356 COMPUTER AIDED ENGINEERING DESIGN
Forλ 1 and λ 2 both > 0, ST∇f may be seen always to be positive. It may be noted that ∇f represents
the direction along which the function increases at a maximum rate in an unconstrained case. Thus
if∇f(X 0 ) is chosen as the search direction at X 0 and a scalar α is used as the step size so that the new
point is expressed as X = X 0 + α∇f(X 0 ), from Taylor series expansion we have
f(X) = f(X 0 ) + α[∇f(X 0 )]T∇f(X 0 )Since [∇f(X 0 )]T∇f(X 0 ) > 0, if α is positive, the function value at the new point will be greater. For
a constrained case, ST∇f represents a component of increment of f along the search direction S. If
ST∇f> 0, the function value increases as we move along S. Thus, for λ 1 and λ 2 both positive, we may
not be able to find any direction in the feasible domain along which the function can be decreased
further. Since the point at which Eq. (12.25) is satisfied is assumed to be optimum, λ 1 andλ 2 have
to be positive. The reasoning can be extended to cases where more than two constraints are active.
The KKT necessary conditions for a minimum can now be written as
∂
∂∂
∂∂
∂Σ
L(, )
=
()
+
()(^0) = 0
=1
XL 0
X
X
X
X
X
fg
i
m
i
λ i
λigi(X 0 ) = 0, i = 1,.. ., m
λi≥ 0, i = 1,.. ., m
Example 12.9
(a)Minimize: ( , ) = ( – 1) + fx x 12 x 1 2 x 22
Subject to: ( , ) (gxx 112 ≡≤x 2 + 2) – x 12 0
We formulate the Lagrangian as
L = (^1 – 1) + + [( + 2) – ]
2
2
2
(^21)
xxxxλ^2
so that the necessary KKT conditions for an optimum are
∂
∂
L
x
xx
1
= 2( – 1) – 2 11 λ = 0
x 2
x 1
g 1 = 0
g 1 > 0
S
g 2 = 0
g 2 > 0
g 1 ,g 2 < 0
∇g (^1) ∇g 2
Figure 12.11 Geometric description of a feasible direction vector S