Dynamic Linear Programming
Propoi, A.I.
IIASA Working Paper
WP-79-038
May 1979
Propoi, A.I. (1979) Dynamic Linear Programming. IIASA Working Paper. IIASA, Laxenburg, Austria, WP-79-038
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Working Paper
DYNAMIC LINEAR PROGRAMMING
Anatoli Propoi
May 1979
WP-79-38
International Institute for Applied Systems Analysis
A-2361 Laxenburg, Austria
The p a p e r p r e s e n t s a s u r v e y of dynamic l i n e a r programming
including discussion of d i f f e r e n t
f o r m a l i z a t i o n s of dynamic l i n e a r programs, models which c a n b e
w r i t t e n iri such a form, d u a l i t y r e l a t i o n s and o p t i m a l i t y cond i t i o n s f o r DLP, n u m e r i c a l methods - - b o t h f i n i t e - s t e p s and
iterative.
( D L P ) models and methods,
DYNAMIC L I N E A R PROGWL!ING*
A.
Propoi**
INTRODUCTION
Dynamic l i n e a r programming (DLP) c a n b e c o n s i d e r e d a s a n e x t
s t a g e o f l i n e a r programming (LP) development [ I -31
Nowadays
it becomes d i f f i c u l t , even sometimes i m p o s s i b l e , t o make dec i s i o n s i n l a r g e s y s t e m s and n o t t o t a k e i n t o a c c o u n t t h e cons e q u e n c e s o f t h e d e c i s i o n o v e r some p e r i o d o f t i m e .
Thus,
a l m o s t a l l p r o b l e m s o f o p t i m i z a t i o n become dynamic, m u l t i s t a g e
ones.
Examples a r e l o n g - r a n g e e n e r g y , w a t e r and o t h e r r e s o u r c e s
supply models, problems of n a t i o n a l s e t t l e m e n t p l a n n i n g , longr a n g e a g r i c u l t u r e i n v e s t m e n t p r o j e c t s , manpower and e d u c a t i o n a l
planning models, r e s o u r c e s a l l o c a t i o n f o r h e a l t h c a r e , etc.
.
N e w p r o b l e m s r e q u i r e new a p p r o a c h e s .
W i t h i n DLP it i s d i f f i c u l t
t o e x p l o i t o n l y LP i d e a s and methods. While f o r t h e s t a t i c LP
t h e b a s i c q u e s t i o n c o n s i s t s of d e t e r m i n i n g t h e o p t i m a l p r o g r a v ,
t h e i m p l e m e n t a t i o n o f t h i s program ( r e l a t e d t o t h e q u e s t i o n o f
t h e f e e d b a c k c o n t r o l o f s u c h a program, i t s s t a b i l i t y and s e n s i t i v i t y , e t c . ) i s no l e s s i m p o r t a n t f o r t h e dynamic p r o b l e m s .
Hence, t h e DLP t h e o r y a n d methods s h o u l d b e b o t h b a s e d on t h e
methods o f l i n e a r programming [ 1 , 2 ] and on t h e methods o f c o n t r o l
t h e o r y , P o n t r y a g i n ' s maximum p r i n c i p l e [ 4 ] and i t s d i s c r e t e
version [5] i n particular.
DLP CANONICAL FORM
I n f o r m u l a t i n g DLP p r o b l e m s it i s u s e f u l t o s i n g l e o u t [ 3 ] :
-
-
s t a t e e q u a t i o n s o f t h e s y s t e m w i t h t h e s t a t e and c o n t r o l
variables;
c o n s t r a i n t s imposed on t h e s e v a r i a b l e s ;
planning horizon T and t h e l e n g t h of each t i m e p e r i o d ;
*On l e a v e from t h e I n s t i t u t e f o r Systems S t u d i e s , MOSCOW, USSR.
**To a p p e a r i n " N u m e r i c a l O p t i m i z a t i o n o f Dynamic Systems"
(Eds. L.C.W. Dixon and G.P. Szeg6) N o r t h - H o l l a n d .
-
o b j e c t i v e f u n c t i o n (performance i n d e x ) , which q u a n t i f i e s
the q u a l i t y of control.
State equations.
S t a t e e q u a t i o n s have t h e f o l l o w i n g form:
x(t+l) = A(t)x(t)
+ B(t)u(t) + s(t)
(1)
. ..
t
x n t 1 defines the s t a t e of
t h e s y s t e m a t s t a g e t i n t h e s t a t e s p a c e X , which i s s u p p o s e d
t o b e t h e n - d i m e n s i o n a l e x c l i d e a n s p a c e E"; v e c t o r
r
u (t) = { u l (t), .. , ur ( t )1 E E s p e c i f i e s t h e c o n t r o l l i n g a c t i o n
where t h e v e c t o r x ( t ) = x
.
. .s
n t1 is a vector defining t h e
e x t e r n a l e f f e c t o n t h e s y s t e m ( u n c o n t r o l l e d , b u t known a p r i o r i
a t s t a g e t ; s ( t )= s
t
i n t h e d e t e r m i n i s t i c model).
0
I t i s a l s o assumed t h a t t h e i n i t i a l s t a t e of t h e s y s t e m i s
given :
PIanninq h o r i z o n T i s supposed t o b e f i x e d .
t = 0,1,.
.. ,T-1.
Thus i n ( 1 ) :
The l e n g t h o f e a c h t i m e p e r i o d may b e t h e
same f o r t h e w h o l e p l a n n i n g h o r i z o n ( s a y , month, y e a r , 5 y e a r s ) ,
o r d i f f e r s f o r d i f f e r e n t periods.
For example, i n long-range
e n e r g y s u p p l y models t h e p l a n n i n g h o r i z o n m i g h t b e 100 y e a r s ,
which f o r m s 1 0 - p e r i o d h o r i z o n w i t h f i v e p e r i o d s e q u a l 6 y e a r s ,
t h r e e p e r i o d s e q u a l 10 y e a r s and two p e r i d s e q u a l 2 0 y e a r s [ 6 ] .
Constraints.
I n r a t h e r g e n e r a l f o r m c o n s t r a i n t s imposed on t h e
s t a t e a n d c o n t r o l v a r i a b l e s may b e w r i t t e n a s
where f ( t ) = i f l ( t ),.
. . , f m ( t )1 E Em
(t = 0,1,-
- =,T-I)d
O b j e c t i v e f u n c t i o n . ( P e r f o r m a n c e i n d e x ) , w h i c h i s t o b e maximized f o r c e r t a i n t y , i s
I n v e c t o r p r o d u c t s t h e r i g h t vector i s a colum.? and t h e l e f t
v e c t o r i s a row.
~ e - f i n i t t c n s . The v e c t o r sequence
u
( u ( O ) , ...,u(T-1) 1 i s a
=
n o n t r o l ( ~ r o g : ? a r n )o f t h e s y s t e m . T h e v e c t o r s e q u e n c e
x = ( x ( 0 ) , x ( 1 ) , . . . , x ( T ) ), w h i c h c o r r e s p o n d s t o c o n t r o l u f r o m
(I),
( 2 ) , i s t h e system's trc,iectory.
The p a i r { u , x ) , which
s a t i s f i e s a l l +-he c o n s t r z i n t s o f t h e p r o b l e m ( e . g .
(1)- (4)) is
a feasible prscess.
A f e a s i b l e p r o c e s s { u * , x * > which maximizes
the objective function ( 5 ) is optimal.
The DLP p r o b l e m i n i t s c a n o n i c a l f o r m i s f o r m u l a t e d a s f o l l o w s .
