Engineering Optimization: Theory and Practice, Fourth Edition

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.