Dynamic Linear Programming

Propoi, A.I.

IIASA Working Paper

WP-79-038

May 1979

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Propoi, A.I. (1979) Dynamic Linear Programming. IIASA Working Paper. WP-79-038 Copyright © 1979 by the author(s).

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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 paper p r e s e n t s a survey of dynamic l i n e a r programming (DLP) models and methods, i nc lud ing d i scuss ion of d i f f e r e n t f o rma l i za t i ons of dynamic l i n e a r programs, models which can be 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 con- d i t i o n s f o r DLP, numerical methods - -bo th f i n i t e - s t e p s and i t e r a t i v e .

DYNAMIC L I N E A R PROGWL!ING*

A . Propoi**

INTRODUCTION

Dynamic l i n e a r programming (DLP) can be cons idered a s a n e x t s t a g e of 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 imposs ib l e , t o make de- c i s i o n s i n l a r g e sys tems and n o t t o t a k e i n t o accoun t t h e con- sequences o f t h e d e c i s i o n over some p e r i o d o f t i m e . Thus, a lmost a l l problems 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 long-range energy , wa te r and o t h e r r e s o u r c e s supp ly models, problems of n a t i o n a l s e t t l e m e n t p l ann ing , long- range a g r i c u l t u r e inves tment p r o j e c t s , manpower and e d u c a t i o n a l p lann ing 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 problems r e q u i r e new approaches. Wi th in DLP it i s d i f f i c u l t t o e x p l o i t on ly 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 de te rm in ing t h e op t ima l p rograv , t h e implementat ion of 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 feedback c o n t r o l o f such 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 , etc . ) i s no less impor tan t f o r t h e dynamic problems. Hence, t h e DLP theo ry and methods shou ld be bo th based 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 v e r s i o n [ 5 ] i n p a r t i c u l a r .

DLP CANONICAL FORM

I n f o rmu la t i ng DLP problems 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 system w i t h t h e s t a t e and c o n t r o l v a r i a b l e s ;

- 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 ; - p lann ing hor i zon T and t h e l e n g t h o f each t i m e p e r i o d ;

*On l eave 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 appear i n "Numerical Opt imizat ion of Dynamic Systems"

(Eds. L.C.W. Dixon and G.P. Szeg6) North-Hol land.

- o b j e c t i v e f u n c t i o n (per formance i n d e x ) , which q u a n t i f i e s t h e q u a l i t y o f c o n t r o l .

S t a t e e q u a t i o n s . 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 )

where t h e v e c t o r x ( t ) = x t . . . x n t 1 d e f i n e s t h e s t a t e o f

t h e system a t s t a g e t i n t h e s t a t e space X , which is supposed

t o b e t h e n-dimensional e x c l i d e a n space 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

a t s t a g e t ; s ( t) = s t . . s n t 1 i s a v e c t o r d e f i n i n g t h e

e x t e r n a l e f f e c t on t h e system ( u n c o n t r o l l e d , b u t known a p r i o r i

i n t h e d e t e r m i n i s t i c model ) . 0

I t is 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 sys tem i s

given :

PIann inq h o r i z o n T i s supposed t o b e f i x e d . Thus i n ( 1 ) :

t = 0 , 1 , . . . ,T-1. The l e n g t h o f each t i m e p e r i o d may b e t h e

same f o r t h e whole p l a n n i n g ho r i zon ( 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 p e r i o d s . For example, i n long-range

energy supp ly models t h e p lann ing ho r i zon might be 100 y e a r s ,

which forms 10-per iod 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 equa l 10 y e a r s and two p e r i d s equa l 20 y e a r s [ 6 ] .

