Page 226 - Numerical methods for chemical engineering

5 Numerical optimization

Many problems in chemical engineering are expressed mathematically as optimization
problems, and involve finding the particular x that minimizes some cost function F(x).
Each component of x may vary either continuously or discretely. In this chapter, we assume
that each x j varies continuously. In Chapter 7, we consider stochastic techniques that can be
used with discretely-varying parameters.
An optimization problem may be unconstrained, in which case each x j can take any real
value, or it can be constrained, such that an allowable x must satisfy some collection of
equality and inequality constraints
g(x) = 0 or h(x) ≥ 0 (5.1)
We consider first unconstrained problems, and then treat constraints. Here, the focus is
upon methods that identify local minima; i.e., points that are lower in cost function than
their neighbors. The stochastic methods of Chapter 7 return (eventually) global minima;
therefore, the reader is referred to that discussion if identifying the global minimum is
necessary.
Numerical optimization problems arise in many contexts. To predict the geometry of a
molecule, we find the conformation of its atoms with the lowest potential energy. In process
design x contains parameters such as equipment sizes, flow rates, temperatures, etc., and
the cost function is a measure of the economic cost of operating the process. We fit a
mathematical model for a system by minimizing the sum of squared differences between
the model predictions and experimental data. In optimal control, we choose the best set of
control inputs to maintain a process at the desired set point.
In addition to a discussion of the basic techniques for identifying local minima in con-
tinuous parameter space, the use of optimization routines in AAB is demonstrated. As
these routines are part of an optional tiiatin tit , an alternative routine is provided
that can be used without the toolkit.

Local methods for unconstrained optimization problems

[0]
We begin by considering iterative techniques that start at some initial guess x , and gen-
[1]
[2]
erate a sequence of estimates x , x ,... that (hopefully) converges to a local minimum
x min of F(x). That is, for x within |x − x min |≤ ε, ε > 0, F(x) ≥ F(x min ). If we envision
F(x) to be a physical elevation, a local minimum lies at the bottom of a “valley,” and
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