F i n d a c o n t r o l u* a n d a t r a j e c t o r y x * , s a t i s f y i n g
with the i n i t i a l s t a t e
a n d t h e conthe s t a t e equations
s t r a i n t s ( 3 ) and ( 4 ) , which maximize t h e o b j e c t i v e f u n c t i o n ( 5 ) .
ProbZen; I ? .
.
I n P r o b l e m 1P v e c t o r s xO, s ( t ) , f ( t ) , a ( t ) , b ( t ) a n d matrices
~ ( t )
~ ,( t ~
) ,( t )
D (,t ) are s u p p o s e d t o be known.
The c h o i c e o f t h e c a n o n i c a l form i s t o some e x t e n t a r b i t r a r y
a n d t h e r e a r e v a r i o u s p o s s i b l e v e r s i o n s , a n d m o d i f i c a t i o n s of
P r o b l e m 1P.
I n p a r t i c u l a r , both s t a t e and c o n t r o l v a r i a b l e s
may be n o n - n e g a t i v e ; s t a t e e q u a t i o n s may i n c l u d e t i m e l a g s ; c o n s t r a i n t s o n s t a t e a n d c o n t r o l v a r i a b l e s may b e s e p a r a t e , g i v e n
a s e q u a l i t i e s o r i n e q u a l i t i e s ; t h e o b j e c t i v e f u n c t i o n may b e
d e f i n e d o n l y by t h e t e r m i n a l s t a t e x ( T ) o f t h e s y s t e m , etc.
[5,7]
.
I t s h o u l d be n o t e d , n o w e v e r , t h a t s u c h m o d i f i c a t i o n s c a n
be e i t h e r r e d u c e d t o P r o b l e m 1P o r t h e r e s u l t s s t a t e d below f o r
P r o b l e m 1P c a n be e a s i l y e x t e n d e d f o r t h e s e m o d i f i c a t i o n s .
Note, t h a t i f T = 1 , t h e n P r o b l e m 1P becomes a c o n v e n t i o n a l LP
problem.
P r o b l e m 1P i t s e l f c a n a l s o be c o n s i d e r e d a s a c e r t a i n
s t r u c t u r e d LP p r o b l e m w i t h c o n s t r a i n t s ( 1 )
-
(4).
I n t h i s case
t h e o p t i m a l c o n t r o l P r o b l e m 1P t u r n s o u t t o b e a n LP p r o b l e m
w i t h t h e s t a i r c a s e c o n s t r a i n t matrix.
S o n e t i m e s d y n a m i c LP p r o b l e m s are f o r m u l a t e d u s i n g o n l y LP l a n guage, a s f o r example, Problem 2 ,
.-
?~o.?Lern 2 .
F i n d v e c t o r s { x * ( 1 ) ,.
[8]
-
[ I 01 :
. ., x * ( T ) 1,
which maxinize
subject t o
One c a n a l s o e x p r e s s t h e s t a t e v a r i a b l e s x ( t ) i n Problem 1P a s
an e x p l i c i t f u n c t i o n o f c o n t r o l .
A s a r e s u l t , t h e f o l l o w i n g LP
problem w i t h a b l o c k - t r i a n g u l a r m a t r i x i s o b t a i n e d .
Problem 3 .
F i n d v e c t o r s {u* ( 0 ) ,
. . .,u* (T-1 ) 1 ,
where t h e v e c t o r s h ( t ), w ( t ) and t h e m a t r i c e s
which maximize
w (t,-r)
depend on
t h e known p a r a m e t e r s of Problem 1P.
Problems 2 and 3 a r e t y p i c a l examples of s t r u c t u r e d LP &,?d a d m i t
v a r i o u s m o d i f i c a t i o n s i n t h e same way a s Problem 1P .(e.g. a b l o c k
diagonal s t r u c t u r e w i t h coupling c o n s t r a i n t s and/or v a r i a b l e s ,
ete.).
They have been s t u d i e d i n t e n s i v e l y (see, e . g .
[1,2,8-101.
But u n l i k e c o n t r o l Problem 1P, s u c h f o r m a l i z a t i o n s of d y n a m i c a l
problems do n o t u s e e x p l i c i t l y t h e s t a t e e q u a t i o n s o f a s y s t e m .
T h e r e f o r e t h i s a p p r o a c h makes i t d i f f i c u l t t o a t t r a c t t h e i d e a s
and methods o f t h e c o n t r o l t h e o r y .
o f Problem 1P w e r e i n t r o d u c e d i n
The DLP p r o b l e m s i n t h e form
[11,5].
DLP MODELS
T h e r e i s a l a r g e and r a p i d l y e x p a n d i g g f i e l d o f a p p l i c a t i o n s of
dynamic l i n e a r p r o g r a m i n g .
H e r e o n l y some of them a r e mentioned.
These a r e b a s i c a l l y extens i o n s of dynamic i n p u t - o u t p u t m o d e l s , when s u b s t i t u t i o n i s
Z??:alr;ic n t l Z t ? : - S PtC
o r e c o n o m t c mode 2 s .
allowed ( s e e f o r exanple [1,2,12-141).
Alongside w i t h d i s c r e t e
t i m e f o r m u l a t i o n c o n t i n u o u s t i m e economy m o d e l s h a v e a l s o b e e n
developed ( e . g . i n [I51 1 .
Enera; niodeis.
following:
For long-range planning t h e problem i s t h e
f o r e x i s t i n g i n i t i a l s t r u c t u r e of t h e secondary
e n e r g y p r o d u c t i o n c a p a c i t i e s and u n d e r g i v e n p r i m a r y e n e r g y a n d
nonenergy r e s o u r c e s s u p p l y c o n s t r a i n t s f i n d a t r a n s i t i o n t o s u c h
a mix o f t h e e n e r g y p r o d u c t i o n o p t i o n s ( f o s s i l , n u c l e a r , e t c . ) ,
which s a t i s f i e d t h e p r o j e c t e d e n e r g y demand a n d m i n i m i z e s t h e
t o t a l c o s t of such t r a n s i t i o n .
These energy supply models have
been s t u d y i n g i n t e n s i v e l y i n t h e r e c e n t y e a r s [6,16-181.
The
next s t a g e i n t h i s d i r e c t i o n i s t h e i n v e s t i g a t i o n o f energyeconomy i n t e r a c t i o n .
T h i s p r o b l e m c a n be a n a l y s e d e i t h e r a s a n
i n t e g r a t e d DLP model o r i n some man-machine i t e r a t i v e mode [ 181
P r i m a r y r e s o u r c e s mode 2 s .
.
These models r e l a t e t o t h e p l a n n i n g
of primary r e s o u r c e s e x t r a c t i o n and/or e x p l o r a t i o n a c t i v i t i e s .
The s u r v e y o f d i f f e r e n t DLP e n e r g y - r e s o u r c e s - e c o n o m y
g i v e n i n [ I 81
Wa%er m o d e l s .
s u p p l y models
models)
,
models i s
.
The DLP m o d e l s f o r w a t e r s y s t e m s a r e t h e w a t e r
(which a r e c o n c e p t u a l y c l o s e t o energy s u p p l y
t h e problems of a l t e r n a t i v e s e v a l u a t i o n i n a r i v e r
basin, etc.
[ I 9-21]
.
Models .for e d u c a t i o ~ a Z and manpower ? l a n n i n ? .
The s i m p l e edu-
c a t i o n a l model a i m s t o f i n d s u c h e n r o l l m e n t s t o d i f f e r e n t educ a t i o n a l i n s t i t u t i o n s w h i c h a s w e l l a s s a t i s f y a l l t h e cons t r a i n t s of t h e systems ( e . g .
t e a c h e r s , b u i l d i n g s and o t h e r
e d u c a t i o n a l c a p a c i t i e s a v a i l a b i l i t y ) and be a s close as p o s s i b l e
t o t h e p r o j e c t e d manpower demand.
