multivariate production systems optimization - Stanford University

Q Q
X2
X1 X1
Negative Definite Positive Definite
Figure 6.1: A Maximum and a Minimum (from Strang 1986).
and will have a minimum. If G is indefinite, the expression is unbounded both above and
below and therefore does not have an extrema.
To find a minimum of an objective function, a positive definite matrix should be
used. Likewise, to find a maximum of an objective function, a negative definite matrix
should be used. However, in all of the literature pertaining to optimization, the convention
is to always frame the problem in terms of finding the minimum of a function. Therefore,
instead of maximizing a function, the convention is to minimize the negative of the
objective function. Thus, even for maximizing a function, we will speak of ‘descending’
to the minimum.
For a problem that is inherently nonlinear, nonlinear approximations of the
objective function allow the solution to be found at a much faster rate than with linear
approximations. In contrast, the much heralded linear programming approximates the
objective function with linear models. Since a linear relationship is unbounded in all
directions (unless it is constant), a minimum or maximum value cannot occur without some
form of constraint. Thus, the solution to a linear programming problem can only occur at
the intersection of a linear approximation and a constraint. This is the principle that the
Simplex algorithm is based upon.
The theory of optimization is very well developed. Optimization routines are
widely available through software libraries such as the Numerical Algorithms Group and
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