This dissertation was presented by Ya-Ping Wang It was defended on December 14, 2012 and approved by the graduate faculty of the School of Engineering of the University of Pittsburgh

1. Duality Theory for Composite
Geometric Programming
Ya-Ping Wang
Department of Industrial Engineering
University of Pittsburgh
December 14, 2012

2. Duality Theory for Composite Geometric Programming
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Geometric Program (GP)
Primal posynomial program (GP)
(GP) 0inf ( ) . . ( ) 1, : {1,..., }m k
R
G s t G k K p

  
0<t
t t
Note that there are
 m design variables, 1
( , , ) , :{1,..., }mt t J m  t 0 ,
 each Gk(t) is a sum of terms indexed by the set [k]:
[ ]
( ): ( ), 0,1, , ,k ii k
G U k p
  t t
Across the p+1 different functions there are a total of
 n terms 1
, , , {1, , };whereU U I nn  
, ( ) : , 0,ija
i i j i ijj J
i I U C t C a R
    t ,
These terms are sequentially distributed into the (p+1)
problem functions as follows:
0 1[0] [1] [ ]; , 0< |[ ]|p kI p n n n n n k       
The “exponent matrix” is given by ,
, 1:[ ]n m
ij i ja A

3. Duality Theory for Composite Geometric Programming
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Equivalent Formulations of (GP)
In the variables = ln , ,j jz t j J
 ln ( ) ln ln , , lni i ij j i i ij J
U C a t c i I c C
       i
t a z
  [ ]
ˆln ( ) ln exp[ ] : ( ), .i
k i ki k
G c g k K
   t a z z
A convex formulation of GP:
(GP)z: 0inf ( ) . . ( ) 0,m k
R
g s t g k K

  
z
z z

4. Duality Theory for Composite Geometric Programming
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Equivalent Formulations of (GP) (cont’d)
In the variables =:ix i Ii
a z,
 [ ]
( ) ln ( ) ln exp( ) : ( )k k
k k i ii k
g G x c geo
    z t x c
where [ ]
( ) : ln[ exp ] : knk
ii k
geo x R R
 x is called a
geometric function (also called logexp(x)), [ ][ ]k
i i kx x
and [ ][ ]k
i i kc c .
A GGP (Generalized Geometric Programming)
formulation of GP: (GP)x
0 0
inf ( ) . . ( ) 0, ,k k
geo s t geo k K     x x c x c x P
where  | m n
R R  Az zP is the column space of A.

5. Duality Theory for Composite Geometric Programming
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Dual Posynomial Program (GD)
 ˆ [ ]
0
1
[ ]
sup ( ) : (Dual function)
. . 0, , 1 (normality condition)
0, (orthogonality conditions)
where : ,
i
n
i k ik K i k
R
i
n
ij i
i
k i
i k
V C
s t i I
a j J

 
 

 
 




   
  
 
 




ˆ.k K







 

Each dual variable i corresponds to a primal term iU .
Degree of difficulty is defined as ( 1)d n m  
The log-dual function
 
0 [ ]
1 1
( ) :ln ( ) ln /
( ln ) ln
p
i i k i
k i k
pn
i i i k k
i k
v V C
c
  
   
 
 
    
  

 
 
is concave on its domain of definition, and differentiable
in the interior of its domain.

6. Duality Theory for Composite Geometric Programming
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Main Lemma of GP
Lemma: If t is feasible for primal program (GP) and  is
feasible for dual program (GD), then
0 ( ) ( )G Vt  .
Moreover, under the same conditions 0 ( ) ( )G Vt  if, and
only if, one of the following two sets of equivalent
extremality conditions holds:
I.
( ) 1, , (1)
ˆ( ) ( ), [ ], (2)
k
k
i k k i
G k K
G U i k k K

 
   

    
t
t t
II.
0( ) ( ), [0] (1)
( ), [ ], (2)
i
i
k i
U G i
U i k k K



 
  
t t
t
in which case t is optimal for primal program (GP) and 
is optimal for dual program (GD).

