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5 is not a characteristic of gradient methods, as will be demonstrated in the next example. 5. 0. 3 which shows contours of constant values of L. Obviously the problem is difficult because of the long narrow valley, as the gradient procedure will tend to choose values for ak alternately on one side and then on the other. t This problem was originally suggested by Rosenbrock (1 960). 22 2. 64) T h e convergence is shown in Table 11. It is very slow. I n fact the optimal solution ayrlt = I , Qt = I is never reached.
First, however, the conjugate direction process will be defined. Consider a quadratic function L(a) that is assumed to have a matrix of second partial derivatives which is negative definite. 75) T h u s the vectors pi are linearly independent and form a basis in the p-dimensional Euclidian space. Hence, any arbitrary p-dimensional vector 01 may be defined in terms of the conjugate set I) a! 76) i=l where the ci are unknown scalar parameters that must be determined. For example, suppose L(a) has the form L(a) = Qa!
3. If Jk+l > J k , set k = k + 1 and go to step 2. 4. If Jk+l < J k , set ~k = ek/2 and return to step 2. 5. Repeat steps 2, 3, and 4 until the desired accuracy is attained. 6. 73) 28 2. 7 Examples T h e two examples used to illustrate the gradient method are repeated so that a comparison of the convergence properties may be made. 6. Maximization of the lift L where L = sin2 01 cos a and the gradient L, is La = 2 sin OL cos2 OL - sin3 a. T h e second derivative L,, is L,, = 2 C O S ~OL - 7 sin2 01 cos 01.
The Computation and Theory of Optimal Control by Dyer