C o n s t r a i n t s . I n r a t h e r g e n e r a l form c o n s t r a i n t s imposed on t h e

s t a t e and 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 as

m where f (t) = i f l ( t ) ,. . . , f m ( t ) 1 E E ( t = 0 , 1 , - - = ,T - I )d

O b j e c t i v e f u n c t i o n . (Pe r fo rmance i n d e x ) , which i s t o b e maxi-

mized 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 is 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 . The v e c t o r s e q u e n c e

x = ( x ( 0 ) , x (1 ) , . . . , x ( T ) ), which c o r r e s p o n d s t o c o n t r o l u f rom

( I ) , ( 2 ) , i s t h e s y s t e m ' s t r c , i e c t o r y . 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 rob lem ( e . g . ( 1 ) - ( 4 ) ) i s

a f e a s i b l e p r s c e s s . A f e a s i b l e p r o c e s s { u * , x * > wh ich maximizes

t h e o b j e c t i v e f u n c t i o n ( 5 ) i s o p t i m a l .

The DLP p rob lem i n i t s c a n o n i c a l fo rm is f o r m u l a t e d a s f o l l o w s .

ProbZen; I ? . F i n d a c o n t r o l u* and a t r a j e c t o r y x* , s a t i s f y i n g

t h e s t a t e e q u a t i o n s w i t h t h e i n i t i a l s t a t e a n d t h e con-

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 ) . . I n Problem 1P v e c t o r s xO, s ( t ) , f ( t ) , a ( t ) , b ( t ) and 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 is t o some e x t e n t a r b i t r a r y

and t h e r e are 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

Prob lem 1P. I n p a r t i c u l a r , b o t h 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 ; con-

s t r a i n t s on s t a t e and 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 or 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 ta te 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 , nowever , 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 Prob lem 1P or t h e r e s u l t s s t a t e d below f o r

Problem 1P c a n be e a s i l y e x t e n d e d fo 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 Prob lem 1P becomes a c o n v e n t i o n a l LP

prob lem. Prob lem 1P i tse l f c a n a lso be c o n s i d e r e d as a c e r t a i n

s t r u c t u r e d LP p rob lem 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 rob lem 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 matr ix .

Sone t imes dynamic LP p rob lems 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 examp le , P rob lem 2 , [8 ] - [ I 01 :

. - ?~o.?Lern 2 . F i n d v e c t o r s {x* ( 1 ) , . . . , x * ( T ) 1, wh ich m a x i n i z e

s u b j e c t t o

One can 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 fo l l ow ing LP

problem w i th a b l o c k - t r i a n g u l a r m a t r i x i s ob ta ined .

Problem 3 . Find v e c t o r s {u* ( 0 ) , . . . ,u* (T-1 ) 1 , which maximize

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 w ( t , - r ) depend on

t h e known parameters 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 admit

va r i ous mod i f i ca t i ons i n t h e same way a s Problem 1P .(e.g. a block

d iagona l s t r u c t u r e w i t h coupl ing c o n s t r a i n t s and/or v a r i a b l e s ,

e t e . ) . 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, such f o r m a l i z a t i o n s of dynamical

problems do n o t use e x p l i c i t l y t h e s t a t e equa t ions o f a system.

Therefore t h i s approach makes it 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 heo ry . The DLP problems i n t h e form

of Problem 1P w e r e i n t roduced i n [11 ,5 ] .

DLP MODELS

There i s a l a r g e and r a p i d l y expandigg 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 on ly some of them a r e mentioned.

Z??:alr;ic ntlZt?:-SPC t o r economtc mode 2 s . These a r e b a s i c a l l y exten-

s i o n s of dynamic i npu t -ou tpu t models, when s u b s t i t u t i o n i s

al lowed ( s e e f o r exanp le [ 1 ,2 ,12 -141 ) . A longs ide w i t h d i s c r e t e

t ime f o rmu la t i on con t i nuous t i m e economy models have a l s o been

deve loped ( e . g . i n [ I51 1 .

