6

Nonlinear Programming II:

Unconstrained Optimization

Techniques

6.1 Introduction

This chapter deals with the various methods of solving the unconstrained minimization

problem:

FindX=











x 1

x 2

..

.

xn











whichminimizesf (X) (6.1)

It is true that rarely a practical design problem would be unconstrained; still, a study

of this class of problems is important for the following reasons:

1.The constraints do not have significant influence in certain design problems.

2.Some of the powerful and robust methods of solving constrained minimization

problems require the use of unconstrained minimization techniques.

3.The study of unconstrained minimization techniques provide the basic under-

standing necessary for the study of constrained minimization methods.

4.The unconstrained minimization methods can be used to solve certain complex

engineering analysis problems. For example, the displacement response (linear

or nonlinear) of any structure under any specified load condition can be found

by minimizing its potential energy. Similarly, the eigenvalues and eigenvectors

of any discrete system can be found by minimizing the Rayleigh quotient.

As discussed in Chapter 2, a pointX∗will be a relative minimum off(X)if the

necessary conditions

∂f

∂xi

(X=X∗) = 0 , i= 1 , 2 ,... , n (6.2)

Engineering Optimization: Theory and Practice, Fourth Edition Singiresu S. Rao 301

Copyright © 2009 by John Wiley & Sons, Inc.