109
NZOR volume 6 number 2
July 1978
This is the first of two articles - one on theory, the other on applications.
The Editor
AVENUES OF GEOMETRIC PROGRAMMING*
T.R. J e f f e r s o n
T h e Un i v e r s i t y
of
and
C.H, S c o t t
N e w So u t h W a l e s , A u s t r a l i a
Summary
Since its initial appearance more than a decade ago, geometric
programming has proved itself to be a powerful means of solving many op­
timization problems, particularly those pertaining to engineering design.
Recently the theory has been generalised to provide a structure for the
analysis of any convex programming problem. Here we trace the develop­
ment of geometric programming from its initial use as a trick for
solving certain cost optimization problems to its current status as a
fully fledged branch of optimization theory. Examples draum from inven­
tory theory reinforce and motivate this development.
1.
Introduction
In the early sixties, C la r e n c e Zener was a D i r e c t o r of
Sci en ce at W e s t i n g h o u s e R e s e a r c h L a b o r a t o r i e s at Pi tt sb ur gh .
In such a p os it io n, a p h y s i c i s t b ec o m e s i n v o lv ed w i t h cost
mod el li ng .
The d if f e r e n c e ,
in Zener's case, was that he o b ­
se rved a curio us m a t h e m a t i c a l p h e n o m e n a w i t h r e g a r d to cost
mode ls ma d e up as a sum of c o m p o n e n t costs
[47,48].
He was
able to o p t i mi ze ce r t a i n such m o del s by m e r e i n s p e c t i o n of
the expon en ts of the de s i g n v ar i a b l e s
(Section 2).
Thus was
b o r n a new and e xc i t i n g a p p r o a c h to m a t h e m a t i c a l p r o g r a m m i n g
which has h e n c e f o r t h b ee n t er me d g e o m e t r i c pr o g r a m m i n g .
Initial r es e a r c h c o n c e n t r a t e d on p o s y n o m i a l
(p olynomial w i th
p os i t i v e coeff ic ie nt s) p r o g r a m s wh e r e bo t h the o b j e c t i v e
fu n c ti on and co n s t r a i n t s are in p o s y n o m i a l form (Section 3).
This r e s ea rc h c u l m i n a t e d in the s eminal b o ok e n t i t l e d " G e o ­
me tr i c P r o g r a m m i n g " by Duffin, P e t e r s o n and Zener
[13].
S u b s e qu en t w o r k d e v e l o p e d in two s i g n i f i c a n t l y d i f f e r e n t
d irections:
S ign om ia l P r o g r a m m i n g
ised G e o me tr ic P r o g r a m m i n g
‘M a n u s c r i p t
submitted
(Section 4) and G e n e r a l ­
(Section 5)
October
28,
1977
[33].
110
Signomial Programming,
W i l de
first d e v e l o p e d by Passy and
[32], s ou g h t to relax the r e s t r i c t i o n of the functions
tre a te d by G e o m e t r i c P r o g r a m m i n g to the form of p ol yn omials.
W it h the r e l a x a t i o n to p o l y n o m i a l s one loses c o n v e x i t y and
h e n c e one can only hope for a local m in i m u m , not a global
m i n i mu m.
G e n e r a l i s e d G e o m e t r i c P r o g r a m m i n g p r o c ee ds to
g e n e r a l i s e the type of f u n c t i o n c o n s i d e r e d from p o sy no mi al
to c onvex.
This g e n e r a l i s a t i o n does m a i n t a i n the c o n ve xi ty
of the p r o b l e m and hen ce one is able to find the global s o l ­
u t i o n to the m a t h e m a t i c a l program.
The r e l a t i o n s h i p among the vari ou s d e v e l o p m e n t s may
o u t w a r d l y see m tenuous.
This is b ec a u s e G e o m et ri c P r o g r a m ­
m i n g s ho ul d not be t h ou ght of as a class of m a t h e m a t i c a l
p r o g r a m s but as a m e t h o d of a n a ly si s of m a t h e m a t i c a l programs.
In this respect, g e o m e t r i c p r o g r a m m i n g has m u c h in common
with dynamic programming.
r e c u r s i v e f unctions.
li ne ar i t y ,
(i)
D y n a m i c P r o g r a m m i n g makes use of
G e o m e t r i c P r o g r a m m i n g mak es use of
s e p a r a b i l i t y , c o nve x i t y , and duality.
Linearity
M o s t m a t h e m a t i c a l p r o g r a m m i n g is c a r r i e d out over
linea r v e c t o r spaces.
d i m e n s i o n a l space
The usual c hoice is E u c l i d e a n n-
(En ) .
Here a v e c t o r is r e p r e s e n t e d by an
n - t u p l e of real com po ne nt s.
T a k i n g any two vect or s x^ and
X 2 b e l o n g i n g to the space, and real scalars a and 3, a
l inear v e c t o r space has the p r o p e r t y that the linear c o m b i n ­
a ti o n a x ^ + g x 2 b e lo ng s to the space.
fu nc tions, a f u n c t i o n £(x)
scal ar s a and 6 , £(ax+By)
= ail (x) + 0J,(y) .
hav e m a n y im p o r t a n t p r o p er ti es .
co ncave.
In the conte xt of
is said to be linear if for any
L in ea r functions
They are both c on ve x and
Further, a first order T ay l o r series e x p an si on
a p p r o x i m a t e s the linear f u n c t i o n exactly.
The value of
l i n e ar it y to m a t h e m a t i c a l p r o g r a m m i n g is best e x e m p l i f i e d by
the fruitful area of linear p r o g r am mi ng .
Ill
(ii) S e p a r a b i l i t y
A fu nc ti o n f ( x ^ , x 2 » • • • *x n ) is s e p a ra bl e if it can be
wr i t t e n as f (x1 ,x2 ,. . . ,xn ) = f x ( x ^ + f 2 (x2 )+ . . . + f n (xn ) .
This p r o p e r t y is i m p o rt an t as it allows one to treat an ndi me ns i o n a l f u n ct io n as n fu nctions of 1-dim en si on .
This is
m uc h simp le r from a c o m p u t a t i o n a l p o i nt of view.
(iii) C o n v ex it y
A set A is c o n v e x if for any
Xx^ + ( 1 - X ) x 2 e A.
x 2 e A and 0 4: \ 4 1,
To e xt e n d the c on c e p t of a fu n c t i o n f(x)
d ef in ed for x e C, a co n v e x set, we re q u i r e the f ol lo wi ng
d efinition.
The e p i g r a p h of a f un c t i o n f d e f i n e d on C is
the set A d e f i n e d by:
A A {(a,x) | a > f(x) , x £ C } .
f(x)
is a co nv ex f u n ct io n iff A is a co n v e x set.
Convex f u n c ­
tions have a nu m b e r of p o w e r f u l pr o p e r t i e s .