Enera; n iode is . For long- range p l a n n i n g t h e prob lem i s t h e

f o l l ow ing : 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 o f t h e secondary

energy p r o d u c t i o n c a p a c i t i e s and under g i v e n p r imary energy and

nonenergy r e s o u r c e s supp 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 such

a mix o f t h e energy 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 energy demand and min imizes t h e

t o t a l c o s t o f such t r a n s i t i o n . These ene rgy supp l y 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

n e x t 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 energy-

economy i n t e r a c t i o n . T h i s problem can 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 . Pr imary 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

o f p r imary 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 su rvey o f d i f f e r e n t DLP energy-resources-economy models i s

g i ven i n [ I 81 . Wa%er m o d e l s . The DLP models f o r wa te r sys tems are t h e wa te r

supp ly models (which a r e concep tua l y c l o s e t o energy supp l y

models) , t h e prob lems o f 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

b a s i n , 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 edu-

c a t i o n a l i n s t i t u t i o n s which a s w e l l a s s a t i s f y a l l t h e con-

s t r a i n t s of t h e sys tems ( 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 ann ing

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 p romot ions . t

There 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 models , b o t h f o r na-

t 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 ] .

., . ...:- - 3 ~ s e tt Z ~ ~ ~ e n t p l a n n i n g . These DLP models re la te t o f i nd -

i n g an o p t i m a l , i n some ' sense, m i g r a t i o n f l ows i n a r e g i o n O r a

cou:1tr-y [ 231 .

3 e ~ Z t h cQre sgstems. The problem o f h e a l t h c a r e sys tem p lan -

n i n g may be reduced t o an a l l o c a t i o n f unds , h e a l t h manpower and

o t h e r r e s o u r c e s o v e r t ime 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 such 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 reduced 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 [ 24 ] .

Agricu Z ture mode is. S e ~ a r a t e DLP models concern p l a n n i n g prob-

l e m s i n l i v e s t o c k b r e e d i n g , c r o p p r o d u c t i o n , r e s o u r c e s u t i l i z a -

' t i o n , e tc . The i n t e g r a t e d models 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 fa rm growth

[25-281 . Production models. These DLP models 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

i n d u s t r y [ 8 , 9 , 2 9 , 3 0 ] .

Dynamic t r a n s p o r t a t i o n problems. Transh ipment and o t h e r n e t -

work prob lems 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 prob lems a r e c o n s i d e r e d i n

[3 ,31 ,32 ] .

Miscellaneous models. I t is wor thwh i le t o ment ion' h e r e some

" i n d i v i d u a l " DLP a p p l i c a t i o n s : conges ted u rban 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 prob lems

L3.51.

THEORY OF DCP

The t h e o r y o f DLP i s connec ted w i t h 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 c o n d i t i o n s 11 1 ,36 ,7 ,5 ] . Problem ID. To f i n d a d u a l c o n t r o l X = { A (T-1) , . . . , X ( O ) and

a d u a l t r a j e c t o r y p = { p ( ~ ) , . . . ,p (O) s a t i s f y i n g t h e c o - s t a t e

( d u a l ) e q u a t i o n C

w i t h t h e boundary c o n d i t i o n

s u b j e c t t o t h e c o n s t r a i n t s

and minimize 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 ) IZ Ern are Lagrange m u l t i p l i e r s f o r con-

s t r a i n t s ( 1 ) - ( 4 ) .

The d u a l Problem I D i s a c o n t r o l - t y p e problem as i s t h e p r i m a l

one 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 and 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. W e have r e v e r s e d t i m e

i n t h e d u a l Problem I D : t = T-1, ..., 1,O.

Theorem I . ( 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 prob lems 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 Jp (u * ) = JD ( A ) . .

L e t u s i n t r o d u c e Hami l ton f u n c t i o n s

f o r t h e p r i m a l and d u a l p rob lems r e s p e c t i v e l y .

Theorem 2 . (The 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* ) and d u a l p rob lems 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 Hami l ton ians 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 of dynamic d u a l p rob lems can be

reduced t o a n a l y s i s o f a p a i r o f s ta 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 nked by t h e s t a t e ( I ) , ( 2 ) and c o - s t a t e ( 6 ) , ( 7 ) e q u a t i o n s .