7. Duality Theory for Composite Geometric Programming
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Dual to Primal Conversion at Optimality
When 0 ( ) ( )G Vt  ,
we get from condition II on p.5 the equations
( ), [0]
( )
, [ ], , and 0
i
i
i k k
V i
U
i k k K

  
 
 
  
t

,
which are log-linear:
 1
ln( ( )), [0]
ln , [ ], for which 0
m
i
ij j i
j i k k
V i
a z c
i k k K

  

  
  


So, knowing the optimal dual solution  an optimal
primal solution can in general be easily recovered by
solving this log-linear system for z and hence for t.

8. Duality Theory for Composite Geometric Programming
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EXAMPLE:
(Maximum Likelihood Estimator of a Bernoulli Parameter)
Suppose that n independent trials, each of which is a
success with probability p, are performed. What is the
maximum likelihood estimator (MLE) of p?
The likelihood function to be maximized is
1, 1
( , | ) (1 ) , with , 0 or 1,n n
ns n s
n n i if x x p p p s x x i
    
Example: n=3, m=2: 1
( , ) 0
inf s.t. 1n ns s n
p q
f p q p q 

  
The exponent matrix and a unique solution from
extremality condition II (2) on p.5:
* *
1
1 *
*2
2 *
1
3
=
1
ˆ1 0 , so
0 1
n n
n n
n
n
p q n
s s n
s S
s p p X
n n
n s
 





  
      
   

9. Duality Theory for Composite Geometric Programming
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Prior Extensions to GP
 Signomial GP which allow some <0iC  non-
convex programs: duality results are not strong
 Peterson’s Generalized GP (GGP)More general
separable convex programs
Extensions to GP investigated in this thesis
 Composite GP (CGP): itself a special case of
Peterson’s GGP. It includes as special cases:
 Exponential GP (EGP)
 Quadratic GP (QGP)
 (lpGP)
We start with Exponential GP (EGP) using a motivating
example…

10. Duality Theory for Composite Geometric Programming
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A Motivational Example
(Maximum Likelihood Estimator of a Poisson Parameter)
Suppose 1, , nX X are independent Poisson random
variables each having mean λ. Determine the MLE of λ.
The likelihood function to be maximized is
1, 1 1
( , | ) / ( ! !), where :n
ns n
n n n ii
f x x e x x s x
  

    
Equivalently, one can 0min ns n
e 
 
  . Although this is
not a posynomial, it can be solved as an EGP:
1
1 1
1
From condition (e) on p.14, we get
1 ˆ/ = , so /
1
n
n n
n
y s
y s n S n X
s

  

 
   
 
This same method also works for Exponential and Normal
parameters…

11. Duality Theory for Composite Geometric Programming
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Exponential Geometric Program (EGP)
In the previous example:
 a posynomial term is multiplied by an exponential
factor of another posynomial term.
This gives rise to an EGP problem where some
posynomial term ( )iU t is multiplied by an exponential
factor of another posynomial ( ): ( )i ll i
E V 
 t t .
Primal EGP problem (EGP): (cf (GP) on p.1)
0inf ( ) . . ( ) 1, ,m kR
G s t G k K
 0
 
<t
t t
where
[ ]
1
ˆ( ) : ( ) exp ( ) , : {0}
( ) , ,and ( ) ,
{1, , } 1 ,| | 0,
ij lj
k i li k l i
a b
i i j l l jj J j J
i n
G U V k K K
U C t i I V D t l L
L r n i r r r r
  
 
      
     
            