I n t h e manpower p l a n n i n g
n o d e l s c o n t r o l s a r e r e c r u i t m e n t s and/or promotions.
t
T h e r e a r e d i f f e r e n t m o d i f i c a t i o n s o f t h e s e m o d e l s , b o t h f o r nat i o n a l and i n s t i t u t i o n a l l e v e l s [ 2 2 ] .
.,
.
s~e tt Z ~ ~ ~ pe l an ntn i n g .
T h e s e DLP m o d e l s r e l a t e t o f i n d i n g an o p t i m a l , i n some ' s e n s e , m i g r a t i o n f l o w s i n a r e g i o n O r a
cou:1tr-y [ 231
...:-
-
3
.
The p r o b l e m o f h e a l t h c a r e s y s t e m p l a n -
3 e ~ Z t hc Q r e s g s t e m s .
n i n g may b e r e d u c e d t o a n a l l o c a t i o n f u n d s , h e a l t h manpower a n d
o t h e r r e s o u r c e s o v e r t i m e among d i f f e r e n t d i s e a s e s t r e a t m e n t
a c t i v i t i e s i n s u c h a way a s t o y i e l d t h e b e s t r e s u l t s i n terms
o f r e d u c e d m o r t a l i t y , m o r b i l i t y and o t h e r l o s e s [ 2 4 ] .
S e ~ a r a t eDLP m o d e l s c o n c e r n p l a n n i n g p r o b -
A g r i c u Z t u r e m o d e is.
l e m s i n livestock breeding, crop production, resources u t i l i z a ' t i o n , e t c . The i n t e g r a t e d m o d e l s r e l a t e t o p l a n n i n g o f d i v e r s i f i e d a g r o - i n d u s t r i a l complexes and t h e problems o f farm growth
[25-281
.
Production models.
T h e s e DLP m o d e l s a r e b a s i c a l l y a s s o c i a t e d
w i t h t h e p r o d u c t i o n s c h e d u l i n g and i n v e n t o r y problems i n
industry [8,9,29,30].
Transhipment and o t h e r n e t -
Dynamic transportation problems.
work p r o b l e m s a r e t h e i m p o r t a n t c l a s s o f L i n e a r programming.
Dynamical a s p e c t s o f t r a n s p o r t a t i o n p r o b l e m s a r e c o n s i d e r e d i n
[3,31,32].
I t i s w o r t h w h i l e t o m e n t i o n ' h e r e some
Miscellaneous models.
" i n d i v i d u a l " DLP a p p l i c a t i o n s :
congested urban road t r a f f i c
c o n t r o l [ 3 3 ] , p l a n n i n g t h e a c q u i s i t i o n and d i s p o s a l o f equipment
i n a t r a n s p o r t f l e e t [ 3 4 ] , m u l t i s t a g e s t r u c t u r a l d e s i g n problems
L3.51.
THEORY O F DCP
The t h e o r y o f DLP i s c o n n e c t e d w i t h d u a l i t y r e l a t i o n s a n d
o p t i m a l i t y c o n d i t i o n s 11 1 , 3 6 , 7 , 5 ]
P r o b l e m ID.
.
To f i n d a d u a l c o n t r o l X = { A (T-1)
...
a d u a l t r a j e c t o r y p = { p ( ~ ) , ,p(O)
(dual) equation
C
with t h e
boundary c o n d i t i o n
,.. ., X ( O )
and
satisfying t h e co-state
subject t o the constraints
and m i n i m i z e t h e o b j e c t i v e f u n c t i o n
Here p ( t )
IZ
En and A ( t )
s t r a i n t s (1)- (4)
IZ
Ern are L a g r a n g e m u l t i p l i e r s f o r con-
.
The d u a l P r o b l e m I D i s a c o n t r o l - t y p e p r o b l e m as i s t h e p r i m a l
o n e 1P.
Here t h e v a r i a b l e A ( t ) i s a d u a l c o n t r o l a n d p ( t ) i s
a c o - s t a t e o r a d u a l s t a t e a t s t a g e t.
i n t h e d u a l Problem I D :
Theorem I .
t = T-1,
W e have r e v e r s e d t i m e
...,1,O.
( T h e DLP NGZobaZ" D u a l i t y T h e o r e m ) .
The s o l v a b i l i t y
o f e i t h e r o f t h e 1P o r I D p r o b l e m s i m p l i e s t h e s o l v a b i l i t y o f
* .
t h e o t h e r , w i t h J p ( u * ) = J D( A )
.
L e t u s i n t r o d u c e Hamilton f u n c t i o n s
f o r t h e primal and d u a l problems r e s p e c t i v e l y .
Theorem 2 .
( T h e DLP " L o c a l u D u a l i t y T h e o r e m ) .
The s o l u t i o n s
of t h e p r i m a l { u * , x * ) a n d d u a l
p r o b l e m s are o p t i m a l if
and o n l y i f t h e v a l u e s of H a m i l t o n i a n s c o i n c i d e :
Thus, t h e s o l u t i o n o f t h e p a i r o f dynamic d u a l p r o b l e m s c a n b e
reduced t o a n a l y s i s o f a p a i r o f s t a t i c l i n e a r programs
max H p ( p ( t + l ) , u ( t )
(10)
G(t)x(t)
+ D(t)u(t)
= f(t) ;
u(t) > 0
min H D ( x ( t ) , A ( t ) )
l i n k e d by t h e s t a t e ( I ) ,
( 2 ) and c o - s t a t e
( 6 ) , (7) equations.
I n p a r t i c u l a r , i t can b e shown, t h a t maximum p r i n c i p Z e f o r
dual
Problem I P and minimum p r i n c i p l e f o r d u a l Problem I D a r e h o l d
[ 5 , 7 ] . An economic i n t e r p r e t a t i o n o f t h e DLP d u a l i t y r e l a t i o n s
i s g i v e n i n [361 .
DLP METHODS
We s h a l l d i s t i n g u i s h f i n i t e and i t e r a t i v e methods f o r s o l v i n g
DLP problem.
F i n i t e - s t e p s methods.
T h e s e methods are t h e e x t e n s i o n o f f i n i t e
Two b a s i c
l a r g e - s c a l e LP methods f o r t h e dynamic p r o b l e m s .
a p p r o a c h e s may b e s i n g l e d o u t h e r e :
p o s i t i o n methods [ 9 , 1 0 , 3 7 , 3 8 :
Compact i n v e r s e m e t h o d s .
compact i n v e r s e and decom-
.
These methods are v a r i a n t s of t h e
s i m p l e x method u s i n g b a s i s f a c t o r i z a t i o n w i t h v a r i o u s s t r a t e g i e s
a s t o t h e v e c t o r p a i r t o e n t , e r and t o l e a v e t h e b a s i s .
The
a p p r o a c h i s b e i n g d e v e l o p e d b o t h f o r s t r u c t u r e d a n d g e n e r a l LP
(in t h e l a t t e r case t h e sparseness of constraint matrix i s
exploited)(see e.g.
[9,10,37,38,39,40]).
For s t a i r c a s e s t r u c t u r e
(Problem 2 ) t h i s a p p r o a c h was u s e d i n [37,41-441
.
