314 THE HANDBOOK OF PORTFOLIO MATHEMATICS
combinations that we would need to try, or 1,002,001 times the objective
function would have to be calculated. If we were then to include three
variables rather than just two, and demand .001 precision this way, we
would then have to evaluate the objective function 1001 (^1001) 1001, or
1,003,003,001; that is, we would have to evaluate the objective function in
excess of one billion times. We are using only three variables and we are
demanding precision to only .001!
Although this crude case of optimizing has the advantage of being the
most robust of all optimization techniques, it is also has the dubious dis-
tinction of being too slow to apply to most problems.
Why not cycle through all variables for the first variable and get its
optimal; then cycle through all variables for the second while holding the
first at its optimal; get the second variable’s optimal, so that you now have
the optimal for the first two parameters; go find the optimal for the third
while setting the first two to their optimal, and so on, until you have solved
the problem?
The problem with this second approach is that it is often impossible to
find the optimum parameter set this way. Notice that by the time we get to the
third variable, the first two variables equal their optimum as if there were no
other variables. Thus, when the third variable is optimized, with the first two
variables set to their optimums, they interfere with the solution of the third
optimum. What you would end up with is not the optimum parameter set
of the three variables, but, rather, an optimum value for the first parameter,
an optimum for the second when the first is set to its optimum, an optimum
for the third when the first is set to its optimum, and the second set to a
suboptimum, but optimum given the interference of the first, and so on.
It may be possible to keep cycling through the variables and eventually
resolve to the optimum parameter set, but with more than three variables,
it becomes more and more lengthy, if at all possible, given the interference
of the other variables.
There exist superior techniques that have been devised, rather than
the two crude methods described, for mathematical optimization. This is a
fascinating branch of modern mathematics, and I strongly urge you to study
it, simply in the hope that you derive a fraction of the satisfaction from the
study as I have.
An extremum, that is the maximum or minimum, can be eitherglobal
(truly the highest or lowest value) orlocal(the highest or lowest value in
the immediate neighborhood). To truly know a global extremum is nearly
impossible, since you do not know the range of values of the independent
variables. If you do not know the range, then you have simply found a local
extremum. Therefore, oftentimes, when people speak of a global extremum,
they are really referring to a local extremum over a very wide range of values
for the independent variables.