Engineering Optimization, 5th Edition

6 Nonlinear Programming II: Unconstrained Optimization Techniques

6.1 Introduction

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

(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 understanding 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 point X ^* will be a relative minimum of f(X) if the necessary conditions

(6.2)

are satisfied. The point X ^* is guaranteed to be a relative minimum if the Hessian ...