Computer Aided Engineering Design

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.