The m o s t i m p o r ­
tant for m a t h e m a t i c a l p r o g r a m s w i t h a c on v e x o bj e c t i v e
fu n c t i o n and a c o n v e x c o n s t r a i n t set is that any local
m i n i m u m is also a gl obal m i ni mu m.
F u rt he r p r o p e r t i e s of
c o n v e x i t y will be g i v e n in S e c t i o n 5.
(iv) D ua l i t y
For a li near space X = E n , l in ea r fu n ct io ns ma y be
w r i t t e n in the form
parameter.
<a,x> = E? ,a.x.,
w h e r e a is a v e c t o r
~ ’~
i= l 1 1 *
The set of all linear f un ct io ns ( p a r a m e t e r i s e d
by the v ec to r a) is a space and is c a l l e d the dual space
(X*) .
The d ua l i t y arises from the fact that <a,x> is
s ymmetric in a and x.
For finite d i m e n s i o n a l spaces, the
set of linear funct io ns on X* turns out to be the o r i gi na l
space X.
Hen ce there is a s y m m etr ic r e l a t i o n s h i p b e t w e e n X
and X * .
Of i m po rt an ce to m a t h e m a t i c a l p r o g r a m m i n g is the p a i r ­
ing of sub- sp ac es in p ri m a l and dual spaces wit h r e sp ec t to
j.
the inner product.
Let X be a s u b sp ace of X.
T h en X =
{y | <y,x> = 0, V x e X} is ca ll e d the o r t h o g o n a l c o m p l e m e n ­
tary su bs pa ce of X.
Any p o i n t in X
poi n t in X by c o n s t r u c t i o n .
If
is o r t h o g o n a l to any
j.
£ x anc* a »6 are
112
scalars, it is eas y to see that a y^ +B }^ e * •
The i m p o r t a n c e of the c on cepts of l in earity, s e p a r a b ­
ility, c o n v e x i t y and d u a l i t y in the d e v e l o p m e n t of g eo me tr ic
p r o g r a m m i n g will re m a i n o b sc ur e in Sec ti on s 2, 3 and 4.
Ho wever,
their p a r a m o u n t role will b e c o m e clear in Sec ti on 5
w h e n we i n t r o d u c e G e n e r a l i s e d G e o m e t r i c Progra mm in g.
S ection
6 uses this g e n e r a l i s e d t he o r y to re - d e r i v e the theory of
po s y n o m i a l pr o g r a m s ,
and the cent ra l role p l a y e d by these
co n ce pt s w i ll be f u r th er cl arified.
T h r o u g h o u t this survey,
th eo ry w ill be r e i n f o r c e d by e x am ple s drawn,
fr o m the a r ea of i n v e n t o r y theory.
in the main,
The e x a mp le s p r e s e n t e d
are r e l a t i v e l y simp le in order to h i g h l i g h t the use of
g e o m e t r i c p r o g r a m m i n g theory.
2.
U n c o n s t r a i n e d Po s y n o m i a l P r o g r a m m i n g
Here we c o n s i d e r the f o l lo wi ng n o n l i n e a r problem:
M i n i m i z e g(t) =
where
n
Z ui= l 1
(1)
, i=l, .. ., n, are p o s y n o m i a l s d ef i n e d by
m
a ..
u, = c.
n t.
1 j=i J
(2)
c. > 0, Vi, t. > 0, Vj , and a-, are a rb i t r a r y real conl
J
t
stants; t = ( t . , . .., tm ) and T den ot es a transpose.
In the
usu al s it ua ti on , g(t) r ep re s e n t s a c o m p os it e cost m ade up of
c o m p o n e n t costs u . , i = l,...,n.
We term this the primal
problem.
In pr i n c i p l e ,
the pr im al p r o b l e m cou ld be s olved by the
m e t h o d s of d i f f e r e n t i a l calculus.
This will give rise to a
set of n o n l i n e a r e qu a t i o n s which,
in general, are di ff ic ul t
to solve.
Hence,
in p ra ct ic e, one w o u l d i n v a r i ab ly resort
to a n u m e r i c a l s o l u t i o n by an itera ti ve d e s c e n t - t y p e method.
Here, we p ro p o s e to c o n s t r u c t a g e o m e t r i c p r o g r a m m i n g dual
p r o g r a m to the primal.
In c e r t a i n cases, this admits a
trivi al s ol u t i o n to the problem.
Further,
this dual
113
pr o b l e m has an in t e r e s t i n g and i n f o r m a t i v e p r a c t i c a l i n t e r ­
pretation.
A central idea here is the a r i t h m e t i c - g e o m e t r i c m e a n
inequality, h e n c e f o r t h t ermed the g e o m e t r i c i n e q u a l i t y .
In
fact, the name "G e o m e t r i c P r o g r a m m i n g " comes f rom the i m p o r ­
tance of this in e q u a l i t y to the orig in al theory.
It states
that
n
n
6.
£ fi.v. * n (v.)
i=l 1 1
i“l
(3)
n
I 6. B 1, 6- » 0, Vi, v. > 0, Vi.
E q u a l i t y is attaini=l 1
1
1
ed w h e n v, = v- = ... * v . It is c o n v e n i e n t to set
1
L
n
Vi =
^i in e q u a t i o n (3) to ob ta i n
where
n
n
6.
Z u. > n (u./6.)
i=l 1
i=l
(4)
In this form, we c an u se the g e o m e t r i c i n e q u a l i t y to
o bt ai n a lower b o u nd on our p ri m a l o p t i m i z a t i o n pr oblem.
Hence, s u b s t i t u t i n g e q u a t i o n
(2) into
(1) and u s i n g (4), we
have that
g (t) *
n (C./5.)
i=i
6i
1 1
If we choos e the w e i g hts ,
n
Z a. .6. = 0,
i=l 1J 1
m
n t.
(5)
j-i J
5., Vi, such that
j = l , . . . ,m
(6)
then the va r i a b l e s t ., j = l , . . . , m on the right h an d side of
in eq ua l i t y (5) m ay be el im in at ed .
In this case, we have that
g(t) >
where v(6) is
n
6n (c •/ 6 •)
= v(6)
i= l
1
(7)
the dual fu n c t i o n w h i c h gives a lower b ou n d
on the m i n i m u m of g(t).
E q u a t i o n (7) i mplies that
m i n g (t) ^ m a x v(<5)
(8 )
114
u n d e r c on d i t i o n s t > 0, 6 % 0,
=
= 1 anc^ e q u a t i o n (6).
It m ay be s h o w n that there exists an opt im al t* (in the
sense of m i n i m i z i n g g(t)) and an o p ti ma l 6* (in the sense of
m a x i m i z i n g v(6))
to s a t i s f y i n e q u a l i t y
(8) at equality.
In
this case the r e l a t i o n b e t w e e n t* and 6* is giv en by
<5i* = u . ( t * ) / g ( t * )
(9)
He n c e the f o l l o w i n g dual p r o g r a m to the or iginal primal
p r o g r a m m a y be c on st r u c t e d :
n
6i
II (c./6-)
i=l 1 1
M a x i m i z e v(6) =
(10)
su bj e c t to the n o r m a l i s a t i o n c o n d i t i o n
n
n .