F o r DLP
Problem 1P t h e a p p r o a c h was p r c p o s e d i n [ 4 5 , 4 6 ] a n d d e s c r i b e d
The method a r i s e d i s a n a t u r a l and s t r a i g h t f o r w a r d
i n (47-491,
e x t e n s i o n o f t h e s i m p l e x method.
method--the
basis--is
The main c o n c e p t of t h e s i m p l e x
r e p l a c e d by t h e s e t o f l o c a l b a s e s , i n t r o -
duced f o r t h e whole p l a n n i n g p e r i o d .
t h e following.
(2),
The i d e a o f t h e method i s
Consider t h e c o n s t r a i n t s (3) f o r t = O .
Using
t h e y can b e w r i t t e n i n t h e form
and a c c o r d i n g t o t h e c o n v e n t i o n a l s i m p l e x p r o c e d u r e b e p a r t i tioned as
(0) is a sqcare non-singular ( m m ) matrix.
If T = 1
0
( s t a t i c : p r o b l e m ) , t h e n onr- c a n s e t u 1 ( 0 ) = 0 a n d f i n d f r o m ( 1 3 )
\.r:rlcre D
.
a Lasic f s a s i b l e s o l u t i o n u0(O) > 0
I n t h e dynz?.nic
case
(T > 1 ) n o t a l l c o m p o n e n t s o f u 1 ( 0 ) a r e z e r o s f o r b a s i c
salutioxs.
T h e s e n o n z e r o c c m p o n e n t s s h o y ~ l db e t r a n s f o r m e d t o
t h e next s t r p s t > 0 .
3 a r t i t i o n i ~ gt h e s t a t e e q u a t i o n s
t = 0 i n t h e s a m e way a s i n
(1) for
(13) and s u b s t i t u t i n g u o ( 0 ) from
(13) t o t h i s p a r t i t i o n e d s t a t e e q u a t i o n s , one can o b t a i n t h e
c o n s t r a i n t s f o r t h e nexE s t e p i n t h e form, s i m i l a r t o
A
j(l);(l)
= 1
)
e (
,
depenscis or, G ( 1 1 , C
h
A
1 =
3 (3)
O
,
0
;
(12):
~ ( 1 ) ; . Tne m s t r i x
6(1)
a 2 2 car! b e a g a i n p a r t i t i o z e d i n t h e
A
h
s a n e 1~2:~: D ( 1 ) = [ D O ( l ) ; D l ( i ) ] , d e t D O ( l ) # 0 and s o o n , u n t i l
tr.c l a s t s t e p u l (T-I) = 0 .
T h e s e t o f n l i n e a r l y i n 6 e p e n d e r . t c o l u r r n s of t h e m a t r i x D ( t ) i s
h
Loccl 3 c s l s a t s t e p t .
T h e s e t of l o c a l b a s e s { D o ( t ) ) f o r t h e
whole plar,r.ing p e r i o d t = 0 , 1 , .
. . ,T-1
5 a s i s i n stan.izrd simplex-method.
plajrs t h e s a m e role as
'
The i n t r o 2 u c t i o n o f l o c a l b a s e s
a l l o w s t o implement a l l s i m p l e x procedures
(selection of vectors
t o b e removed f r a x a n d t o b e i n t r o d u c e d i x t o t h e b a s i s , p r i c i n g ,
I t s h o u l d be a l s o n o t e d , t h a t
updating) very effectively
[49].
t h e dynamic simplex-method
g i v e s n o t o n l y compact s u b s t i t u t e
f o r b a s i s i n v e r s e , b u t u s e s o n l y a p z r t of t h e l o c a l b a s e s
r e p r e s e n t a t i o n a t each i t e r a t i o n .
T h e e x t e n s i o n o f t h e d y n a m i c s i m ~ l e x - m e t h o d t o DLP p r o b l e i n s w i t h
s t a t e v a r i a b l e s as w e l l a s d u a l ver-
t i m e l a g s o r ncn-negative
s i o n s o f t h e method are c o n s i d e r e d i n
[46,491.
2ecor?=,csi t i o r . ns rkods.
T h e s e m e t h o d s f o l l o w s fror;l r e p e a t e d a p p l i c a t i o n of d i f f e r e n t decomposition methods ( f i r s t of a l l , ~ a n t z i c j W o l f e d c c o m ; ~ o s i t i o n ~ r i ~ c i 2 l et oj t h e DLP ~ r o b l e m , a n d t h e s t a i r c a s e
s t r u c t u r e ( ? r o b l e m 2 ) was
described i n
[ ?,50-541
F o r D L ? P r o b l e m 1P t h i s a 2 p r o a c h w a s u s e d i n [ S S ] .
s e q u e n c e o f LP p r o b l e m s
max
( t = T- 1 , .
p(t+l)x(t+J)
. ., I , 0 ) :
.
Consider t h e
where R t ( x o ) i s t h e s e t o f a l l s t a t e s x ( t ) f e a s i b l e from i n i t i a l
state
x0 f o r t s t z p s [ 5 1 ; p ( t + l ) i s d u a l s t a t e v e c t o r .
S o l u t i o n s o f t h e s e LP problems w i t h o p t i m a l v a l u e s of p * ( t + l )
g i v e an o p t i m a l s o l u t i o n o f t h e o r i g i n a l Problem 1P. Any v e c t o r
x ( t ) of R t ( x o ) can b e r e p r e s e n t e d a s a l i n e a r convex combinat i o n o f i t s e x t r e m e p o i n t s ( a s s u m i n g boundness o f R t ( x o ) ) .
Thus, problem ( 14 ) can b e r e w r i t t e n as
max
p(t+l)[LA(t)xi(t)pi(t)
-i
+ B(t)u(t) + s(t)l
Problem ( 1 5 ) i s a master problem a t s t e p t .
For i t s s o l u t i o n
c o n v e n t i o n a l column g e n e r a t o r p r o c e d u r e can b e a d o p t e d .
Other
methods f o r s o l u t i o n o f Problem 1P b a s e d o n d i f f e r e n t decomposit i o n and p a r t i t i o n p r i n c i p l e s w e r e c o n s i d e r e d i n [56-601.
T h e DLP i t e r a t i v e m e t h o d s .
The a p p l i c a t i o n o f t h e LP f i n i t e
methods t o t h e dynamic p r o b l e m s may c a u s e c e r t a i n d i f f i c u l t i e s ,
e s p e c i a l l y f o r l a r g e p l a n n i n g peri'ods T .
The r e a s o n f o r i t i s
t h a t s o l u t i o n p a t h s i n t h e s e methods c o n s i s t o f a number o f
v e r t i c e s of a f e a s i b l e p o l y h e d r a l s e t ( i n some s p a c e ) , and t h i s
number w i l l i n c r e a s e e x p o n e n t i a l l y w i t h T.
The i t e r a t i v e methods seem t o b y - p a s s t h e s e d i f f i c u l t i e s .
They
a l s o c h a r a c t e r i z e d by low demand t o c o m p u t e r ' s c o r e memory,
s i m p l i c i t y o f t h e computer i m p l e m e n t a t i o n , low s e n s i t i v i t y t o
disturbances.
D i s a d v a n t a g e s o f t h e s e methods may b e low con-
vergence r a t e , y e i l d i n g o n l y approximate s o l u t i o n .
.\
We s h a l l s i n g l e o u t t h e f o l l o w i n g i t e r a t i v e methods.
--;
Li
1
_ _ : r ~ c r ~ ! !t: ~mn? t i ; o J ; ; .
