Engineering Optimization: Theory and Practice, Fourth Edition

8.9 Primal and Dual Programs in the Case of Less-Than Inequalities 513

Table 8.2 (continued)

Primal program Dual program

the exponentsakijare real numbers, and
the coefficientsckjare positive numbers.

∑N^0

j= 1

λ 0 j= 1

∑m

k= 0

∑Nk

j= 1

akijλkj= 0 , i= 1 , 2 ,... , n

the factorsckjare positive, and the

coefficientsakijare real numbers.

Terminology

g 0 =f=primal function

x 1 , x 2 ,... , xn=primal variables

gk≤1 are primal constraints

(k= 1 , 2 ,... , m)

xi> 0 , i= 1 , 2 ,... , npositive restrictions.

n=number of primal variables

m=number of primal constriants

N=N 0 +N 1 + · · · +Nm=total number

of terms in the posynomials

N−n− 1 =degree of difficulty of the

problem

ν=dual function

λ 01 , λ 02 ,... , λmN m=dual variables

∑N^0

j= 1

λ 0 j= 1 is the normality constraint

∑m

k= 0

N∑k

j= 1

akijλkj= 0 , i= 1 , 2 ,... , nare the

orthogonality constraints

λkj≥ 0 , j= 1 , 2 ,... , Nk;

k= 0 , 1 , 2 ,... , m

are nonnegativity restrictions

N=N 0 +N 1 + · · · +Nm

=number of dual variables

n+1 number of dual constraints

v(λ)=

∏^1

k= 0

∏Nk

j= 1

(

ckj

λkj

∑Nk

l= 1

λkl

)λkj

=

N∏ 0 = 3

j= 1



c^0 j

λ 0 j

N∑ 0 = 3

l= 1

λ 0 l





λ 0 jN

∏^1 =^1

j= 1

(

c 1 j

λ 1 j

N∑ 1 = 1

l= 1

λ 1 l

)λ 1 j

=

[

c 01

λ 01

(λ 01 +λ 02 +λ 03 )

]λ 01 [

c 02

λ 02

(λ 01 +λ 02 +λ 03 )

]λ 02

·

[

c 03

λ 03

(λ 01 +λ 02 +λ 03 )

]λ 03 (

c 11

λ 11

λ 11

)λ 11

(E 1 )

subject to the constraints

λ 01 +λ 02 +λ 03 = 1

a 011 λ 01 +a 012 λ 02 +a 013 λ 03 +a 111 λ 11 = 0