» i,
6 - => o
i= l 1
(ii)
1
and the o r t h o g o n a l i t y c o n d i t i o n
n
I a 6
i= 1 J
At o p t i m a l i t y g(t*)
= 0,
j = l , . . . ,m
(12)
= v(<5*).
F ro m e q u a t i o n (9), we see that we can i nt er pr et the
dual v a r i a b l e s at op ti ma l i t y ,
6 ^ , Vi, as the rel a ti ve c o n ­
t r i b u t i o n of each c o m p o n e n t cost u^ to the c o m p o si te cost
g(t).
Further, we ob t a i n the f o l lo wi ng r el a t i o n s h i p b e t we en
the p ri ma l and dual v a r i a bl es
In (6.*v(5*)/c.)
=
m
Z a., log t.*
j=l 1J
J
(13)
This is a s ys t e m of linear e q uat io ns in log tj ,
j = l , . . . , m w h i c h are r e a d i l y s o lv ab le once the dual p r o g r a m
has b e e n solved.
We note that the dual p r o g r a m m a x i m i z e s a n o n l in ea r
fu n c t i o n s u bj ec t to line ar e q u a l i t y const r ai nt s.
A dual
v a r i a b l e 6^ is a s s o c i a t e d w i t h each term i= l , . . . , n in the
pr i m a l f or mu lation.
In the case that the n um be r of terms in
115
the pr im al n is equal to the n u m b e r of v a r i a b l e s in the
prima l m plus one, i.e., n=m+l, the li ne ar c o n s t r a i n t s admit
a un iq ue so l u t i o n 6 * and the o p t i m i z a t i o n p r o b l e m in the
dual is trivial.
The q u a n t i t y n-(m+l)
is te rm ed the de gr ee
of d i f f i c u l t y of a g e o m e t r i c p r o g r a m and is, in some sense,
a m ea s u r e of the c o m p u t a t i o n a l c o m p l e x i t y of the dual
program.
E xample
We c on s i d e r the wel l k n ow n " E c on om ic Lot Size" p r o b l e m
from in v e n t o r y theory.
Items are w i t h d r a w n c o n t i n u o u s l y
from i n v e nt or y at a k no w n c o n s t a n t rate a.
Items are o r d e r ­
ed in equal nu mbers, Q, at a time and p r o d u c t i o n is i n s t a n ­
taneous.
The p r o b l e m is to d et e r m i n e h o w o f t e n to m a ke a
p r o d u c t i o n run and h o w m u c h to order, Q, each time to
m i n i m i z e the cost, C, p er u n i t time, w h e r e C * aK/Q + hQ/2
+ ac, and K is the fixed se t-u p cost, h is the i nv e n t o r y
ho l d i n g cost per item per u ni t time, and c is the v a r i a b l e
p r o d u c t i o n cost per item.
N e g l e c t i n g the c o n s t a n t part, ac,
we r eq u i r e to
minimize
* aK/Q ♦ hQ/2
This is an u n c o n s t r a i n e d p o s y n o m i a l p ro gram.
c o r r e s p o n d i n g dual p r o gra m,
from e qu a t i o n s
(1 0 ),
The
(1 1 ) and
(1 2 ), is to m a x i m i z e
6
(aK/6 ^
6
1 (h / 2 6 2 ) 2
s ub je ct to 5^ + 62 = 1
-6i
62 = 0
61 ^ 0,
(normality)
( or thogonality)
62 > 0
Here the c o n s t r a i n t eq u a t i o n s can be u n i q u e l y s ol v e d to
yie ld 6-^* = 62* = h and there is no m a x i m i z a t i o n probl e m. We
have a p r o g r a m w i t h zero d egr ee of d i ff ic u lt y.
The i n t e r ­
p r e t a t i o n of this re su lt is that at op t i m a l i t y , the se t-up
116
cost per unit time and the h o l d i n g cost per un i t time c o n t r i ­
b ute equal a mounts to the op t i m a l cost,
i.e., the optimal
d i s t r i b u t i o n of cost is an i nv a r i a n t w i t h r es pe ct to the
c os t c o e f f i c i e n t s .
Hence, from e q u ati on s
(10) and (13), the
op t i m a l cost is / 2 a K h , and the o pt im al ord e r q u a n t i t y is
JTaTTK.
S up po se n o w that the p r o d u c t i o n pr o c e s s is a lengthy
one so that the a s s u m p t i o n of i n s t a n t a n e o u s a v a i l a b i l i t y is
no lo ng er a c c e pt ab le .
F u rth e r m o r e ,
the size of the lot d e ­
t e r m in es the l e n g t h of the p r o d u c t i o n pr oc es s, and this in
t ur n d e t e r m i n e s the i n - p roc es s i n v e n t o r y ho l d i n g cost.
In
this case a m o d i f i e d e co n o m i c lot size mod el could be to
minimize
= aK/Q + hQ/2 + IQT/2, whe re I is i n- process i n ­
v e n t o r y h o l d i n g cos t pe r unit time, and T is the length of
the p r o d u c t i o n p r oc es s, g i ve n by T(Q) = nQ + m, wh er e n and
m are e m p i r i c a l l y d e t e r m i n e d con st an ts .
In this case, we
h a v e the u n c o n s t r a i n e d p o s y n o m i a l program:
Minimize
= aK/Q + Q( h/ 2+ I m / 2 )
+ In Q^/2
T he c o r r e s p o n d i n g dual p r o g r a m is given by
6
Maximize
6
6
( a K/ 6 ^) * ( ( h + Im )/ 2 6 2 ) 2 (m/2 6 3 ) 3
subj ec t to
6i + 6 2 + 63 = i
-61 + 62 + 26j = 0
6j_ > 0 , 6 2 > 0 , 63 > 0
Here we have two e q u a t io ns and three un knowns;
e ffect, a d eg r e e of d i f f i c u l t y of one.
in
One could p r o c e e d to
solve this p r o b l e m as an u n c o n s t r a i n e d m a x i m i z a t i o n p r o b l e m
in one v a r i a b l e or as a c o n s t r a i n e d ma xi mi z a t i o n .
However,
i ns i g h t m ay be g a i n e d w i t h o u t going into this detail.
Mani­
p u l a t i o n of the c o n s t r a i n t e qu a t i o n s gives the follow in g i n ­
e q u a li ti es :
0 .5 $ <5^ s 1. 0 , and 0 . 3 3 ^ 6 ^ ^ 0 . 6 6 ,
that at op ti ma l i t y ,
w hi c h imply
the set-up cost makes a c o n t r i b u t i o n
117
b e t w e e n 50% and 66% to the o pt im al c o m p o s i t e cost.
Si m i l a r
insights m ay be o b ta in ed from the o th e r dual vari ab le s.
