Engineering Optimization: Theory and Practice, Fourth Edition

8.6 Primal–Dual Relationship and Sufficiency Conditions in the Unconstrained Case 503

where

w=











w 0

w 1

..

.

wn











, =











 1

 2

..

.

N











, λ=











λ 0

λ 1

..

.

λN











(8.37)

withλdenoting the vector of Lagrange multipliers. At the stationary point ofL, we
have

∂L

∂wi

= 0 , i= 0 , 1 , 2 ,... , n

∂L

∂j

= 0 , j= 1 , 2 ,... , N

∂L

∂λi

= 0 , i= 0 , 1 , 2 ,... , N

(8.38)

These equations yield the following relations:

1 −

∑N

j= 1

λj= 0 or

∑N

j= 1

λj= 1 (8.39)

∑N

j= 1

λjaij= 0 , i= 1 , 2 ,... , n (8.40)

λ 0 −

λj

j

= 0 or λ 0 =

λj

j

, j= 1 , 2 ,... , N (8.41)

∑N

j= 1

j− 1 = 0 or

∑N

j= 1

j= 1 (8.42)

−ln

j

cj

+

∑n

i= 1

aijwi−w 0 = 0 , j= 1 , 2 ,... , N (8.43)

Equations (8.39), (8.41), and (8.42) give the relation

∑N

j= 1

λj= 1 =

∑N

j= 1

λ 0 j=λ 0

∑N

j= 1

j=λ 0 (8.44)

Thus the values of the Lagrange multipliers are given by

λj=

{

1 for j= 0

j for j= 1 , 2 ,... , N

(8.45)