I n p a r t i c u l a r , i t can be shown, t h a t maximum p r i n c i p Z e fo r dua l

Problem I P and minimum p r i n c i p l e f o r dual Problem I D a r e ho ld

[ 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 given 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. These methods are t h e ex tens ion o f f i n i t e

l a rge -sca le LP methods f o r t h e dynamic problems. Two b a s i c

approaches may be s i n g l e d o u t here: compact i n v e r s e and decom-

p o s i t i o n methods [9,10,37,38: . Compact i n v e r s e methods. These methods are v a r i a n t s of t h e

s implex method us ing 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 en t ,e r and t o l eave t h e b a s i s . The

approach i s be ing developed both f o r s t r u c t u r e d and g e n e r a l LP

( i n t h e l a t t e r c a s e t h e s p a r s e n e s s o f c o n s t r a i n t m a t r i x i s

e x p l o i t e d ) ( s e e 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 approach was used i n [37,41-441 . For DLP

Problem 1P t h e approach was prcposed i n [45,46] and desc r ibed

i n (47-491, 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

ex tens ion o f t h e s implex method. The main concept of t h e s implex

method--the b a s i s - - i s 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 lann ing pe r iod . The i d e a of t h e method i s

t h e fo l lowing. Cons ider t h e c o n s t r a i n t s ( 3 ) f o r t = O . Using

( 2 ) , they can be w r i t t e n i n t h e form

and accord ing t o t h e convent iona l s implex procedure be p a r t i -

t i oned a s

\.r:rlcre D ( 0 ) i s a s q c a r e n o n - s i n g u l a r ( m m ) m a t r i x . I f 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 u1 ( 0 ) = 0 a n d f i n d f r o m ( 1 3 )

a L a s i c f s a s i b l e s o l u t i o n u 0 ( O ) - > 0 . I n t h e dynz?.nic c a s e

(T > 1 ) n o t a l l componen ts o f u 1 ( 0 ) a r e z e r o s f o r b a s i c

s a l u t i o x s . 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 d be 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 ~ g t h e s t a t e e q u a t i o n s ( 1 ) f o r

t = 0 i n t h e same way a s i n ( 1 3 ) a n d s u b s t i t u t i n g u o ( 0 ) f r o m

( 1 3 ) 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 , o n e c a n 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 ( 1 2 ) : A

j ( l ) ; ( l ) = 1 ) e ( 1 = , 0 ; ~ ( 1 ) ; . Tne m s t r i x 6(1) depenscis or, G ( 1 1 , C , 3 ( 3 ) a22 car! be a g a i n p a r t i t i o z e d i n t h e

h A O h A

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 .

h

The 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

Loccl 3 c s l s a t s t e p t . The s e t of l o c a l b a s e s { D o ( t ) ) f o r t h e

w h o l e p la r , r . i ng p e r i o d t = 0 , 1 , . . . ,T-1 p l a j r s t h e s a m e r o l e a s '

5 a s i s i n s t a n . i z r d s i m p l e x - m e t h o d . 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 i m p l e m e n t a l l s i m p l e x procedures ( s e l e c t i o n o f v e c t o r s

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 ,

u p d a t i n g ) v e r y e f f e c t i v e l y [ 4 9 ] . I t s h o u l d be a l s o n o t e d , t h a t

t h e d y n a m i c s imp lex -me thod g i v e s n o t o n l y c o m p a c t 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 e a c h i t e r a t i o n .

The e x t e n s i o n o f t h e dynamic s i m ~ l e x - m e t h o d t o DLP p rob le i ns w i t h

t i m e l a g s o r n c n - n e g a t i v e 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-

s i o n s o f t h e method are c o n s i d e r e d i n [ 4 6 , 4 9 1 .

2ecor?=,cs i 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 ap- - p l i c a t i o n o f d i f f e r e n t d e c o m p o s i t i o n m e t h o d s ( f i r s t o f a l l , ~ a n t z i c j -

Wol fe d c c o m ; ~ o s i t i o n ~ r i ~ c i 2 l e j t o t h e DLP ~ r o b l e m , and 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 d e s c r i b e d 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 was u s e d i n [ S S ] . C o n s i d e r t h e

s e q u e n c e o f LP p r o b l e m s ( t = T- 1 , . . . , I , 0 ) :

max p ( t + l ) x ( t + J )

where R (xo) i s t h e s e t of 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 t s t a t e 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 .