3.
C o n s t r a i n e d P o syn o m i a l P r o g r a m m i n g
Here we c o n s i d e r the m i n i m i z a t i o n of a p o s y n o m i a l form
subject to c o n s t r a i n t s w h ic h are also p o s y n o m i a l s .
This is
the p rimal formulation.
M i n i m i z e gg(t) s ubj ec t to gk (t) $ 1,
m
,P (14,15)
1,... ,m
t. > 0,
J
J , k-Q,.
a. .
a..
wh er e g,.(t)
c.
n t.
g k (j) ■ I
ie [k]
j-1 J
k - 1 ,..
(16)
(17)
k - 0 f..
and
(18)
m Q -l, m 1 - n 0 + l i . . . tm pp -npp-1
_1 +l,’ n np -n
{ lk]}, k * 0 , l , . . . , p is a s eq u e n t i a l p a r t i t i o n o f the in tegers
1 to n.
As be fo r e a^j are a r b i t r a r y real e xp o n e n t s and the
c o e f f i c i e n t s c^ are posit iv e.
In o rd e r to h a n d l e c o n s t r a i n t s , we n ee d a g e n e r a l i s a t ­
ion of the ge om et ri c i n e q u a l i t y in w h i c h the <5^'s are not
no rm al is ed .
We let
(19)
X* = ie
-Z [k]
rv,4 i
for a p a r t i c u l a r c o n s t r a i n t k.
Hen ce the g e o m e t r i c i n e q u a l ­
ity, e q u a t i o n (4) m a y be w r i t t e n in a c o n v e n i e n t f orm as
(2 0 )
(I
u.) k > n
(u./6.) 1 X,
ie[k]
ie[k]
Co m b i n i n g e q u a ti on s
(15),
A-*
i » g k (t) k *
(17) and (20) we have that
in
n
(c.
ietk]
a. .
n t. 1J/6±)
j=l J
6.
\
t
(2 1 )
for any c o n s t r a i n t k =l ,. .. ,p .
For the o b j e c t i v e f unction,
we no r m a l i s e the 5^, i e [0].
Hence
118
g n (t) >
n
(c
ietO]
and
n
a ..
6.
II t. XJ/6 -) 1
1 j=l J
1
(22)
z
6-:=l
i e [0]
C o m b i n i n g res u lts
(23)
(21) and (22), we o b t a i n the
i ne q u a l i t y
n
g0 Ct)
n (C./5.)
i=l
6. p
n X.
X.
(24)
k-1 1
Hence, u si n g the same lines of r e a s o n i n g as for the u n ­
c o n s t r a i n e d pr o b l e m ,
the dual p r o b l e m is g i v en by:
M a x i m i z e v(6) =
n
6. p
X,
n (c./6.)
II X.
i=l
1 1
k=l
(25)
s u b j e c t to the li n e a r co n s t r a i n t s
Z
6- = 1
ie [0]
n
Z a . .6- = 0,
i=l
1
5i * 0,
(normalisation)
(26)
j = l , . . . , m ( o r t h o g o n a l i t y ) (27)
i-l,...,n
(28)
We n ote that the p r i m a l p r o b l e m is a h i g h l y n on l i n e a r
m a t h e m a t i c a l p r og ra m, wh e r e a s the dual is the m a x i m i z a t i o n
of a co nc a v e f u n c t i o n s ub je ct to linear c on st ra in ts .
again,
if n=m+ l,
Once
the p r o g r a m has zero de gr ee of d if ficulty,
and the dual p r o g r a m is trivial.
It ma y be shown
[13] that
at o p t i m a l i t y the p ri ma l v ar i a b l e s t* and the dual varia b le s
6* are r e l a t e d by
c,
m
a ..
n t.* 1;) =
1 j=rj
6^*v(6*),
i e [0]
l<Si * /X k (6*) , i e [k], Xk (6* )> 0 ,k -l ,. .. ,p
(2 9 )
Hence 6 ^ , i e [0] gives the re la ti ve c o n t r i b u t i o n of
each c om p o n e n t c ost to the c o m p o s i t e cost.
o p t im al it y,
Also at
the pr im al and dual o bj e c t i v e f u nctions are
119
equal in value,
i.e., g Q (t*) * v(6*).
Example
We c o n si de r an Econo mic Lot Size model w i t h m u l t i ­
pro du ct s and a res ou rc e const ra in t.
In this case we have
the f ol lo wi ng c o n s t r a i n e d p o s y n o m i a l program:
N
Z (a.K./Q. + h.Q./2)
i=l
1 1 1
11
M in i m i z e
subject to a res ou rc e c on s t r a i n t
N
Z b.Q. * W
i= l 1 1
wh er e b^ and W are g i ve n cons ta nt s.
From e qu a t i o n s
(25) to
(28), we find the c o r r e s p o n d i n g dual program:
N
n (a-K./6-)
Maximize
i=l
s ubject to
1
6.
2N
6.
n
(h-/26.)
i-N+1
1
1
1
3N
(b-/W6.)
i=2N+l
1
1
XA
Z 5- = 1
i*1
3N
X -
Z
<5i-2N+l
’6 i + 6N + i + 6 2N+i " °*
6^ £ 0,
4.
6. .
n
»• • • »N
i=l,...,3N
Signo mi al P r o g r a m m i n g
While p o s y n o m i a l p r o g r a m m i n g has found a p p l i c a t i o n in a
v a r i e t y of areas, m a n y p r o b l e m s of i n t e re st h ave f al le n o u t ­
side the po s y n o m i a l form.
O f t e n it was the p o s i t i v i t y c o n ­
d i t i o n on the c o e f f i c i e n t s w h i c h was vi ol at ed .
H o we ve r,
the
b e n e fi ts of p os y n o m i a l p r o g r a m m i n g w ere too te m p t i n g to be
p a s s e d up.
Passy and W i l de
[32] i n t r o d u c e d a g e n e r a l i s a t i o n
of p o s y n o m i a l p r o g r a m m i n g , w h e r e the c o e f f i c i e n t s of the
terms wer e not r e q u i r e d to be p os it iv e.
grams is c a l l e d si gn omial.
This class of p r o ­
In s i g n om ia l p r o g r a m m i n g , a
global m i n i m u m c an n o t be g u a r an te ed .
Ho we ve r ,
ties of the dual s u b s p a c e are ma in ta i n e d .
the p r o p e r ­
120
D u f f i n and P e t e r s o n
[11,12] have d e v e l o p e d m e th od s for
the a na l y s i s of si g n o m i a l s and an a l g o r i t h m for solving them
(at least for a local s olution,
if not for a global solution).
To use the an al ysi s, s i g n o m i a l p r o g r a m s must first be c o n ­
v e r t e d into p o s y n o m i a l p r o g r am s.
