Engineering Optimization, 5th Edition

1 Introduction to Optimization

1.1 Introduction

Optimization is the act of obtaining the best result under given circumstances. In design, construction, and maintenance of any engineering system, engineers have to take many technological and managerial decisions at several stages. The ultimate goal of all such decisions is either to minimize the effort required or to maximize the desired benefit. Since the effort required or the benefit desired in any practical situation can be expressed as a function of certain decision variables, optimization can be defined as the process of finding the conditions that give the maximum or minimum value of a function. It can be seen from Figure 1.1 that if a point x* corresponds to the minimum value of function f (x), the same point also corresponds to the maximum value of the negative of the function, −f (x). Thus, without loss of generality, optimization can be taken to mean minimization, since the maximum of a function can be found by seeking the minimum of the negative of the same function.

Figure 1.1 Minimum of f (x) is same as maximum of −f (x).

In addition, the following operations on the objective function will not change the optimum solution x* (see Figure 1.2):

1. Multiplication (or division) of f (x) by a positive constant c.
2. Addition (or subtraction) of a positive constant c to (or from) f (x).

Figure 1.2 Optimum ...