material and many references to the techniques of linear and nonlinear programming which have not been discussed here. These techniques are particularly useful for optimization problems characterized by an abundance of inequality constraints.
Complexity of Computation by R. Karp