T h i s g r o u p o f m e t h o d s i s bclscd
I
-
o n f i n d i n g extremum o f f u n c t i o n s :
'
p
(:,
max
=
X f U
L(ufx ; A ,p)
L
;
;( u , ~ )=
min
L(ufx
pix > 0
0
;
;\,PI
(16!
i s t h e L a g r a n g e f u n c t i o n o f P r o b l e m 1P ( I D ) .
where L(u,x,A,p)
Minimization of Y is equivalent to t h e solution of t h e dual
P r o b l e m ID, w h i l e t h e maximizati.on o f
s o l u t i o n o f p r i m a l P r o b l e m 1P [ 7 ] .
c$
is e q u i v a l e n t t o t h e
A s t h e s e f u n c t i o n s a r e non-
d i f f e r e n t i a b l e b y n a t u r e , t h e g e n e r a l i z e d g r a d i e n t t e c h n i q u e i611
is n e e d e d .
The a p p l i c a t i o n o f t h i s a p p r o a c h t o t h e s o l u t i o n o f
d u a l P r o b l e m ID l e a d s t o t h e f o l l o w i n g a l g o r i t h m .
W e consider
Problem 1P w i t h a d d i t i o n a l c o n s t r a i n t s o n c o n t r o l v a r i a b l e s :
u ( t ) EUt
Ut
;
! u ( t ) I ~ ( t ) u ( t2
) r ( t ), u ( t ) 2 0 )
=
.
.
v
( t )j ( t = 0 , 1 , . . ,T-1 )
(1)
Choose a r b i t r a r y d u a l c o n t r o l { A
(2)
Cernpute d u a l s t a t e t r a j e c t o r y ~ ~ ' ;
( ftr o)m t h e d u a l
.
.
state e q u a t i o n s (6) w i t h boundary c o n d i t i o n (7)
v
F o r p\J ( t + l ) s o l v e T LP p r o b l e m s : rnax H p ( p ( t + l ) . u ( t ))
(3)
u ( t )E U t
.
L e t uV( t ) b e a s o l u t i o n o f t h e s e p r o b l e m s .
Compute p r i m a l s t a t e t r a j e c t o r y x
(4)
.
v
( t l from ( 1 ) and
(2) f o r u V ( t ) .
where h v ( t ) = f ( t )
:.
* *
) ~ ' +
I , 1
--
.
-
~(t)x'(t)- ~(t)u\;(t).
v+ 1
C o m p u t e new d u a l c o n t r o l A
:
. v+l = A v - a
v
v
( '
3 ) I(
(6)
- r.. . 3 : 1 i . ? ~
v
v
S Y ( A ) = h ( t ),
Compute g e n e r a l i z e d g r a d i e n t o f Y :
(5)
Let nV-tO;
= {A
* (t)
v - + ~ ;
1 av
:>=o
-
( V
-.
=
(17)
O,l,2 ....)
T h e n ?' ( i V ) !'(A
(18)
*
-+
)
,
is a n o p t i m a l c o n t r o l o f d u a l P r o b l e m I D .
The a l g o r i t h m ( 1 ) - ( 6 ) g i v e s a d u a l s o l u t i o n o f t h e p r o b l e m . To
o b t a i n t h e p r i m a l s o l u t i o n o n e c a n a p p l y t h e same a p p r o a c h t o
function
:( u )
in
,,--,:*.c..
... ,
L Q C L : : n 3 2 i i :C
..-,
(16)
.
-
-
~uY~c?.~
. - s: e;z:, : z : [ 6 2 6 3 1
.
T h i s a p p r o a c h com-
b i n e s t h e a d v a n t a g e s o f b o t h f i n i t e and i t e r a t i v e methods:
under
a i n i m a l r e q u i r e m e n t s f o r s t o r a g e i t e n s u r e s monotonic c o n v e r cjsnce i n a f i n i t e number of s t e p s . I n a s e n s e , t h i s method c-n be
c o n s i d e r e d a s a "smooth" a l t e r n a t i v e t o t h e nonsmooth a p p r o a c h ,
L e t t h e L a g r a ~ q ef ~ i c t i o n5 ( u , ) . ) o f P r o b l s m 1 2
dsscribeci above.
( I D ) b e ailqnented by q ! ~ a d r a t l cte~rix:
where 0 > 0 , v e c t o r h ( t ) i s d e f i n e d from ( 1 7 ) .
Now i n s t e z d of
( 1 6 ) t h e f o l l o w i n g p a i r of problems can . b e
cor?siders2
\U
f1
( A ) = rriax i ? ( u , A ) ; t , ( u )
,
O p p o s i t e t o ?' ( A )
m
in M(u,X!
X
t h e f u ~ c t i o nP (?, j
f u n c t i o n ; d e r i v a t i v e o f '?
a'? M ( E , ) / a A ( t )
=
.-
U
i s a s a ~ o t hc o n c a v e
m ( A ) i s a c o n t i n u o u s f 7 t ? c t i o n of X m d
= h ( t ) where h ( t ) i s computed from ( 1 7 1 , when
u* = u i ( A ) i s a s o l u t i o n o f t h e l e f t p r o b l e m i n ( 2 0 ) ( 6 2 1 .
Problems n i n Y ,(1) and max $ ( u ) a r e t h e p a i r o f m o d i f i e d d u a l
M
)I
~ r o b l e ~ . sa s, s o c i a t e ? w i t h o r i g i n a l Problem 1P.
I n [621 i s
shown, t h a t t h e m o d i f i c a t i o n d o e s n o t c h z n g e t h o g e n e r a l d u a l
relations.
Thus i;l o r d e r t o o b t a i n a s o l u t i o n o f Troblem
one can s o l i r e e i t h e r t h e d u a l
smooth fu-rlction
p r o b l e ~min ? 0.) with t h e no2-
o r t h e modified Zual problen n i n ? ( A ) with
Y
>!
t h e smooth fuirlction
'!
M'
The l a t t e r g i v e s t h e f o l l o w i n s a l g o r i t h m :
( 1 ) Choose a r b i t r a r y d u a l c o n t r o l A V .
( 2 ) Solve t h e l e f t problen i n
(20).
T h i s ?robLen h a s l i n e a r s t a t e e q u a t i o n s ( 1 )
t i v e function ( 1 9 )
(e.9.
I)
-< u(t) <
c o n t r o l t2chnique
,
g u a d r a t i c objec-
and s i m p l e c o n s t r a i n t s on c o n t r o l u ( t )
c ( t )1 .
Therefore f o r i t s solution o p t i n a l
(iv-hich r e d u c e s i n t h i s c a s e t o s o l u t i c n of
n a t r i x A i c c a t i e q ~ a t i o n )can b e e f f e c t i v e l y 2 ~ 2 l l e d [ 6 3 1 .
2
(3) Let u
v
be a s o l u t i o n o f t h e abolre p r o b l m .
Ccm~ate
h" ( t ) f r o m ( 1 7 ) f o r t h i s u V a n d new d u a l c o n t r o l X
v+ 1
from
p + 1 (t) = Xv (t)- 4hv(t)
T k e o r ~ s ,4 ~.
For any
e
(\J
> 0,
*
a n d b e g i n n i n g f r o m some f i n i t e v O : X v €11
&
where A T I
I
= 0,1,2,.-.)
*
, u;' EU , v > \
J ~ ,
U-, a r e s o l u t i o n sets o f d u a l a n d p r i m a l problems
I D a n d 1P r e s p e c t i v e l y .
Theorem 4 s t a t e s t h a t , o p p o s i t e t o t h e g e n e r a l i z e d g r a d i e n t
a l g o r i t h m ( 1 ) - ( 6 ) , t h e a l g o r i t h m ( 1 )- ( 3 ) c o n v e r g e s f o r a f i n i t e
n u m b e r o f s t e p s a n d t h e v a l u e s o f t h e m o d i f i e d f u n c t i o n '? ( A )
m o n o t o n o u s l y d e c r e a s e s w i t h v ( i n f a c t so d o e s t h e n o n n o d y f i e d
d u a l f u n c t i o n Y ( A ) a l s o 1621 ) . M o r e o v e r , t h i s a l g o r i t h m g i v e s
s i m u l t a n e o u s l y d u a l A* and ?rimal u* o p t i m a l s o l u t i o n s .
