Engineering Optimization: Theory and Practice, Fourth Edition

512 Geometric Programming

Table 8.2 Corresponding Primal and Dual Programs

Primal program Dual program

FindX=











x 1

x 2

..

.

xn











so that

g 0 (X)≡f (X)→minimum

subject to the constraints

x 1 > 0

x 2 > 0

..

.

xn> 0 ,

g 1 (X)≤ 1

g 2 (X)≤ 1

..

.

gm(X)≤ 1 ,

Findλ=





















































λ 01

λ 02

..

.

λ 0 N 0

· · ·

λ 11

λ 12

..

.

λ 1 N 1

· · ·

..

.

·· ·

λm 1

λm 2

..

.

λmN m





















































so that

v(λ)=

∏m

k= 0

N∏k

j= 1

(

ckj

λkj

∑Nk

l= 1

λkl

)λkj

→ maximum

with subject to the constraints

g 0 (X)=

∑N^0

j= 1

c 0 jx

a 01 j

1 x

a 02 j

2 ·· ·x

a 0 jn

n

g 1 (X)=

∑N^1

j= 1

c 1 jx

a 11 j

1 x

a 12 j

2 ·· ·x

a 1 jn

n

g 2 (X)=

∑N^2

j= 1

c 2 jx

a 21 j

1 x

a 22 j

2 ·· ·x

a 2 jn

n

..

.

gm(X)=

∑Nm

j= 1

cmjx

am 1 j

1 x

am 2 j

2 ·· ·x

amnj

n

λ 01 ≥ 0

λ 02 ≥ 0

..

.

λ 0 N 0 ≥ 0

λ 11 ≥ 0

..

.

λ 1 N 1 ≥ 0

..

.

λm 1 ≥ 0

λm 2 ≥ 0

..

.

λmNm≥ 0

(continues)