Engineering Optimization: Theory and Practice, Fourth Edition

74 Classical Optimization Techniques

As an example, consider the functionf (x, y)=x^2 −y^2. For this function,

∂f

∂x

= 2 x and

∂f

∂y

= − 2 y

These first derivatives are zero atx∗= 0 andy∗=. The Hessian matrix of 0 f at

(x∗,y∗) is given by

J=

[

2 0

0 − 2

]

Since this matrix is neither positive definite nor negative definite, the point (x∗= , 0

y∗= 0)is a saddle point. The function is shown graphically in Fig. 2.5. It can be seen

thatf (x, y∗) =f(x, 0) has a relative minimum andf (x∗, )y =f ( 0 , y)has a relative

maximum at the saddle point (x∗,y∗). Saddle points may exist for functions of more

than two variables also. The characteristic of the saddle point stated above still holds

provided thatxandyare interpreted as vectors in multidimensional cases.

Example 2.5 Find the extreme points of the function

f (x 1 , x 2 )=x 13 +x^32 + 2 x 12 + 4 x^22 + 6

SOLUTION The necessary conditions for the existence of an extreme point are

∂f

∂x 1

= 3 x^21 + 4 x 1 =x 1 ( 3 x 1 + 4 )= 0

∂f

∂x 2

= 3 x^22 + 8 x 2 =x 2 ( 3 x 2 + 8 )= 0

Figure 2.5 Saddle point of the functionf (x, y)=x^2 −y^2.