CONCLUSION
A s h o r t s u r v e y h a s b e e n g i v e n a b o v e o f DL? m o d e l s , t h e o r y a n d
methods.
T h e r e i s n o t much l i t e r a t u r e o n c o m p u t e r i m p l e m e n t a t i o n o f t h e methods.
E x p e r i m e n t s , h o w e v e r , show t h a t t h e s e
m e t h o d s c a n b e much m o r e e f f i c i e n t i n c o z p a r i s o n w i t h t h e
s t a n d a r d ( s t a t i c ) approach (see, f o r i n s t a n c e , [ 3 9 , 5 3 1 ) . Theref o r e c o m p u t e r t e s t i n g a n d e v a l u a t i o n o f t h e a l g o r i t h m s a r e now
a n important problem.
Other important d i r e c t i o n s of research
a r e t h e problem o f t h e implementation o f o p t i m a l c o n t r o l and
p o s t - o p t i m a l a n a l y s i s o f t h e model ( f e e d b a c k v s program o p t i m a l
c o n t r o l , s e n s i t i v i t y a n d s t a b i l i t y a n a l y s i s , i n t e r r e l a t i o n between s h o r t and long-term p r o b l e m s , e t c . )
I t should a l s o be noted t h a t n o t a l l d y n a i c a l optimization
p r o b l e m s c a n be k e p t w i t h i n t h e f r a m e w o r k o f DLP.
Therefore,
t h e e x t e n s i o n o f DLP f o r s t o c h a s t i c , m a x m i n , n o n l i n e a r d y n a m i c
Some w o r k
p r o g r z m i n g problems is o f l a r g e p r a c t i c a l i n t e r e s t .
h a s Seen a l r e a d y done i n t h i s d i r e c t i o n (see, e . g . [64-6611.
References
D a n t z i g , G.B., Lineal2 Programming and E x t e n s i o n s , P r i n c e t o n
U n i v e r s i t y P r e s s , P r i n c e t o n , N . J . , 1363.
K a n t o r o v i t c h , L . V . , -The B e s t Use o f Economic R e s o u r c e s ,
H a r v a r d U n i v e r s i t y P r e s s , C a m b r i d g e , Mass., 1 9 6 5 .
P r o p o i , A . I . , Problems o f Dynamic L i n e a r Programming,
Rb;-76-78,
I r i t e r n a t i o n a l I n s t i t u t e f o r Applied Systems
A n a l y s i s , Laxenburg, A u s t r i a , 1976.
P o n t r y a g i n , L.S., e t a l . Mathematical Theory o f Optimal
P r o c e s s e s , I n t e r s c i e n c e , New York, 1962.
P r o p o i , A . , Elementy T e o r i i Optimalnykh D i s c r e t n y k h
P r o t s e s s o v ( E l e m e n t s o f t h e T h e o r y o f O p t i m a l Discrete
S y s t e m s ) , Nauka, Moscow, 1 9 7 3 ( i n R u s s i a n ) .
M a r k u s e , W . , e t a l . A Dynamic Time-Dependent Model f o r t h e :
A n a l y s i s o f Alternative E n e r g y P o l i c i e s , i n
O p e r a t i o n a l R e s e a r c h ' 7 5 ( e d . K.B. H a l e y ) , N o r t h H o l l a n d P u b l i s h i n g Company, 1 9 7 6 , p p . 647-667.
P r o p o i , A . , Dual S y s t e m s o f ~ y n a h c ine ear Programming,
RR-77-9, I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s
A n a l y s i s , Laxenburg, A u s t r i a , 1977.
D a n t z i g , G.B., On t h e S t a t u s o f M u l t i s t a g e L i n e a r Programming P r o b l e m s , Management S c i e n c e , Vol. 6 ( 1 9 5 9 ) ,
pp. 53-72.
.,
L a s d o n , L. S O p t i m i z a t i o n Theory f o r Large S y s t e m s ,
MacP-lillan, 1 9 7 0 .
I~ladsen;O., S o l u t i o n o f LP-Problems w i t h S t a i r c a s e S t r u c t u r e , RZ-26-1977, I n s t i t u t e o f M a t h e m a t i c a l S t a t i s t i c s a n d O p e r a t i o n a l R e s e a r c h , D e n n a r k , 1977.
I v a n i l o v , Yu. P . a n d A.I. P r o p o i , The L i n e a r Dynamic P r o gramming, DokZ. Acad. Nauk SSSR, Vol. 1 9 8 , N o . ' 5
(1971) pp. 1011-1014 ( i n R u s s i a n ) .
A g a n b e g i a n , A.G., K.A. B a g r i n o v s k i i a n d A.G. G r a n b e r g ,
Systema Modelei Narodnohoziastvennogo P l a n i r o v a n i a
( S y s t e m s of Models o f N a t i o n a l P l a n n i n g ) , Moscow,
"Idys Z " , .I972 ( i n R u s s i a n )
.
D a n t z i g , G.B., O p t i m a l s o l u t i o n of a ~ y n a m i cL e o n t i e f
Flodel w i t h S u b s t i t u t i o n , Econometrics, V o l . 2 3
( 1 9 5 5 ) , pp.295-302.
K o e h l e r , G . , A. Whinston and G. Ylright, O p t i m i z a t i o n Over
L e o n t i e f S u b s t i t u t i o n S y s t e m s , North-Holland
P u b l i s h i n g . Company, 1975.
Dukalov, A. N . , Yu.N. I v a n o v a n d V.V. T o k a r e v , C o n t r o l
Theory and Zconomic Systems. A v t o m a t i k a i
Te Zemekhanika, No. 5 (1974) ( i n R u s s i a n )
.
'
tlakarov A.A. and L.A. M e l e n t j e v , Metody I s s l e d o v a n i a
i O p t i n i z a t z i i Energeticheskaso Khoziastva
( A n a l y s i s and O p t i n i z a t i o n Methods o f Energy S y s t e n s ) ,
Nauka, N o v o s i b i r s k , 1 9 7 3 ( i n R u s s i a n )
.
H a e f e l e , W. a n d A.S. Manne, S t r a t e g i e s o f a T r a n s i t i o n
from F o s s i Z t o NucZear F u e l s , RR-74-07, I n t e r n a t i o n a l
I n s t i t u t e f o r A?plied Systems A n a l y s i s , Laxenburg,
A u s t r i a , 1974.
P r o p o i , A . and I . Z i n i n , E n e r g y - B e s o u r c e s - E c o n o m y DeveZopment ModeZs, I n s t i t u t e f o r A p p l i e d Systems A n a l y s i s , .
Laxenburg, A u s t r i a , 1978 ( f o r t h c o m i n g ) .
Agarkov, S . , I . D r u z h i n i n , 2. Konovalenko, A. K u z n e t s o v ,
T. Naumova, V. Ushakova, I . G e r a r d i , N . Dorfman,
R. Kogan, E. Kokorev and D. R ~ s ~ u ~ o vThe
,
Optima2
A l l o c a t i o n o f W a t e r Consuming P r o d u c t i o n . Vodnye
R e s u r s y (Water R e s o u r c e s ) , Vol. 3 (1957) , p p . 5-17.
P a r i k h , S . C . , L i n e a r Decomposition Programming o f O p t i m a l
Long-Range O p e r a t i o n o f a M u l t i p l e M u l t i p u r p o s e
.