An y s ignomial f(t) can be
w r i t t e n as a d i f f e r e n c e of two p o s y n o m i a l s , say r(t) and
s(t) and h ence
f(t) j? 1 -<-+r(t)-s(t) «: 1 +-*■r (t) s s (t)+l
f(t) > 1 +-*-r(t)-s(t) > 1 +-*• r (t) » s ( t ) +1
f(t) $ 0 +-vr(t)-s(t)
0 +-*■r (t) ^ s(t)
Since r(t), s(t) and s(t)+l are all n o n - n e g a t i v e for
all v al u e s of t, in each case a n e w v a ri ab le x can be i n t r o ­
d u ce d so that
r(t) H v < s (t)+l «-+ r (t ) / x < 1
and
l * s ( t ) / x + l/x
r (t) > t » s (t) +1 -*->• r (t) / x >,1
and
1 ^ s ( t ) / x + l/x
r (t) ^ x ^ s (t)
and
1
*-*■ r ( t ) / x ^ 1
s(t)/x
T he s e t r a n s f o r m a t i o n s on a s ig no mi al p ro d u c e a p o s y n o m ­
ial p r o g r a m w i t h r e v e r s e d c o n s tr ai nt s:
(30)
M i n i m i z e gQ(t)
su b j e c t to
S 1
g k (t) > 1
k=l,...,p
(31)
k=p + l , . . . ,r
(32)
t > Q
w h e r e the f u n ct io ns g^(t) are p o s y n o m i a l s d ef in ed by
e q u a t i o n (17).
The d i f f e r e n c e in this f o r m u l a t i o n is the
reversed constraints
(32).
W h i le the c on s t r a i n t s
(31) d e ­
fine a c o n v e x region, the r e v e r s e d c o n s t r a i n t s make the
re gi o n n o n - c o n v e x .
by
The dual to the pr ima l p r o g r a m def in ed
(30) to (33) is giv en by:
121
6.
Maximize v(S) = II (c./6.)
ie[0]
b
6,
n
n
(c.X,/6.) K
IT
II (c.X,/<5.) 1
k=l ie[k] 1 K 1
k*p+l ie[k]
(34)
subj ec t to same c o n s t r a i n t s as for p o s y n o m i a l s , n a m e l y
E
6. = 1
ie [0]
Ak "
E
(35)
6i»
n
iElk>
E a . .6• - o ,
13
i= l
6^ i 0,
k-l,...,r
(36)
j = l , . . . ,m
(37)
i « l , . . . ,n
(38)
We note that log v(6) is c o n c a v e in the v a r i a b l e s a s s o c ­
i ated w i t h the r eg u l a r c o n s t r a i n t s k * l , . . . , p and the o b j e c t ­
ive function.
Ho we ve r, it is c o n v e x in the v a r i a b l e s a s s o c ­
iated w it h the r e v e r s e d c o n s t r a i n t s k ap + l ,... , r .
In the
p o s y n o m i a l case, the dual p r o g r a m r e q u i r e d finding the m a x ­
imum of a con ca ve f u n c t i o n s ubj ec t to li ne ar c o n s tr ai nt s.
Here we are look in g for the m a x i m u m of a n u m b e r of saddle
point s of a c o n c a v e - c o n v e x f u n c t i o n s ub je ct to linear c o n ­
straints.
If the p r o g r a m has de gr ee of d i f f i c u l t y of zero,
it is p o s s i b l e that a s o l u t i o n can r e a d i l y be f o u nd but in
the g eneral case,
the p r o g r a m is very d i f f i c u l t to solve.
or de r to cope w i t h this p ro b l e m , D uf f i n and P e t e r s o n
In
[11]
apply the a r i t h m e t i c - h a r m o n i c m e a n i n e q u a l i t y to c o n v e r t the
r e v e r s e d co n s t r a i n t s into r e g u l a r c o n s tr ai nt s.
This a p p r o x ­
ima ti on has the a d v a n t a g e that the l i near c o n s t r a i n t s in the
dual are the same for all such a p p r o x i m a t i o n s .
As well, the
sequ en ce of all r eg ul ar p o s y n o m i a l p r o g r a m s g e n e r a t e d is
m o n o t o n e d e c r e a s i n g in the v al u e of the o b j e c t i v e function.
Thus one can onl y improve any f e a si bl e solution.
The a r i t h m e t i c - h a r m o n i c in eq ua li ty , g i v en p a r a m e t e r s
a •, i e [ k ] , p o s i t i v e and
( E
u .) 1
ie[k]
E
a- = 1, is
i£[k]a
n
(a./u,) 1 «
E
a i 2/u-j
i e[k]
1
ie[k]
(39)
122
with equality holding when
d; = U. / ( Z
U.)
ietk] 1
The m e a ns in e q u a t i o n
(40)
(30) are the ha rm on ic , ge om et ri c and
arithmetic respectively.
C o n s i d e r any r ev e r s e d c on st r a i n t
g k (t) > 1
or
(gk ( t ) ) _1 ^
1
2
We set g v (t,ct)
Z
a- / u - . This is a p os yn o m i a l for
k ~ ~
ie [ k] 1
1
fi xe d a*
We note that g k (t,a) ^ 1 implies that gk (t) > 1.
U s i n g e q u a t i o n (40) for any t such that g k (t)
g e n e r a t e an a such that gk (t,a) $ 1.
1, one can
Hence, by use of the
a r i t h m e t i c - h a r m o n i c in eq ua l i t y , we m a y reduce a p os yn o m i a l
p r o g r a m w i t h r e v e r s e d c o n s t r a i n t s to a re g u l a r po sy no m i a l
p r o g r a m p a r a m e t e r i s e d w i t h r es p e c t to a, viz.
M i n i m i z e g0 C't)
s u b j e c t to
Sk*--) < 1 »
k = l ,. . . ,p
1,
(42)
k - p + 1 ,...,r
t > 0
(43)
(44)
To o b t a i n a m o n o t o n e d e c r e a s i n g s e q ue nc e of pr ograms,
we solve the p r o g r a m (41) to (44) for a p a r t i c u l a r a and
the n reset the a's u s i n g e q u a t i o n
(40).
E xa mp le
We r e t u r n to the m u l t i p r o d u c t eco no mi c lot size p ro b l e m
g i v e n in S e c t i o n 3.
t r i b u t i o n effort.
Here we add c o n s i d e r a t i o n of the d i s ­
Sales are n ow a f u n ct io n of the a l l o c a t ­
ion of e ff or t to d i s t r i b u t i o n , and we seek to m a x im i ze
prof it .
The p r o b l e m becomes:
Maximize
s ub j e c t to
N
d.
d.
Z C(Pi -ci ) a iD i 1 -D i -K ia iD i 1/ Q i 'hiQ i/3)
N
Z b.Q. s W
i= l 1 1
123
Q. > 0,
D i * 0,
Vi
Heje Pj*c^ is the sales p r i ce minus the cost for p r o d u c t
i,
of
1 are the sales g e n e r a t e d by a d i s t r i b u t i o n e ffort
and a^ and d^ are p a ra me te rs .
b ee n d e f in ed p r e v io us ly .