 
 
t t t
t t

 
[ ]:ljB b r m 

12. Duality Theory for Composite Geometric Programming
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Equivalent formulations of (EGP)
In the variables
A convex form of (EGP): (EGP)z
0inf ( ) . . ( ) 0,m k
R
g s t g k K

   
z
z z , where
  [ ]
ˆ( ) :ln ( ) ln exp exp ,i
k k i li k l i
g G c d k K  
        l
z t a z b z +
In the variables : , : ,i lx i I l L   i l
a z, b z,
[ ] [ ]
( ) ( , ) with exp,k k k k
k lg geo h l L      z x c ξ d
where
 [ ]
[ ]
( , ): ln exp exp( )k k
i li k l i
geo x   
    x ξ
:
k
kn r
R R R  is called an exponential geometric
function, [ ]
[ ] [ ]
[ ] , [ ] , :irk i i k
i k l l i ii k
R r r   
    ξ ξ ξ ,
and [ ]k
d is similarly defined.
= ln , ,j jz t j J

13. Duality Theory for Composite Geometric Programming
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Equivalent formulations of (EGP) (cont’d)
A GGP formulation of (EGP):
(EGP)x,ξ
0 0 [0] [0]
( )
[ ] [ ]
inf ( , )
. .
( , ) 0, ,( )
n r
R R
k k k k
geo
s t
geo k K
 
  


      
x,
x c ξ d
x c ξ d x,ξ



P
where  | m n r
M R R 
  z zP is the column space of the
composite exponent matrix
A
B
M
 
  
 
.

14. Duality Theory for Composite Geometric Programming
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Dual of EGP program (EGD) (cf (GD) on p.4)
ˆ( ) ( , ) [ ]
0
1 1
sup ( , ) :
. . 0, , 0, , 1,
0, (Orthogonality Conditions)
0
i l
n r
y
i k l i
R R i k l ik K i Ii l
i l
n r
ij i lj l
i l
i
C D y
V e
y
s t y i I l L
a y b j J
y



 


    
 
   
     
   
      
  
 
   
 
,
y
y 

[ ]
0, , (*) p.16
ˆwhere : , , : =
l
k i l
i k l L
l i i I
y k K  

 










       

   

 
Note that ( ), ( )i i l ly U V  t t
and degree of difficulty ( 1)d n r m    →p.4
The log-dual function (cf p.4)
   
ˆ [ ]
( , ) ln / ln /i i k i l l i l
i k l ik K i I
v y C y D y   
   
       y 
is usc proper concave in( , )y  , and differentiable in the
interior of its domain.

15. Duality Theory for Composite Geometric Programming
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Main Lemma of EGP
Lemma If t is feasible for primal program (EGP) and
( , )y  is feasible for dual program (EGD), then 0 ( ) ( , )G V y t 
Moreover, under the same conditions, 0 ( ) ( , )G V y t  if
and only if one of the following two sets of equivalent
extremality conditions hold: (cf I & II on p.5)
I' Condition (e) & I
( ) 1, ,
ˆ( ) ( ), [ ],
k
k
i k k i
G k K
y G U i k k K


  

  

 
t
t t
II' Condition (e) & II
0( ) ( ), [0]
( ), [ ],
i
i
k i
U G i
y
U i k k K
 
 
 


t t
t
where (e) ( ), ,l i lyV l i i I 
   t
in which case t is optimal for primal program (EGP) and
( , )y  is optimal for dual program (EGD).
 p.9 Solve the previous motivational Example.

16. Duality Theory for Composite Geometric Programming
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Composite Geometric Program (CGP)
Primal CGP Program (CGP) (cf (EGP) on p.10)
0inf ( ) . . ( ) 1,m kR
G s t G k K
 0
 
<t
t t , where
[ ]
ˆ( ) ( ) exp (ln ( )) ,k i l li k l i
G U h V k K  
       t t t ,
:lh R R is a differentiable and strictly convex function.
Note that:
(QGP): if 21
2( ) ,lh l L   
(lpGP): if ( ) | | / , where 1,lp
l l lh p p l L    

