Post on 03-Apr-2018
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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