All ot h e r s ymbols have
This is a signo mi a l p r o g r a m and is
e q u i v a l e n t to the fo llo wi ng program:
Minimize
1/P
N
Z b.Q./W < 1
i= 1 1 1
N
d.
d.
.^1 (Pi -ci ) a iD i 1 - D. - K i a iD i 1/ Q i - h . Q i/2 >. P
subje ct to
Q i » 0,
> 0,
(45)
Vi
P is a n ew v a r i a b l e in d i c a t i n g p rofit.
T a k i n g the
ne g a t i v e terms in e q u a t i o n (45) to the right hand side and
i nserting a n e w v a r i a b l e R b e t w e e n the sides of the i n e q u a l ­
ity as in the theory, we o b t a i n the program:
Minimize
1/P
N
E b-Q./W < 1
i*l 1 1
d.
P/R ♦ I ( D i/R + K ia iD i 1/(Q.R) ♦ h . Q. /( 2 R) ) «: 1
s ub je ct to
d.
E ( p i -ci ) a iD i X/R ^ 1
i
Q i ^ 0,
Di * 0
(46)
Vi
Our or ig in al si gn om ia l p r o g r a m is n o w in the form of a
po s y n o m i a l p r o g r a m w i t h one r e v e r s e d c o n s t r a i n t
(46) .
This
ma y be h a r m o n i s e d to beco me
?
d.
Z a i ZR / ( ( p i - c i ) a iD i x ) « 1
(47)
where the ct^ > 0 (and Za^ = 1) are p a r a m e t e r s to be varied.
To find a local solu ti on , we take a f e as ib le set of
124
d i s t r i b u t i o n e f fo rt s
to ev a l u a t e ou w h i c h is gi ve n by
(see e q u a t i o n (40))
= (pi -ci)a.D.
d.
d.
1/(i:(Pi -c i) a iD i x ) , V i
i
(48)
We n o w solve the p o s y n o m i a l p r o g r a m wi t h e qu a t i o n (46)
r e p l a c e d by (47).
we reset
W i t h the r e s u l t i n g v a lu e of
f ro m e q u a t i o n
this n e w v a l u e of a^.
obtained,
(48) and re s o l v e the p r o g r a m w i t h
We r epeat this p r o c e d u r e until there
is no s i g n i f i c a n t ch a n g e in the solution.
5.
Generalised Geometric Programming
G e n e r a l i s e d g e o m e t r i c p r o g r a m m i n g deals w it h the a n a l y ­
sis of c o n v e x m a t h e m a t i c a l progra ms .
C o m m o n l y these appear
in the form:
Minimize
s u bj ect to g^(z) ^ 0 ,
i e I
z e C
(51)
w h e r e C is a c on v e x set and g^, i e {0} U
f unctions.
(50)
I are convex
In the g e n e r a l i s e d g e o m e tr ic p r o g r a m m i n g
ap proach, we first s e p ar at e the ar gu me nt s of the con st ra in ts
and o b j e c t i v e functions.
We i nt ro du ce the following
notation:
z = x° = x*,
X = X^
V i £ I
X X* £ X.
i£l
X A {x|x° = x * , V i e I}, a su bspace of X
C. = C, for i e {0} U
The p r o g r a m d e f i n e d by e qu at io ns
m ay no w be r e w r i t t e n as:
I
(49) , (50) and (51)
125
Minimize
go(*°)
sub je ct to g ^ C x 1 ) $ 0
x1 e Ci
i e I
(53)
i e {0} U I
(54)
x e X
(55)
This f o r m u l a t i o n s e p a ra te s the c o n s t r a i n t s and the o b ­
ject iv e fu nc ti on ex c e p t i n g for the su b s p a c e con di ti on ,
e q ua ti on (55) w h i c h ties the p r o b l e m together.
The s u b ­
space ca pt ur es the l i n ear it y of the problem.
At this stage we r e q ui re f u rt he r ideas r e l at in g to c o n ­
ve xi t y .
Recall from S e c t i o n l(iii), the d e f i n i t i o n of an
ep ig r aph to a pai r
conve x set C.
[g,C] of a c o n v e x f u n ct io n g d e f i n e d on a
Each n o n v e r t i c a l h y p e r p l a n e that supp or ts the
e pi g r a p h of a c o n v e x f un c t i o n g at a b o u n d a r y poi nt
(x',g(x')) p ro d u c e s a s u b g r a d i e n t of g at x', i.e., a v e c t o r
£ e Rn w ith
g(x')
♦ <£,x-x’
> « g(x) ,
V x e C
(56)
The s u b g r a d i e n t set 3g(x') that c o n sis t s of all such
v ec to rs ^ is g e n e r a l l y a c l o s e d c on v e x s ubset of E n w h i c h
c on t a i n s only a single v ec t o r iff g is d i f f e r e n t i a b l e at x'.
In this case the single v e c t o r is the usual g r a d i e n t v e c t o r
Vg(x') .
A c on v ex f u n c t i o n
[g,C]
ep i g r a p h is a cl o s e d set.
is said to be c l o s e d if its
We shall a ss um e that all functions
are closed.
The c o n j u g a t e t r a n s f o r m
f u n ct io n
[h,D] of an a r b i t r a r y c on v e x
[g, C] is d ef i n e d by:
h(y) = sup (<y,x> - g (x ))
xeC
and
D =* (y|sup (<y,x> - g(x))
xeC
We note that h(y) = <y,x'> - g(x')
By c o n s t r u c t i o n
[h,D]
fined on a c onvex set.
(57)
< +<=°}
(58)
for each y e 8g(x') .
is a cl os ed co nv e x f un c t i o n d e ­
A f ur t h e r c o n s e q u e n c e of the
126
c o n j u g a t e t r a n s f o r m is the c o n j u g a t e i n e q u a l i t y w h i c h states
that
g(x) + h(y) > < x , p
for x e C and y
ently x
e
e
(59)
D, w i t h e q u a l i t y iff y
8g(x) or e q u i v a l ­
e
3h(y).
To h an d l e c o n s t r a i n t s we r eq ui re the c o n j u g a t e t r a n s ­
form of a p a r t i c u l a r f u n c t i o n
[0,g(x) < 0, x e C].
t r a n s f o r m is r e l a t e d to the t r a n s f o r m of
the p o s i t i v e h o m o g e n e o u s e x t e n s i o n of [h,D].
d e n o t e d by
This
[g,C] and is c alled
T his is
[h+ ,D+ ] w h e r e
h (y,X)
=
[Ah(y/A) ,
^
"
[sup <y,x>,
X > 0
(60)
X = 0
xeC
and
D + = { (y ,X) |y/XeD ,X>0} U { (y ,0) |sup<y ,x> < <=°
}
xeC
(61)
T he c o n j u g a t e i n e q u a l i t y in this case is g i v en by
0 + h + ( y ,X) $ <x,y>
for (y,X)
x
e
e
(62)
D + and x e C w i t h e q u a l i t y iff y
e
X 3g(x) or
9h + (y,X) .