17. Duality Theory for Composite Geometric Programming
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Composite Geometric Program (CGP) (cont’d)
A Convex Form of (CGP): (CGP)z
0inf ( ) . . ( ) 0,m kR
g s t g k K
   z
z z , where
  [ ]
( ): ln ( ) ln exp i l
k k i l li k l i
g G c h d  
        z t a z b z
A GGP Form of (CGP): (CGP)x,
0 [0] [ ]
0
( )
inf ( , ) . . ( , ) 0, , ( )n r
k k
k
R R
f s t f k K
 
   ξ ξ ξ
x,
x x x, P

with   [ ]
( , ) ( , ) : ln expk i l li k l i
f geo x h   
          

18. Duality Theory for Composite Geometric Programming
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Dual program (CGD) (cf (EGD) on p.13)
*
ˆ( ) [ ]
0
1 1
sup ( , ): exp ( / )
. . 0, , 1, (Normality Condition)
0, (Ortho
i
n r
y
i k
l l i l l i
R R l ii kk K i Ii
i
n r
ij i lj l
i l
C
V d y h y
y
s t y i I
a y b j J

 


    
 
   
       
  
   
  
 
 
y,η
y 
[ ]
gonality Conditions)
, , (**)
ˆwhere : , .
l i l
k i
i k
y J l i i I
y k K














     

   


where *
lh , the conjugate of lh , has domain interval lJ .
(**) 0l  in (EGD) on p. 13 and that in any (CGD)
(*) 0 0, ,i ly l i i I 
       
The log-dual function (cf p.13)
  *
ˆ [ ]
( , ) ln / ( / )i i k i l l i l l i
i k l ik K i I
v y C y d y h y  
   
          y 
is usc proper concave in( , )y  , and differentiable in the
interior of its domain.

19. Duality Theory for Composite Geometric Programming
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Main Lemma of CGP
Lemma: If t is feasible for primal program (CGP) and
( , )y  is feasible for dual program (CGD), then
0 ( ) ( , )G V t y  .
Moreover, under the same conditions, 0 ( ) ( , )G V t y  if
and only if one of the following two sets of equivalent
extremality conditions hold: (cf I’ & II’ on p.14)
I'' Condition (c) & I
( ) 1, ,
ˆ( ) ( ), [ ],
k
k
i k k i
G k K
y G U i k k K


  

  

 
t
t t
II'' Condition (c) & II
0( ) ( ), [0]
( ), [ ],
i
i
k i
U G i
y
U i k k K
 
 
 


t t
t
where (c) '
(ln ( )), ,ll i ly h V l i i I 
   t
in which case t is optimal for primal program (CGP) and
( , )y  is optimal for dual program (CGD).
(lpGP): (c) 1
| ln ( ) | sgn(ln ( )), ,lp
l i l ly V V l i i I  
   t t

20. Duality Theory for Composite Geometric Programming
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First Duality Theorem of CGP
NOTE: The statements for EGP, QGP, lpGP cases are
almost the same.
Theorem: Suppose that primal program (CGP) is super-
consistent. Then the following three conditions are
equivalent:
1) t’ is a minimal solution to (CGP).
2) There exists a vector ' p
R for z′ (where z′= ln t′)
such that ( ', ')z  forms a saddle point of ( , )l z  .
where ( , )l z  is the Lagrangian of (CGP)z.

21. Duality Theory for Composite Geometric Programming
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3) There exists a vector ' p
R for z′ (where z′= ln t′)
such that ( ', ')z  satisfies the KKT conditions for
(CGP)z:
in which case the set of all such vectors 'λ is a non-empty
compact convex subset of p
R , and the dual program
(CGD) also has a maximum solution ( ', ')y  such that
0min(CGP) ( ') ( ', ') max(CGD)G V   t y 
and they satisfy the extremality conditions I''& II'' on
p.18.
(Perfect Duality)
' '
' '
0ˆ
( ) 0, ( ) 0, ( ) 0,
( ) ( , ') ( ) , 1
k k k k
z k kk K
a g g k K
b l g where
 
 
    
     0
 
 
z' z'
z' z'

