Engineering Optimization: Theory and Practice, Fourth Edition

500 Geometric Programming

or

x 1 ∗= 1

Finally, we can obtainx 3 ∗by adding Eqs. (E 14 ) (E, 15 ) and (E, 16 ) sa

w 3 = n 1l +ln 2+ln 1=ln 2=lnx∗ 3

or

x 3 ∗= 2

It can be noticed that there are four equations, Eqs. (E 13 ) o (Et 16 ) n three unknownsi

w 1 , w 2 , andw 3. However, not all of them are linearly independent. In this case, the

first three equations only are linearly independent, and the fourth equation, (E 16 ) can,

be obtained by adding Eqs. (E 13 ) (E, 14 ) and (E, 15 ) and dividing the result by, −2.

8.5 SOLUTION OF AN UNCONSTRAINED GEOMETRIC
PROGRAMMING PROBLEM USING
ARITHMETIC–GEOMETRIC INEQUALITY
Thearithmetic mean–geometric mean inequality (also known as the arithmetic–
geometric inequalityorCauchy’s inequality) is given by [8.1]

 1 u 1 + 2 u 2 + · · · +NuN≥u

 1

1 u

 2

2 · · ·u

N

N (8.20)

with

 1 + 2 + · · · +N= 1 (8.21)

This inequality is found to be very useful in solving geometric programming problems.

Using the inequality of (8.20), the objective function of Eq. (8.3) can be written as (by

settingUi=uii, i = 1 , 2 ,... , N )

U 1 +U 2 + · · · +UN≥

(

U 1

 1

) 1 (

U 2

 2

) 2

·· ·

(

UN

N

)N

(8.22)

where Ui=Ui( X), i= 1 , 2 ,... , N, and the weights  1 ,  2 ,... , N, satisfy

Eq. (8.21). The left-hand side of the inequality (8.22) [i.e., the original functionf(X)]

is called theprimal function. The right side of inequality (8.22) is called thepredual

function. By using the known relations

Uj=cj

∏n

i= 1

x

aij

i , j=^1 ,^2 ,... , N (8.23)

the predual function can be expressed as

(

U 1

 1

) 1 (

U 2

 2

) 2

·· ·

(

UN

N

)N

=









c 1

∏n

i= 1

x

ai 1

i

 1









 1 







c 2

∏n

i= 1

x

ai 2

i

 2









 2

·· ·









cN

∏n

i= 1

x

aiN

i

N









N