download searchable PDF of Circuit Design book - IEEE Global ...

Conjugate Gradient Search 127
choices will be mentioned, especially the relaxation method (varying each
variable in turn) and the steepest descent strategy, which selects the steepest
slope direction at each turning point. After considering several more important
properties of quadratic functions, conjugate vectors and conjugate direction
search methods will be defined. Finally, the Fletcher-Reeves conjugate
gradient search direction formula will be discussed with examples.
5.2.1. Linear Search. At the point in variable space (x) where a new search
direction (s) has been selected, some clear description of the next linear search
must be available. The common notation is
i= J,2, ... , (5.39)
where the superscript denotes that this is the ith linear search or iteration. The
search parameter is the single variable ,,;, which determines the distance of
x;+' from x;. For well-posed problems, there will be some optimum a; that
determines the lowest value of F(x) in the s; direction; in that sense, the linear
search is concerned with a function of only a single variable, namely F(oJ
Consider the nonquadratic surface over two-variable space previously introduced
in (5.1)-(5.5). Suppose that the starting point X=(7,3)T is selected,
where the gradient turns out to have the value g=(72080, 159976)T. Since the
gradient is the set of coordinates describing the direction of maximum
function increase.. a reasonable choice for a linear search might be to the
"southwest," i.e., in search direction s=(-I, -I? Table 5.2 summarizes a
set of moves in this direction according to (5.39) using Program A5-1. A graph
of this function of ,,; is shown in Figure 5.12. A new turning point is in the
vicinity of x=(5.25, 1.25?, and a new search direction must be obtained,
preferably by a more effective procedure than illustrated. Some simple alternatives
are considered in the next section. A particular linear search strategy will
be considered in some detail in Section 5.3.
Before continuing, examples based on quadratic functions can be implemented
much easier if the linear search parameter" is obtained in closed form
for these cases. Consider the standard quadratic function defined by (5.6) and
write F(x;+ ') by substituting (5.39):
(5.40)
Table 5.2. Searching to the Southwest on (5.1) From XI =7, x2=3 Using (5.39)
"
as, as, x, x, F 'I',F 'I',F
0 0 0 7 3 160,016 72,080 159,976
1 -1 -I 6 2 20,740 20,704 43,212
2 -2 -2 5 1 1,972 -4,824 - 8,764
1.5 . _1.5 -1.5 5.5 1.5 1,740 5,170 10,400
1.75 - 1.75 -7.75 5.25 1.25 38 -462 -728