We n o w a p p ly these ideas of c o n j u g a t e tr a n s f o r m theory
to the p r o g r a m d e f i n e d by eq u a t i o n s
(52) to (55) .
p r o g r a m is the pr imal m a t h e m a t i c a l program.
tive fu nc ti on , we have from eq u a t i o n
This
For the o b j e c ­
(59) that
r 0.
V r O', ^ . 0 0^
g 0 (x ) + h 0 (y )
<x ,y >
,,,,
(63)
and for eac h c o ns tr ai nt , f rom e q u a t i o n (62), we have that
0 + h i+ (y1 ,Xi ) » <x 1 ,y1 > ,
A dd i n g the in equ a l i t i e s
g 0 (x°) + h 0 (y°) +
i £ I
(64)
(63) and (64), we have that
Z h i+ (y1 ,^i ) » <x,y>
i£l
(65)
127
wh e re x = x°
X
x 1 a n d y = y°
and y e X , e d i t i o n
X
y1*
By r e s t r i c t i n g x e X
(65) b e c o m i i 1
g 0 (x°) ♦ h 0 (y°) + Z h. + (y1 ,X .) j» 0
u '
u *
ie I
1
(66)
w i t h e q u a l i t y w hen
y° e 3 g 0 (x°)
and
or
x° e 8h0 (y°)
(67)
y 1 e A - ^ g ^ x 1 ) or x 1 e a h / (y1 ,Aj) , i e I
(68)
Hence the g eo m e t r i c p r o g r a m m i n g dual to the prima l
p r o g r a m (equations
(52) to (55)) is g i v e n by:
Minimize
hg(y°) +
E h . + (y*,A.)
ie I
s ub je ct to y° e D g , (y1 , ^ )
x
(69)
e D i+ » * e 1 (^0)
y e x
The dual p r o g r a m ca n have c e r t a i n a dv a n t a g e s o ve r the
or i ginal prim al program:
(i)
The n o n l i n e a r c o n s t r a i n t s are i n c o r p o r a t e d into the
ob j e c t i v e function.
(ii)
If X is of d i m e n s i o n n and X is of d i m e n s i o n m, then
i
X , the o rt h o g o n a l c o m p l e m e n t of X, is of d i m e n s i o n
n-m.
Wit h the p o s s i b i l i t y of n - m b e in g m u c h sm a l l e r
than m and the e l i m i n a t i o n of e x p l i c i t n o n - l i n e a r
c o n s t ra in ts ,
the dual may turn out to be a m u ch
simpl er p r o b l e m to solve.
(iii)
The fact that the pr i m a l and dual ob j e c t i v e s sum to
zero at o pt im a l i t y ,
+
(y1 »
(k
.
,
(k
= 0, p r ov ide s a good sto pp in g
c r i t e r i o n for an algo rit hm .
t i m a li ty are
,
i.e., g Q (x ) + hp(y )
The c o n d i t i o n s for o p ­
128
g i U i ) * 0, x 1 e C i
e D i+ *
X E X
i e I
y e X
x° £ 3 h 0 (y°)
or y
x1 e
Or y 1 e X i 3 g i (x1 ) , i e I
9 h i (y1 ,Xi)
e
8 g 0 (x )
Fro m these c o n d i t i o n s one m a y c a l c u l a t e the optimal
s o l u t i o n for one p r o b l e m from the o pt im al sol ut io n of
the other.
It is i m p o r t a n t to note the roles pl a y e d by the four
c o n c ep ts of lin ear it y,
s e pa ra bi li ty , c o n v e x i t y and dual it y
in g e n e r a l i s e d g e o m e t r i c p r o g r a m m i n g theory.
L i n e a r i t y may
o c cu r n a t u r a l l y in the p r o b lem , e.g., as linear c on st raints,
or it m a y be i nd uc ed by the n e ed to se pa ra te the va r ia bl es
as in the p r o g r a m at the b e g i n n i n g of this section.
Any
l i n e a r i t y is u s u a l l y c o n v e n i e n t l y c a p t u r e d in the subspace
co n d i t i o n , e q u a t i o n (55).
S e p a r a b i l i t y is n e c e s s a r y to f a c ­
i li ta te a s im pl e c o m p u t a t i o n of c on j u g a t e tr ansforms.
vexity
Con­
(and closure) g u a r a n t e e s that there is no d u a li ty gap,
i.e., the pr im al and dual o bj e c t i v e s sum to zero at o p t i m a l ­
ity.
D u a l i t y is the goal of g e n e r a l i s e d g e o m et ri c p r o g r a m ­
m i n g theory.
Ex a m p l e
A well k n o w n p r o b l e m in i nv en to ry control is to select
a set {xt ; t=l, .. ., T} of p r o d u c t i o n levels to mi ni mi ze , over
a p l a n n i n g h o r i z o n of lengt h T, the sum of p r o d u c t i o n and
h o l d i n g costs, w h i l s t m e e t i n g demand.
F o r ma ll y the pr o b l e m
m a y be p o s e d a s :
Minimize
T
Z c(x ) + h y
t= l
1
sub j ec t to the i nv e n t o r y b a l a n c e dynamics
(7 2 )
129
y1 - *1 - dj
y t ■ y t .1 = x t -
t = 2 , . . . ,T-1
(73)
-yT .j = x-p - d^,
and the n o n - n e g a t i v i t y c o n s t r a i n t s
y
» 0,
t * l ,... ,T
x t 5 0,
t * l , . . . ,T
(74)
Here y t denot es the in v e n t o r y level in p e r i o d t, d^ is
the de mand in p e r i o d t, c ( x t ) is the p r o d u c t i o n cost ( a s s u m ­
ed c onvex and s t ri ct ly m o n o t o n i c a l l y increasing) and h^ is
the h ol di ng cost per unit in p e r i o d t.
To invoke the theory of g e n e r a l i s e d g e o m e t r i c p r o g r a m ­
ming, we n ee d to put the c o n s t r a i n t eq u a t i o n s
subspace.
(73) into a
Hen ce we i n t r od uc e a n e w v a r i a b l e a^, t =l , ...,T,
and r e s tr ic t it to a one p o i nt d o m a i n {dt >, t=l,...,T. This
v a r i a b l e is then a s s o c i a t e d w i t h an a d d i t i v e c o m p o n e n t of
the o b j e ct iv e f un c t i o n w h i c h is i d e n t i c a l l y zero.
