Engineering Optimization: Theory and Practice, Fourth Edition

346 Nonlinear Programming II: Unconstrained Optimization Techniques

Example 6.10 Show that the Newton’s method finds the minimum of a quadratic

function in one iteration.

SOLUTION Let the quadratic function be given by

f (X)=^12 XT[A]X+BTX+C

The minimum off (X)is given by

∇f=[A]X+B= 0

or

X∗= −[A]−^1 B

Theiterative step of Eq. (6.86) gives

Xi+ 1 =Xi−[A]−^1 ([A]Xi+B) (E 1 )

whereXiis the starting point for theith iteration. Thus Eq. (E 1 ) gives the exact solution

Xi+ 1 =X∗= −[A]−^1 B

Figure6.16 illustrates this process.

Example 6.11 Minimizef (x 1 , x 2 )=x 1 −x 2 + 2 x^21 + 2 x 1 x 2 +x 22 by taking the start-

ing point asX 1 =

{ 0

0

}

.

SOLUTION To findX 2 according to Eq. (6.86), we require [J 1 ]−^1 , where

[J 1 ]=











∂^2 f

∂x^21

∂^2 f

∂x 1 ∂x 2

∂^2 f

∂x 2 ∂x 1

∂^2 f

∂x 22











X 1

=

[

4 2

2 2

]

Figure 6.16 Minimization of a quadratic function in one step.