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128 Gradient Optimization
20E3
F(QI)
10E3
OL-
...L__-'-_--'---""'_"""---..JL-_--'
1.25 1.5 1.75 2.0
Figure 5.12.
x_(7,3)T.
A plot of the function in Equation (5.1) in the (-l,-l)T direction from point
But (5.40) is just a function of the single variable a during the linear search.
To find the minimum in the search direction s, it is necessary to differentiate
F(a) in (5.40) with respect to a and to equate this to zero. The result is:
- (Si{g'
. T .'
(s') As'
(5.41)
where gi = g(x i ). This provides an exact value of the linear search scalar a to a
minimum from any point Xl on a quadratic surface in an arbitrary search
direction Si.
Example 5,1. Consider the central sample quadratic function (5.8) shown in
Figure 5.5. Suppose that a linear search is to begin in the negative gradient
direction from the point xi=(lO, lol. i To find ai and then the minimum point
X + l in that direction, (5.41) will require g' and the Si= _gi arbitrarily chosen
for this example. The quadratic function gradient vector was defined generally
by (5.13) and, for this example, by (5.15) and (5.16). Using Program A5-2 for
Xl = 10 and x 2 = 10, find g~(100,28)T. Appendix Program A5-4 solves realvariable
inner products as in the numerator of (5.41) and conjugate forms as
in the denominator of (5.41). The significance of the latter will be discussed in
Section 5.2.4. As previously noted, the matrix A is described for this example
function by a = b = 26 and k = - 10. The sequence 26, - 10, and 26 is entered
into an HP-67 calculator with program A5-4 running, and key B is pressed to
input thes;;"-data. The sequence -100, -28, -100, and -28 is input using
key A. The (5.41) numerator inner product is found using key D (10784), and
the (5.41) quadratic form is found using key E (224384). Then (5.41) yields