Engineering Optimization: Theory and Practice, Fourth Edition

8.6 Primal–Dual Relationship and Sufficiency Conditions in the Unconstrained Case 501

=

(

c 1

 1

) 1 (

c 2

 2

) 2

·· ·

(

CN

N

)N







(n

∏

i= 1

xiai^1

) 1 (n

∏

i= 1

xiai^2

) 2

·· ·

(n

∏

i= 1

xaiiN

)N





=

(

c 1

 1

) 1 (

c 2

 2

) 2

·· ·

(

cN

N

)N{(

x

∑N

j= 1 a^1 jj

1

)(

x

∑N

j= 1 a^2 jj

2

)

·· ·

(

x

∑N

j= 1 anjj

n

)}

(8.24)

If we select the weightsj so as to satisfy the normalization condition, Eq. (8.21),

and also the orthogonality relations

∑N

j= 1

aijj= 0 , i= 1 , 2 ,... , n (8.25)

Eq. (8.24) reduces to

(

U 1

 1

) 1 (

U 2

 2

) 2

·· ·

(

UN

N

)N

=

(

c 1

 1

) 1 (

c 2

 2

) 2

·· ·

(

cN

N

)N

(8.26)

Thus the inequality (8.22) becomes

U 1 +U 2 + · · · +UN≥

(

c 1

 1

) 1 (

c 2

 2

) 2

·· ·

(

cN

N

)N

(8.27)

In this inequality, the right side is called thedual function,v( 1 ,  2 ,... , N) The.

inequality (8.27) can be written simply as

f≥v (8.28)

A basic result is that the maximum of the dual function equals the minimum of the

primal function. Proof of this theorem is given in the next section. The theorem enables

us to accomplish the optimization by minimizing the primal or by maximizing the dual,

whichever is easier. Also, the maximization of the dual function subject to the orthog-

onality and normality conditions is a sufficient condition forf, the primal function, to

be a global minimum.

8.6 PRIMAL–DUAL RELATIONSHIP AND SUFFICIENCY
CONDITIONS IN THE UNCONSTRAINED CASE
Iff∗indicates the minimum of the primal function andv∗denotes the maximum of
thedual function, Eq. (8.28) states that

f≥f∗≥v∗≥v (8.29)