He n c e we
o b t a i n a s u b sp ac e c o n d i t i o n
y l ' X 1 + al “ 0
y t ' y t-l " x t + a t “ °» t = 2 »---»T -1
" y T-l
(7 S )
" X T + aT " ®
It is c o n v e n i e n t to treat the n o n - n e g a t i v i t y c o n s t r a i n t s
x t £ 0, y
> 0, t = l ..... T, in an im pl ic it way,
C q = (x t |xt » 0, y t > 0, d t = a t , t=l,... ,T }.
i.e.,
Our p r o b l e m
is n ow in a form w h ic h is d i r e c t l y s u i ta bl e for a p p l i c a t i o n
of the theory.
We note that in this p r o b l e m there are no
e x p l ic it co n s t r a i n t s as in e q u a t i o n (53) .
ive is g i ve n by
The dual o b j e c t ­
130
sup
E (xtu t + y tv t + a t e t - c ( x t ) - h ty t )
x t ,yt »at t
xt > 0
yt > o
=
sup
E (x u
- c(x )) + sup E a 6
xt ^ 0 t
tt:
1
at t
+
sup E(y v -h y )
y t ^0 t 1 1 1 r
■ I c«(ut) * d t8t
w h e r e c*(u..) =
sup x u
- c(xt )
xt * 0 z r
r
u t e 3c(xt)
and
vt
h
U s u a l l y c ( x t ) is a q u a d r a t i c f u nc ti on and c * ( u t ) is
r e a d i l y c a lc ul at ed .
Furth er we r eq ui re the or t ho g o n a l c o m ­
p l e m e n t to the s u b s p a c e de fi n e d by e q u a ti on s
{ ( u t ,vt ,Bt ) | E ( u tx t + v ty t + 0 ^ )
s a t i s f y i n g eq u a t i o n s
(75)}.
(75), i.e.
= 0, V x t ,yt ,at
A straightforward calculation
shows that
V t = Pt ‘ p t + l ’
u t - ~P ■
£>
t - 1 , ... ,T
Bt = P t ,
t = l , . . . ,T
H e n c e the dual p r o b l e m is
T
Minimize
E c * ( - p t) + d p
t-1
su b j e c t to p^ - P t + -^ - h t ^ 0, t = l,...,T-l
For c o n s t a n t h^, we have a m i n i m i z a t i o n over a monotonicall y i nc r e a s i n g set of d e c i s i o n vari abl es .
131
6.
The A n a l y s i s of Po s y n o m i a l P r o g r a m m i n g by G e n e r a l i s e d
G eo me tr ic P r o g r a m m i n g
Recall that in the p r e c e d i n g s e c t i o n we w er e able to
use the c o n v e x i t y of the funct i ons of a c o n v e x m a t h e m a t i c a l
p r o g r a m by a p p l yi ng s e p a r a b i l i t y and then u s in g l i n ea ri ty
and d ua l i t y to d er i v e a p o t e n t i a l l y s i mp ler p ro blem.
To
br in g this s t r u ct u re out in p o s y n o m i a l p r o g r a m m i n g , we c o n ­
sider the f ollowing c onv ex function:
E 6. log
i e [k ] 1
d ef i n e d on 5- >, 0, i e [k]
1
6./c-
(76)
1
and
E
&■ = 1
Here the p a r a m e t e r s c^ > 0, i e [ k ] .
(77)
ie [k]
The c on j u g a t e
t r a n s f o r m of the f u n ct io n d e f i n e d by (76) t a k en o v e r the set
d ef i n e d by e qu a t i o n s
log
(77) m a y be s hown to be
E c. e x p (:■)
ie [k]
(78)
Hence the c on j u g a t e i n e q u a l i t y from e q u a t i o n (59) is:
£ <5. log 6. /c • ♦ log(
E c. exp ( z .)) >,
E 5.z.
ie[k] 1
1 1
ie[k] 1
1
ie[k] 1 1
(79)
with e q u a l i t y w h e n
6 i ■ c- e x p (z i )/(
E c i e x p ( z i ))
ie [k]
(80)
In or de r to ha nd l e c o n s t r a i n t s of the form
log
E
c- exp(z-) ^ 0
ie[k]
1
(81)
we req ui re the p o s i t i v e h o m o g e n e o u s e x t e n s i o n (see e q u a t i o n
(60)) of f u n c t i o n (76) d e f i n e d over the set
(77).
This is
g i v e n by
E 6 i log
ietk]
(5i/ ( X k c i ))
(82)
132
d e f i n e d for A^ =
£ 6^, 6^ £ 0, i e [k]
ietk] x
(83)
To r el at e f u n c t i o n s of the f o rm (78) to p o s y n o m i a l g e o ­
m e t r i c p r o g r a m s d i s c u s s e d in S e c t i o n 3 we set
Xj = log t . ,
(84)
V j
m
and
(85)
in e q u at io ns
(14) and (15) .
Equation
(85) is the subspace
c o n d i t i o n in the g e n e r a l i s e d t heory (see e q u a t i o n (55)).
M a k i n g the a bo v e s u b s t i t u t i o n and taking logs, our p os yn om ia l
p r o g r a m is as follows:
M i n i m i z e log
s u b j e c t to
log
Z c^ exp(z^)
ie [0]
Z c- exp(z.)
ietk]
1
0,
i = l,...,p
m
In the ab ov e form, the th eo ry of g e n e r a l i s e d geome tr ic
p r o g r a m m i n g m a y be i nvoked since we have a lr e a d y c a l c u l a t e d
the r e l e v a n t c on j u g a t e t ra ns f o r m s
(82) and (83)).
(see e q u at io ns
Hence, u sing e q u a t io ns
(69),
(76),
(77),
(70) and (71),
the dual to a p o s y n o m i a l p r o g r a m is gi v en by
Minimize
n
Z 1 6-1 log
•
i=l
s ub je ct to
6 -/ c 1- 1-
p
Z 1A, 1 log
A,K
K
k=l
Z 6- = l
is [0]
6. » 0,
Vi
n
i= l
The above o b j e c t i v e f u n c t i o n is e qu i v a l e n t to ma x i m i z i n g
133
n
i= l
w h i c h is the o r i gi na l form of the dual o b j e c t i v e f un c t i o n
for p o s y n o m i a l p r o g r a m m i n g
(see e q u ati on s
(25) to (28)).
Furth er it is s t r a i g h t f o r w a r d to o b t a i n the o p t i m a l i t y c o n ­
dit io ns be t w e e n the primal and dual va ri ab l e s , e q u a t i o n (29)
from the general o p t i m a l i t y c on di t i o n s in Se ct i o n 5.
7.
Ap p l i c a t i o n s
In the pr e v i o u s s e c t ion s we have shown the d e v e l o p m e n t
of the theor y of g e o m e t r i c p ro gr a m m i n g .
I n cl ud e d in the
r ef er en ce s are v ar i ous books and pa p e r s d e s c r i b i n g a p p l i c ­
ations of g e o m e t r i c p r o g r a m m i n g .
pa pe r
In the sequel to this
(appearing in the next issue of NZOR) we p r e s e n t in
some detail the m a j o r areas of a p p l i c a t i o n of ge o m e t r i c
programming.
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