The Leverage Space Model 313
independent variable(s) make the answer of this equation equal to 0 (these
are theroots). There are traditional root-finding techniques, such as the
Newton-Raphesonmethod, to do this.
It would seem that root finding is related to mathematical optimiza-
tion in that the first derivative of an optimized function (i.e., extremum
located) will equal 0. Thus, you would assume that traditional root-finding
techniques, such as the Newton-Rapheson method, could be used to solve
optimization problems (careful to use what is regarded as an optimization
technique to solve for the roots of an equation can lead to a Pandora’s box
of problems).
However, our discussion will concern only optimization techniques and
not root finding techniques per se. The single best source for a listing of
these techniques isNumerical Recipesand much of the following section
on optimization techniques is referenced therefrom.^3
Optimization Techniques
Mathematical optimization, in short, can be described as follows: You have a
function (we call itG), the objective function, which depends on one or more
independent variables (which we callfl...fn). You want to find the value(s)
of the independent variable(s) that results in a minimum (or sometimes,
as in our case, a maximum) of the objective function. Maximization or
minimization is essentially the same thing (that is one person’sGis another
person’s−G).
In the crudest case, you can optimize as follows: Take every combina-
tion of parameters, run them through the objective function, and see which
produce the best results. For example, suppose we want to find the optimal
ffor two coins tossed simultaneously, and we want the answer to be precise
to .01. We could, therefore, test Coin 1 at the 0.0 level, while testing Coin 2
at the 0.01 level, then .01, .02, and proceed until we have tested Coin 2 at the
1.0 level. Then, we could go back and test with Coin 1 at the .01 level, and
cycle Coin 2 through all of its possible values while holding Coin 1 at the
.01 level. We proceed until both levels are at their maximum, that is, both
values equal 1.0. Since each variable in this case has 101 possible values (0
through 1.0 by .01 inclusive), there are 101101 combinations which must
be tried, or 10,201 times the objective function must be evaluated.
We could, if we wanted, demand precision greater than .01. Suppose
we wanted precision to the .001 level. Then we would have 1,0011,001
(^3) William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling,Nu-
merical Recipes: The Art of Scientific Computing, New York: Cambridge University
Press, 1986.