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Geometric Programming Problem

using arithmetic-geometric (Cauchy) inequality method

Unconstrained GP Problem

Find X that minimize

f (X) = U1 + U2 + …….+ U N =

 N 

 j

 j X U 1

)(

 

 

 

 

 N 

 j

n

i

a

i j

ij

 xc1 1

 N 

 j

a

n

aa

 jnj j j  x x xc

1

21 )......( 21

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Unconstrained GPP using Cauchy’s inequality method 

The arithmetic –geometric inequality or Cauchy’s inequality is

with

Set

Using Cauchy’s inequality, objective function is

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

Left hand =objective function f (X) = Primal function

Right hand = Predual function

 N 

 N  N  N  uuuuuu

............

21

212211

1.........21 N 

 N....1,2,i , iii uU 

 N 

 N 

 N 

 N  N 

U U U 

uuu

 

 

 

 

 

 

 

 

 

 

 

 

............

21

2

2

1

1

2211

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Predual function

using know relation

 N 

iN ii

 N 

 N 

n

i

a

i N 

n

i

a

i

n

i

a

i

 N 

 N 

 xc xc xcU U U 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

  

 

 

  

 

1

2

1

2

1

1

1

2

2

1

1 ..... .

2

2

1

1

21

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 N 

iN ii

 N  n

i

a

i

n

i

a

i

n

i

a

i

 N 

 N   x x xccc

1112

2

1

1 .. ... .

2

2

1

1

21

....N1,2, j ,1

n

i

a

i  j  jij xcU 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N 

 j j j N 

a N 

 j j ja

 N 

 j j ja N 

 N 

 N 

 N   x x xccc 11

21

121

....... 21

2

2

1

1

Unconstrained GPP using Cauchy’s inequality method 

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Select weight  j so as to satisfy

 Normalization condition

Orthogonality condition

Cauchy’s inequality 

Right side is called dual function, v(

1,

2, …….

 N)

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

  

 

 

  

 

N 

 j j j N 

a N 

 j j ja

 N 

 j j ja N 

 N 

 N 

 N   x x xccc 11

21

121

....... 21

2

2

1

1

. . . . .1,2,i ,01

 N 

 j jija

1.........21 N 

 N  N 

 N 

 N 

 N 

 N  cccU U U 

 

  

 

 

  

 

 

  

 

 

  

 

 

  

 

 

  

 

......

2121

2

2

1

1

2

2

1

1

 N 

 N 

 N  N 

cccU U U 

 

  

 

 

  

 

 

  

 

..........

21

2

2

1

121

Unconstrained GPP using Cauchy’s inequality method 

 N 

 N 

 N ccc

 

 

 

 

 

 

 

 

 

 

 

 

...

21

2

2

1

1

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Maximum of the dual function = minimum of the primal function

Minimizing the given original function (primal function) is equal to

maximization of the dual function subject to the orthogonally and

normality condition. It is a sufficient condition for f (X), the primal

function, to be a global minimum.

Unconstrained GPP using Cauchy’s inequality method 

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Unconstrained GPP using Cauchy’s inequality method