22. Duality Theory for Composite Geometric Programming
Page 21 of 27
Second Duality Theorem of EGP
From linear programming duality theory:
Of the following two linear systems, exactly one has a
solution (where  :
A
B
M n r m
 
  
 
):
(I) Find z with 0 0
 
  
 
A
B
z
(II) Find >0 with T T
A B
 
 
 
 
y
y 0

We say that program (EGD) is canonical if system (II)
has a solution when M is the program’s composite
exponent matrix.
Theorem: Suppose that primal program (EGP) is
consistent. Then the minimum set of program (EGP)z is
non-empty and bounded if, and only if, dual program
(EGD) is canonical, in which case program (EGP) has a
minimum solution t′.

23. Duality Theory for Composite Geometric Programming
Page 22 of 27
Second Duality Theorem of lpGP*
* This theorem also applies to its special case QGP.
Again, from linear programming duality theory
Of the following two linear systems exactly one has a
solution (where  :
A
B
M n r m
 
   
 
):
(I) Find z with ,  0 Az 0 Bz 0
(II) Find with , and T T
A B
 
 
 
  
y
y 0 y 0


We say that program (lpGD) is canonical if system (II)
has a solution when M is the program’s composite
exponent matrix.
Theorem 7.3.2 Suppose that primal program (lpGP) is
consistent. Then the minimum set of program (lpGP)z is
non-empty and bounded if and only if its dual program
(lpGD) is canonical.

24. Duality Theory for Composite Geometric Programming
Page 23 of 27
SUMMARY OF CONTRIBUTIONS
 Extension of traditional GP models to
o to the more general Exponential GP models,
o to the even more general Composite GP models,
which include as important special cases:
 (EGP)
 (lpGP)
 (QGP)
 Showing that all of these are special cases of
Peterson’s GGP models (for which he has given a
Main Lemma but not a First or a Second duality
theorem).

25. Duality Theory for Composite Geometric Programming
Page 24 of 27
CONTRIBUTIONS (cont’d)
 For each extension:
o multiple direct proofs of the Main Lemma, with a
2nd set of equivalent Extremality Conditions.
 For the First Duality Theorem, proof that
o a superconsistent primal program (CGP) has a
minimal solution 't if, and only if, there exists a
vector ' p
R such that ( ', ')z  forms a saddle
point of the Lagrangian ( , )l z  and if, and only
if, it satisfies the KKT conditions for (CGP)z,
o in which case the set of all such Lagrange
multiplier vectors λ is a non-empty compact
convex subset of p
R , whereas the original
theorem is a “If…then” statement and only
showed the existence ofλ for the (GP) case only.

26. Duality Theory for Composite Geometric Programming
Page 25 of 27
 For the Second Duality Theorem, proof that
o for the special (and more important) cases of
(QGP), (lpGP) and (EGP),
 the minimum set of a consistent primal
program is non-empty and bounded if and
only if its dual program is canonical
 the original theorem showed the existence of
a primal solution for the (GP) case only,
when its dual program is canonical.
 Sensitivity Analysis: If *
z is optimal for (CGP)z:
* * *
* * *0 0 0
* *
* * * *0 0
( ) ( ) ( )
, , ,
( ) ( )
, , where ln
i l k
i l k
i j l j k k
ij lj
g g g
y
c d b
g g
y z z b B
a b
 

  
   
  
 
  
 
  
 
z z z
z z
 Dual to primal conversion of optimal solutions
when there is no duality gap: 0 ( ) ( , )G V y t  .

27. Duality Theory for Composite Geometric Programming
Page 26 of 27
Directions for Future Research
1.Development of proofs of a strong version of the
First Duality Theorem of (CGP): If a primal program
(CGP) is superconsistent and has a finite infimum,
then the dual program (CGD) has a maximum
solution ( ', ')y  such that
inf(CGP) max(CGD) ( ', ')V  y 
2.Development of computational algorithms.

28. Duality Theory for Composite Geometric Programming
Page 27 of 27
QUESTIONS