2.4 Multivariable Optimization with Equality Constraints 75
These equations are satisfied at the points
( 0 , 0 ), ( 0 ,−^83 ), (−^43 , 0 ), and (−^43 ,−^83 )
To find the nature of these extreme points, we have to use the sufficiency conditions.
The second-order partial derivatives offare given by
∂^2 f
∂x 12
= 6 x 1 + 4
∂^2 f
∂x 22
= 6 x 2 + 8
∂^2 f
∂x 1 ∂x 2
= 0
The Hessian matrix off is given by
J=
[
6 x 1 + 4 0
0 6x 2 + 8
]
IfJ 1 = | 6 x 1 + 4 |andJ 2 =
∣
∣
∣
∣
6 x 1 + 4 0
0 6x 2 + 8
∣
∣
∣
∣,the values ofJ^1 andJ^2 and the nature
of the extreme point are as given below:
PointX Value ofJ 1 Value ofJ 2 Nature ofJ Nature ofX f(X)
(0, 0) + 4 + 32 Positive definite Relative minimum 6
( 0 ,−^83 ) + 4 –32 Indefinite Saddle point 418/27
(−^43 , 0 ) –4 –32 Indefinite Saddle point 194/27
(−^43 ,−^83 ) –4 + 32 Negative definite Relative maximum 50/3
2.4 Multivariable Optimization with Equality Constraints
In this section we consider the optimization of continuous functions subjected to equal-
ity constraints:
Minimizef=f (X)
subject to
gj( X)= 0 , j= 1 , 2 ,... , m
(2.16)
where
X=
x 1
x 2
..
.
xn
