The Leverage Space Model 315
There are a number of techniques for finding the maximum or min-
imum in such cases. Usually, in any type of mathematical optimization,
there areconstraintsplaced on the variables, which must be met with
respect to the extremum. For example, in our case, there are the con-
straints that all independent variables (thefvalues) must be greater than
or equal to zero. Oftentimes, there are constraining functions that must
be met [i.e., other functions involving the variable(s) used which must
be above/below or equal to certain values].Linear programming, includ-
ing thesimplex algorithm, is one very well developed area of this type
of constrained optimization, but will work only where the function to be
optimized and the constraint functions are linear functions (first-degree
polynomials).
Generally, the different methods for mathematical optimization can be
broken down by the following categories, and the appropriate technique
selected:
1.Single-variable (two-dimensional) vs. multivariable (three- or more di-
mensional) objective functions.
2.Linear methods vs. nonlinear methods. That is, as previously mentioned,
if the function to be optimized and the constraint functions are linear
functions (i.e., do not have exponents greater than one to any of the terms
in the functions), there are a number of very well developed techniques
for solving for extrema.
3.Derivatives. Some methods require computation of the first derivative of
the objective function. In the multivariable case, the first derivative is a
vector quantity called thegradient.
4.Computational efficiency. That is, you want to find the extremum as
quickly (i.e., with as few computations) and easily (something to consider
with those techniques which require calculation of the derivative) as
possible, using as little computer storage as possible.
5.Robustness. Remember, you want to find the extremum that is local
to a very wide range of parameter values, to act as a surrogate global
extremum. Therefore, if there is more than one extremum in this range,
you do not want to get hung up on the less extreme extremum.
In our discussion, we are concerned only with the multidimensional
case. That is, we concern ourselves only with those optimization algorithms
that pertain to two or more variables (i.e., more than one scenario set).In
searching for a singlefvalue, that is, in finding thefof one market system
or one scenario set, parabolic interpolation, as detailed in Chapter 4,
Portfolio Management Formulas,will generally be the quickest and most
efficient technique.