So lu t i ons o f t h e s e LP problems w i th op t ima l va lues of p * ( t + l )

g i v e an op t ima 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 (xo) can be r e p r e s e n t e d a s a l i n e a r convex combina- t t i o n o f i t s extreme p o i n t s (assuming boundness o f R ( x o ) ) . t Thus, problem (1 4 ) can be r e w r i t t e n as

max p ( t + l ) [ L A ( t ) x i ( t ) p i ( t ) + B ( t ) u ( t ) + s ( t ) l i -

Problem (15) i s a master problem a t s t e p t. For i t s s o l u t i o n

convent ional column g e n e r a t o r procedure can be adopted. Other

methods f o r s o l u t i o n o f Problem 1P based on d i f f e r e n t decomposi-

t 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 cons idered i n [56-601.

The 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 problems may cause 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 lann ing peri'ods T. The reason f o r it 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 po l yhedra l set ( 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 by-pass 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 computer ' s co re memory,

s i m p l i c i t y of t h e computer implementat ion, low s e n s i t i v i t y t o

d i s tu rbances . Disadvantages of 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 fo l l ow ing i t e r a t i v e methods.

; 1 L i : r ~ c r ~ ! ! : ~ n t m?t i ;oJ ; ; . 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 ex t remum o f f u n c t i o n s :

' (:, p = max L ( u f x ; A , p ) ; ; ( u , ~ ) = min L ( u f x ; ; \ ,PI ( 1 6 ! X f U L 0 p i x > 0 -

w h e r e L ( u , x , A , p ) 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 ) .

M i n i m i z a t i o n o f Y i s e q u i v a l e n t t o t h e s o l u t i o n o f t h e d u a l

P r o b l e m ID, w h i l e t h e m a x i m i z a t i . o n o f c$ i s e q u i v a l e n t t o t h e

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 ] . 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 c o n s i d e r

P r o b l e m 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 ; U t = ! u ( t ) I ~ ( t ) u ( t ) 2 r ( t ) , u ( t ) 2 0 ) .

v ( 1 ) C h o o s e a r b i t r a r y d u a l c o n t r o l { A ( t ) j ( t = 0 , 1 , . . . ,T-1 ) . ( 2 ) Cernpute d u a l s t a t e t r a j e c t o r y ~ ~ ' ( t ) ; f r o m t h e d u a l

s t a t e e q u a t i o n s ( 6 ) w i t h b o u n d a r y c o n d i t i o n ( 7 ) . v

( 3 ) 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 ) ) . 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 .

v ( 4 ) Compute p r i m a l s t a t e t r a j e c t o r y x ( t l f r o m ( 1 ) and

( 2 ) f o r u V ( t ) .

v v (5) Compute g e n e r a l i z e d g r a d i e n t o f Y : S Y ( A ) = h ( t ) , w h e r e h v ( t ) = f ( t ) - ~ ( t ) x ' ( t ) - ~ ( t ) u \ ; ( t ) . ( 1 7 )

v+ 1 ( 6 ) Compute new d u a l c o n t r o l A : . v+l v v v

= A - a ( ' 3 ) I ( ( V = O , l , 2 . . . . ) ( 1 8 )

- - - * r . . 3 : 1 i . ? ~ : . L e t n V - t O ; v - + ~ ; - . 1 a v - -. Then ?' ( i V ) -+ ! ' (A ) ,

:>=o * * *

) ~ ' + . I , 1 = {A ( t ) 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

f u n c t i o n : ( u ) i n ( 1 6 ) .

