78 Classical Optimization Techniques
we indicate its salient features through the following simple problem withn=2 and
m=1:
Minimizef (x 1 , x 2 ) (2.17)
subject to
g(x 1 , x 2 )= 0 (2.18)
A necessary condition forf to have a minimum at some point (x∗ 1 ,x 2 ∗) is that the total
derivativeoff (x 1 , x 2 ) ith respect tow x 1 must be zero at (x 1 ∗,x 2 ∗). By setting the total
differential off (x 1 , x 2 ) qual to zero, we obtaine
df=
∂f
∂x 1
dx 1 +
∂f
∂x 2
dx 2 = 0 (2.19)
Sinceg(x∗ 1 , x∗ 2 ) = 0 at the minimum point, any variationsdx 1 and dx 2 taken about
the point (x 1 ∗,x∗ 2 ) are calledadmissible variationsprovided that the new point lies on
the constraint:
g(x∗ 1 + dx 1 , x∗ 2 + dx 2 )= 0 (2.20)
The Taylor’s series expansion of the function in Eq. (2.20) about the point(x∗ 1 , x∗ 2 )
gives
g(x 1 ∗+ dx 1 , x 2 ∗+ dx 2 )
≃g(x 1 ∗, x 2 ∗)+
∂g
∂x 1
(x 1 ∗, x∗ 2 ) dx 1 +
∂g
∂x 2
(x 1 ∗, x 2 ∗) dx 2 = 0 (2.21)
wheredx 1 and dx 2 are assumed to be small. Sinceg(x∗ 1 , x∗ 2 ) = 0 , Eq. (2.21) reduces
to
dg=
∂g
∂x 1
dx 1 +
∂g
∂x 2
dx 2 = 0 at (x 1 ∗, x∗ 2 ) (2.22)
Thus Eq. (2.22) has to be satisfied by all admissible variations. This is illustrated
in Fig. 2.6, wherePQindicates the curve at each point of which Eq. (2.18) is sat-
isfied. IfAis taken as the base point(x 1 ∗, x 2 ∗) the variations in, x 1 andx 2 leading
to pointsB andC are calledadmissible variations. On the other hand, the varia-
tions inx 1 andx 2 representing pointDare not admissible since pointDdoes not
Figure 2.6 Variations aboutA.