Overview

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

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...