By Harry Katzan
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Extra resources for APL Programming and Computer Techniques (VNR Computer Science Series)
37 g~. >0 S(x~ + ~d~). xj+l = xj + c~dj. Evaluate gj+l. sj = x j + l - - x j ; Yj = g j + l - - g j . i-1. ~--"5-Yj _ l ttj --1 (59) This is precisely the Hestenes-Stiefel conjugate gradient direction (1:35), but now it has been derived in a conceptually more general and satisfactol T way. 4) are identical for arbitrary smooth functions, which can be seen as follows: yf_ldj-1 = -gf_ldj_l, because g f dj_l -= O = g L ~ g ~ - , , because gT-'e~-~ = O. Thus (59) can be written in the equivalent Polak-Ribiere form @,gJ dj = - g j -4- g f _ l g j _ l d j - 1 9  (60) Under the foregoing assumptions, the Fletcher-Reeves choice, which we may also note is always nonnegative, is not equivalent to the other two.
However, should we seek to satisfy the equation IIw($*)]12 = A, the optimal solution would be dominated by the columns of Qm and we would not have a direction of descent. We must be content with the descent direction p($*) defined by (22). An alternative is to choose a large but finite value for A and proceed as in case c) (ii). Thus, in all four cases we have found a suitable search direction of descent. The heart of the procedure is an eigendecomposition of H and a zero-finding procedure to solve a nonlinear equation in one variable, namely, I l w ( ~ ) ] 1 2 - z~ = o, where w(A) is defined by (17).
Note that the objective function is nonnegative and replacing it by its square does not alter the optimal solution of (34)-(35). t. (R + uvT)s = (R + uvT)-T y. (36) This can be solved with the help of the standard Lagrangian optimality conditions. However, the special form of (36) also makes the problem amenable to solution by the following three-stage procedure that is especially tailored to it: 1. t. where the third constraint follows directly from the previous two. 2. t. (R + ~vT)T~ = V- (42) Denote the optimal solution of (41)-(42) by u(w) and v(w) and the corresponding optimal objective value by [u(~) T~(~)][~(~)~(~)].
APL Programming and Computer Techniques (VNR Computer Science Series) by Harry Katzan