R e s e r v o i r S y s t e m , i n The P r o c e e d i n g s o f t h e F o u r t h
I n t e r n a t i o n a l Conference on OperationaZ Research,
( e d s . D . H e r t z and I . :.felese) W i l e y , 1966.
pp. 763-777.
Biswas, A . K . ,
(Ed. S y s t e m s Approach t o W a t e r Management,
McGraw-Hill P u b l i s h i n g Company, 1976.
P r o p o i , A . , Models f o r E d u c a t i o n a l and Manpower P l a n n i n g :
A Dynamic L i n e a r Programming Approach, RM-78-20,
' I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d Systems I u l a l y s i s ,
Laxenburg, A u s t r i a , 1978.
.
P r o p o i , A. a n d F. ~ 7 i l l e k e n s ,A DLP Approach t o N a t i o n a l
S e t t l e m e n t S y s t e m P l a n n i n g , RE-77-8, I n t e r n a t i o n a l
I n s t i t u t e f o r A p p l i e d Systems A n a l y s i s , Laxenburg,
A u s t r i a , 1977. A l s o see E n v i r o n m e n t and PZanning A ,
Vol. 10 ( 1 9 7 8 ) , pp.
561-576.
P r o p o i , A . , Dynamic L i n e a r Programming Models i n H e a l t h
C a r e S y s t e m s i n Modeling H e a l t h Care S y s t e m s , e d s .
E. S h i g a n and R. G i b b s , CP-77-8, I n t e r n a t i o n a l
I n s t i t u t e f o r A p p l i e d Systems A n a l y s i s , Laxenburg,
A u s t r i a , 1977.
S w a r t , PI. a.0. E x p a n s i o n P l a n n i n g f o r a L a r g e D a i r y Farm,
i n S t u d i e s i n L i n e a r Programming, Eds. H . S a l k i n a n d
J.' S a h a , North-Holland P u b l i s h i n g Company, N . Y . , 1975.
Dynamic L i n e a r Progrcmming ModeZs f o r L i v e s s t o c k Farms, 311-77-29, I n t e r n a t i o n a l I n s t i t u t e f o r
Propoi, A.,
Applied Systems A n a l y s i s , Laxenburg, A u s t r i a , 1973.
Dyncmic L i n e a r Programming Model f o r A g r i c u Z t u r a Z I n v e s t n e n t and R e s o u r c e s U t i l i z a t i o n P o Z i c i e s ,
Csaki, C.,
RM-77-36, I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s
A n a l y s i s , L a x e n b u r g , A u s t r i a , 1977.
C . C s a k i a n d A . P r o p o i , P l a n n i n g Long-Range
A g r i c u Z t u n a Z I n v e s t m e n t P r o j e c t s : A Dynamic L i n e a r
P r o g r a n m i n g A p p r o a c h , W-77-38, I n t e r n a t i o n a l
Carter, H.,
I n s t i t u t e f o r Applied Systems A n a l y s i s , Laxenburg,
A u s t r i a , 1977.
G l a s s e y , C . R . , Dynamic L i n e a r P r o g r a m s f o r P r o d u c t i o n
S c h e d u l i n g , O p e r a t i o n a l R e s e a r c h , V o l . 1 9 , 1 (1971)
pp. 45-56.
.
.
-
P e g e l s , C . C . , S y s t e m s AnaZysiis f o r P r o d u c t i o n O p e r a t i o n s ,
. G o r d o n a n d B r e a c h P u b l i s h i n g Company, London, 1 9 7 6 .
".
E k l o f a n d G.L. Nemhauser, T r a n s h i p m e n t
B e l l m o r e , PI. , W
Algorithm f o r a M u l t i p e r i o d T r a n s p o r t a t i o n Problem,
Naval R e s e a r c h L o g Z s t i c s Q u a r t e r l y , V o l . 19, 4
(1969).
Madsen, O . ,
A Case S t u d y i n P r o b l e m S t r u c t u r i n g , i n S e c o n d
E u r o p e a n C o n g r e s s on O p e r a t c o n a l R e s e a r c h ,
(ed. M . Roubens), North-Holland
1 9 7 6 , p p . 287-293.
P u b l i s h i n g Company,
Tamura, H. M u l t i s t a g e L i n e a r Programming f o r Discrete
Optimal C o n t r o l w i t h D i s t r i b u t e d Lags.
Avtomatica,
V o l . 1 3 ( 1 9 7 7 ) , p p . 369-376.
New,
, Transport F l e e t Planning f o r Multiperiod
Operational Resecrch Quarterly,
Operations.
V o l . 2 6 , 1 , ii ( 1 9 7 5 ) , p p . 151-166.
C. C.
H o t J.K.,
O p t i m a l D e s i g n of M u l t i - S t a g e S t r u c t u r e s : a
N e s t e d ~ e c o m p o s i t i o nA p p r o a c h .
Computers and
S t r u c t u r e s , V o l . 5 ( 1 9 7 5 ) , p p . 249-255.
I v a n i l o v , Pu. P . a n d A . P r o p o i , D u a l i t y R e l a t i o n s i n
Dynamic L i n e a r Programming, A v t o m a t i c a i
T e l e m e k h a n i k a , V o l . 1 2 , ( 1 9 7 3 ) , p p . 100-108
( i n R u s s i a n ) _..
.
B u l a v s k i i , V . A . , X.A. Z v i a g i n a a n d ;i.A. Y a k o v l e v a ,
C h i s l e n n y e Metody L i n e i n o g o P r o g r a m r n i r o v a n i a
( N u m e r i c a l Methods o f L i n e a r Programming) , "Nauka I f ,
bloscow, 1 9 7 7 ( i n R u s s i a n )
.
Beer, K., Ldsung Grosser L i n e a r e r ~ p t i m i e r u n g s a u f g a b e n ,
V E B ~ e u t s c h e rV e r l a g d e r W i s s e n s c h a f t e n , B e r l i n ,
1977 ( i n German)
.
W i n k l e r , C . , B a s i s F a c t o r i z a t i o n f o r Block Angular
L i n e a r Programs:
U n i f i e d Theory o f P a r t i t i o n i n g
and D e c o m p o s i t i o n Using t h e S i m p l e x Method,
RR-74-22, I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s
A n a l y s i s , Laxenburg, A u s t r i a , 1974.
G i l l , P.E.
a n d W. M u r r a y , L i n e a r l y - C o n s t r a i n e d P r o b l e m s
I n c l u d i n g L i n e a r and Q u a d r a t i c Programming, i n The
S t a t e o f t h e A r t i n Numerical A n a l y s i s ,
( e d . D . J a c o b s ) , Academic P r e s s , 1 9 7 7 , p p . 313-363.
D a n t z i g , G.B.,
Compact B a s i s T r i a n g u l a r i z a t i o n f o r t h e
S i m p l e x Method i n R e c e n t A d v a x c e s i n M a t h e m a t i c a l
Programming ( E d s . R. G r a v e s a n d P. Wolfe) , I-lcGrawH i l l P u b l i s h i n g Company, N . Y . ,
1 9 6 3 , p p . 125-132.
a n ' d J. S a n d e e , S p e c i a l S i m p l e x
H e e s t e r m a n , A.R.G.
A l g o r i t h m f o r L i n k e d P r o b l e m s , Management S c i e n c e ,
V o l . 1 1 , N o . 3 ( 1 9 6 5 ) p p . 420-428.
K a l l i o , M. a n d 'E. P o r t e u s , T r i a n g u l a r F a c t o r i z a t i o n a n d
G e n e r a l i z e d Upper Bounding T e c h n i q u e s , O p e r a t i o n s
R e s e a r c h , V o l . 2 5 , 1 ( 1 9 7 7 ) , p p . 89-99.