,,--,:*.c.. - - - ... , .. -, L Q C L : : n 3 2 i i :C ~ u Y ~ c ? . ~ s ; : . - : e z , : z : [ 6 2 631 . 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 a n d i t e r a t i v e m e t h o d s : u n d e r

ain imal requirements f o r s t o r a g e i t ensures monotonic conver-

cjsnce i n a f i n i t e number of s t e p s . I n a sense , t h i s method c-n be

considered a s a "smooth" a l t e r n a t i v e t o t h e nonsmooth approach,

dsscr ibeci above. Le t t h e L a g r a ~ q e f ~ i c t i o n 5 ( u , ) . ) o f Problsm 1 2

( I D ) be ailqnented by q ! ~ a d r a t l c te~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 fo l l ow ing p a i r of problems can . be

cor?s iders2

\U ( A ) = rriax i?(u ,A) ; t , ( u ) = in M(u,X! f1 . -

U X

Opposi te t o ?' ( A ) , t h e f u ~ c t i o n P (?, j i s a s a ~ o t h concave m f u n c t i o n ; d e r i v a t i v e o f '? ( A ) i s a con t inuous f7 t?c t i on of X m d m a'? ( E , ) / a A ( t ) = h ( t ) where h ( t ) i s computed from (171, when M

u* = ui(A) i s a s o l u t i o n o f t h e l e f t problem i n ( 2 0 ) (621.

Problems n i n Y ,(1) and max $ ( u ) a r e t h e p a i r of modi f ied d u a l )I M

~ r o b l e ~ . s , a 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 does n o t chznge t h o g e n e r a l d u a l

r e l a t i o n s . 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 sol ire e i t h e r t h e d u a l p r o b l e ~ min ? 0.) with t h e no2-

smooth fu-rlction Y o r t h e modi f ied Zual p rob len n i n ? ( A ) w i t h >! 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 lgor i thm:

( 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 p r o b l e n i n ( 2 0 ) .

Th is ?robLen has l i n e a r s t a t e e q u a t i o n s ( 1 ) , g u a d r a t i c ob jec-

t i v e f u n c t i o n ( 1 9 ) and s imp le c o n s t r a i n t s on c o n t r o l u ( t )

( e . 9 . I) - < u(t) - < c ( t ) 1 . There fo re f o r i t s s o l u t i o n o p t i n a l

c o n t r o l t2chnique (iv-hich reduces i n t h i s c a s e t o s o l u t i c n of 2

n a t r i x A i c c a t i e q ~ a t i o n ) can be e f f e c t i v e l y 2 ~ 2 l l e d [ 6 3 1 .

v ( 3 ) L e t u b e a s o l u t i o n o f t h e abo l re p r o b l m . C c m ~ a t e

h" ( t ) f r o m ( 1 7 ) f o r t h i s uV and new d u a l c o n t r o l X v+ 1

f r o m

p + 1 ( t ) = X v ( t ) - 4 h v ( t ) ( \ J = 0 , 1 , 2 , . - . )

T k e o r ~ s , ~ 4 . F o r a n y e > 0 ,

* * and b e g i n n i n g f r om some f i n i t e vO : X v €11 , u;' EU , v > \ J ~ ,

& -

I

where A T I 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 p r o b l e m s

I D and 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 number 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 ?r imal 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 and m e t h o d s . 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 imp lemen ta - t i o n o f t h e m e t h o d s . 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 me thods c a n b e much more 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 ) a p p r o a c h (see, f o r i n s t a n c e , [ 3 9 , 5 3 1 ) . T h e r e - f o r e c o m p u t e r t e s t i n g and 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 i m p o r t a n t p r o b l e m . O t h e r i m p o r t a n t d i r e c t i o n s o f r e s e a r c h a r e t h e p r o b l e m o f t h e i m p l e m e n t a t i o n o f o p t i m a l c o n t r o l a n d p o s t - o p t i m a l a n a l y s i s o f t h e mode l ( f e e d b a c k v s p rog ram 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 be- tween s h o r t and l o n g - t e r m p r o b l e m s , e t c . )

I t s h o u l d a l s o b e n o t e d t h a t n o t a l l d y n a i c a l o p t i m i z a t i o n 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 ramework o f DLP. T h e r e f o r e , 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 , maxmin, n o n l i n e a r dynamic p r o g r z m i n g p r o b l e m s i s o f l a r g e p r a c t i c a l i n t e r e s t . Some w o r k h a s S e e n a l r e a d y d o n e i n t h i s d i r e c t i o n (see, e . g . [64 -6611 .

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