multivariate production systems optimization - Stanford University

Chapter 6
NONLINEAR OPTIMIZATION
There are numerous problems for which people strive to find an optimal solution. These
problems might be minimizing the fuel consumption of an engine, maximizing the
efficiency of a process, or minimizing the volume occupied by a cluster of objects. It is
trivial to optimize a problem that depends on a single variable. It is when several variables
are optimized simultaneously that matters become complicated.
Mathematics can provide a powerful tool to optimize a problem, regardless of the
number of variables involved. Numerical optimization is the location of the extrema of a
mathematical model. The mathematical model, referred to as the objective function, accepts
multiple decision variables as input and returns a single value, the objective variable, as
output. Optimization strives to locate the minimum or maximum value of the objective
function.
Nonlinear optimization methods based on Newton’s technique locate the extrema by
approximating the objective function with a nonlinear quadratic model. Let the objective
function, F, be a nonlinear function of the vector of decision variables, x.
F = f x (6.1)
Newton’s method approximates the nonlinear objective function, F, with a quadratic
model, Q, which is also a function of the vector of decision variables, x.
The quadratic approximation
Q x ≈ F x (6.2)
Q x = c T x + 1 2 xT G x (6.3)
may be conceptualized, in two dimensions, as a bowl. If the matrix G is negative-definite,
the expression is bounded above and unbounded below (the bowl is upside-down) and will
therefore have a maximum (See Figure 6.1). Conversely, if the matrix G is positive
definite, the expression is bounded below and unbounded above (the bowl is right-side-up)
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