W o l l m e r , R . D . , A S u b s t i t u t e I n v e r s e f o r t h e as is 6 f a
S t a i r c a s e S t r u c t u r e L i n e a r Programs, Mathematics of
O p e r a t i o n s R e s e a r c h , V o l . 2,' 3 . ( 1 9 7 7 ) , p p . 230-239.
.
K r i v o n o z h k o , V . I . a n d A . I . P r o p o i , A Method f o r L i n e a r
Dynamic P r o g r a m s w i t h t h e U s e of B a s i s M a t r i x
Factorization, i n Abstracts o f t h e I X International
Symposium o f M a t h e m a t i c a l Programming, B u d a p e s t ,
August 1976.
K r i v o n o z h k o , V . I . a n d A . I . P r o p o i , S i m p l e x Method f o r
Dynamic L i n e a r P r o g r a m S o l u t i o n , i n O p t i m i z a t i o n
Technique, Proceedings o f t h e 8 t h IFIP Coxference,
Kflrzburg 1 9 7 7 , ( e d . J. S t o e r ) , S p r i n g e r - V e r l a g ,
1978,
P r o p o i , A. a n d V. K r i v o n o z h k o , The Dynamic S i m p l e x Method,
RM-77-24, I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s
A n a l y s i s , Laxenburg, A u s t r i a , 1977.
K r i v o n o z h k o , V.E. a n d A . I . P r o p o i , C o n t r o l S u c c e s s i v e
Improvement Method i n Dynamic L i n e a r Programming
P r o b l e m s I , 11, I z v e s t i a AN SSSX, T e c h n i c h e s k y a
K i b e r n e t i k a , V o l . 3 , 4 , 1 9 7 8 , p p . 5-16,5-14 ( i n X u s s i a n ) .
P r o p o i , A. a n d V. K r i v o n o z h k o , S i m p l e x . Method f o r Dynamic
L i n e a r Programs, RR-78-14, I n t e r n a t i o n a l I n s t i t u t e
for Applied Systems A n a l y s i s , Laxenburg, A u s t r i a .
Cobb, R.H. a n d J . C o r d , D e c o m p c s i t i o n A p p r o a c h e s f o r
S o l v i n g L i n k e d P r o g r a m s , in P r o c c e Z i n g s o f t h e
P r i n c e t . 0 ~Symposium o n b i a t l z e n a t i c a l Programming
( e d . H.W. K u h n ) , P r i n c e t o n u n i v e r s i t y P r e s s , N . J .
1967.
[51]
Ho, J . K . a n d A.S. Manne, N e s t i d D e c o m p o s i t i o n f o r Dynamic
M o d e l s , M a t h e m a t i c a l Programming, V o l . 6 , 2 ( 1 9 7 4 ) ,
pp. 121-140.
[52]
Glassey, C . R . , Nested Decomposition and Multi-Stage
L i n e a r P r o g r a m s , Management S c i e n c e , V o l . 2 0 ,
pp. 282-292.
[53).
N e s t e d D e c o m p o s i t i o n o f a Dynamic E n e r g y Model,
Ho, J . K .
Ma?zagement and S c i e n c e , V o l . 2 3 , ' 9 ( 1 9 7 7 ) ,
p p . 1022-1026.
[54]
K a l l i o , M . , D e c o m p o s i t i o n o f ~ r b o r e s c e n tL i n e a r P r o g r a m s ,
M a t h e m a t i c a l Programming, Vol. 1 3 ( 1 9 7 7 ) ,
pp. 348-356.
[55]
K r i v o n o z h k o , V.E., D e c o m p o s i t i o n A l g o r i t h m f o r Dynamic
L i n e a r Programming P r o b l e m s , I z v e s t i a AN SSSR,
T e c h n i c h e s k y a K i b e r n e t i k a , V o l . 3 ( 1 9 7 7 ) , p p . 18-25
( i n Russian).
[56]
I v a n i l o v , Yu. P. a n d Yu. E . M a l a s h e n k o , Method f o r L i n e a r
Dynamic P r o g r a m i n g S o l u t i o n , ~ i b e r n e t i k a , V o l . 5
( 1 9 7 3 ) , p p . 42-49 ( i n R u s s i a n ) .
[57]
B e l u k h i n , V.P., P a r t i t i o n P r i n c i p l e a n d P r i m a l - D u a l Method
f o r Dynamic L i n e a r Programming P r o b l e m s .
Zhurnal
; ' 3 c h i s l i t e l n o i Matematiki i Matematicheskoi P h y s i k i ,
Vol. 1 5 , 6 ( 1 9 7 5 ) , pg. 1424-1435.
[58]
B e l u k h i n , V.P., P a r a m e t r i c Method f o r Dynamic L i n e a r
A v t o m a t i c a i Te Zemekhanika,
Programming S o l u t i o n .
V o l . 3 ( 1 9 7 5 1 , pp. 95-103.
[591
Aonuma, T., A T w o - l e v e l A l g o r i t h m f o r Two-stage L i n e a r
Journal o f t h e Operations Research S o c i e t y
Programs.
o f gapan, V o l . 21, 2 ( 1 9 7 8 ) , p p . 171-187.
[601
F u k a o , T., A C o m p u t a t i o n Method f o r Dynamic L i n e a r
Programming, J o u r n a l o f t h e O p e r a t i o n s R e s e a r c h
S o c i e t y o f Japan, Vol. 2 , 3 ( 1 9 6 0 ) .
-
--- -
-.
-
.
[611
Nonsmooth O p t i m i z a t i o n , P r o c e e d i n g s o f IIASA Workshop,
March-April 1 9 7 7 , e d s . C . L e m a r e c h a l , R. M i f f l i n ,
Pergamon P r e s s , 1 9 7 8 .
[62]
P r o p o i , A . I . a n d A.B. Yadykiri, M o d i f i e d Dual R e l a t i o n s
i n Dynamic L i n e a r Programming, A v t o m a t i k a i
Tezemekhanika, 5 ( 1 9 7 5 ) , p p . 106-1 14 ( i n R u s s i a n ) .
-.
.
1631
.
. .
-
P r o p o i , A . I . a n d A.B. Y a d y k i n , P a r a m e t r i c I t e r a t i v e
!.lethods f o r Dynamic L i n e a r Programming.
I . Nondegenerate C a s e , Avtomatika i Telemekhcnika,
V o l . 12 ( 1 9 7 5 ) , 11. G e n e r a l C a s ~ .L v t o m a t i k a i
T e l e m e k h a n i k a , V o l . 2 ( 1 9 7 6 ) , pp. 127-135, 136-144.
[64]
Ermol j e v , Yu. 14. , Metody S t o k h a s t i c h e s k o g o Prograrnmirovania
(Methods o f S t o c h a s t i c P r o g r a m m i n g ) , Nauka, M o s c o w ,
1976 ( i n R u s s i a n ) .
[65]
I v a n i l o v , Yu.P. a n d A . I . P r o p o i , Convex Dynamic Programming. Zhurnal Vychis lite Znoi Matematiki i
Matematicheskoi Phyziki, V o l . 1 2 , 3 ( 1 9 7 2 )
( i n Russian).
[66]
P r o p o i , A . I . and A . B . Yadykin, The P l a n n i n g Under
U n c e r t a i n Demand, 1,II. Avtomatika i Telemekhanika,
V o l . 2 , 3 ( 1 9 7 4 ) , pp. 78-83, 1 0 2 - 1 0 8 . ( i n R u s s i a n ) .