354 COMPUTER AIDED ENGINEERING DESIGN
Fig. 12.10 Function (thin lines) and constraint (thick line)
curves for Example 12.8
C
D
B
A
y
5 4 3 2 1 0
–1
–2
–4 –2 0 2 4
x
i = 1,... , m. These constraints can be converted to equality constraints by adding slack variables yi
such that
gyii( ) + X i^2 = 0, = 1,... , m (12.20)
The slack variables ensure that gi(X),i= 1,... , m are all smaller than or equal to zero. The
minimization problem can be solved using the method of Lagrangian multipliers discussed above.
ForY = [y 1 ,y 2 ,... , ym], the augmented Lagrangian can be written as
LL(xx 12 , ,... , xnmm, 12 , ,... , , , yy 12 ,... , y) ( , , ) = ( ) + f i=1 [g( ) + y^2 ]
m
λλ λ ≡ XY XΛΛ ΣλiiXi
(12.21)
Noting that we have additional m variables Y, employing the necessary conditions in Eq. (12.14)
gives
∂
∂
∂
∂
∂
∂
L( , , )
=
()
+
()
(^0) =
=1
XY 0
X
X
X
X
X
ΛΛ fg
i
m
i
Σλ i 0
(12.22a)
∂
∂
≡
L( , , )
( 0 ) +^2 = 0, = 1,... ,
XY
X
ΛΛ
ΛΛ
gyi mi i (12.22b)
and
∂
∂
≡
L( , , )
2 = 0, = 1,... ,
XY
Y
ΛΛ
λiiyi m (12.22c)
which is a system of n + 2m equations in the same number of unknowns X,ΛΛΛΛΛ and Y. From
Eq. (12.22c), if yi = 0, then from (12.22b) gi(X 0 ) = 0 and the constraint is said to be active^3. In such a
case, the corresponding Lagrange multiplier λi may or may not be zero. If yi≠ 0, then λi has to be zero
andgi(X 0 ) is strictly smaller than zero.
(^3) The inequality constraints satisfied with the equality sign g
j(X) = 0 at the optimum X 0 are called active
constraints while those satisfied with the strict inequality sign gj(X) < 0 are called inactiveconstraints.