PROBLEMS O F DYNAMIC L I N E A R PROGRAMMING
A . I . P r o p o i
N o v e m b e r 1 9 7 6
Research Memoranda are interim reports o n research being con- ducted by the international I n s t i t ~ t e for.Applied Systenls Analysis, and as such receive only limited scientific review. Vicws or opirl- ions contained herein do not necessarily represent thosc o f the Institute or o f the National Member Organizations supporting the fnstitutc.
Preface
The paper is devoted t o t h e problems of dynamic linear programming (models and formalizations, theory and computer methods, extensions and applications).
It contains a brief survey and discusses the necessities and possibilities for research in the area.
Problems of Dynamic Linear Programming
Abstract
Dynamic linear programming (DLP) can be considered as a new stage of linear programming (LP) development.
Nowadays it becomes difficult, maybe even impossible, to make decisions in large systems and not take into account the consequences of the decision over a long- range period. Thus, almost all problems of optimal decision making become dynamic, multi-stage ones.
New problems require new approaches. With DLP it is difficult to exploit only LP ideas and methods:
even having ,found the optimal program, we often do not know how to use it.
This paper represents in some sense the statement of the problem; although it contains a brief survey of DLP, it is focused on the things to be done, rather than on those already being tackled.
1 . I n t r o d u c t i o n
The impact of linear programming (LP) [ 1 , 2 ] models and methods on the practice of decision making is well known. However, both the LP theory'itself, and the basic range of its application are of a one-stage, static nature; that is in this case the problem of the best allocation of limited resources is considered at some fixed stage of development of a system.
However, when the system to be optimized is developing (and not only in time, but possibly, in space as well), and this devel- opment is to be planned, a one-stage solution is inadequate. In this case a decision should be made several stages in advance and the problem of optimization becomes a dynamic,. multi-stage one, for example, problems in long- and sl7ort-range planning, or gen- erally speaking, in programming of a system development.
In fact, any static LP model may have its own dynamic variant, the latter being of growing importance because .of -the increasing
role of planning in decision making. It leads to the emergence of a general problem of dynamic linear programming (DLP), dynamic transportation and distribution problems, dynamic integer pro- gramming, etc.
With a new quality of DLP, new problems arise. While for the static LP the basic question consists of determining the optimal program, the realization of this program (related to the questions of the feedback control of such a program, its stability and sen- sitivity, etc.) is no less important for the dynamic problem.
Hence, the DLP theory and methods should be both based on the classical methods of linear programming and on the methods of con- trol theory, Pontryagin's maximum principle [3] and its discrete version [4] in particular. One should distinguish in the DLP theory two basic, closely related problems: determination of an optimal program and its realization, i.e., control of the pro- gram.
2 . DLP C a n o n i c u l Form
In formulating DLP problems it is useful to single out:
1) state (development) equations of the system with the distinct separation of state and control variables; 2) constraints imposed on these variables; 3) planning period (horizon) T I that is the number of stages, during which the system is considered; 4) per- formance index, which quantifies the quality of control.
State E q ? ! a t i o n s . State equations have the following form:
where the vector x (t) = {xl (t)
. . .
(t) 1 defines the state of the xnsystem at stsge t in the state space X; vector u(t) = {ul (t) # . .
.,
ur(t)} specifies the controlling action at stage t; s(t) =
{slit), ...,
sn(t)} is a vector defining the external effect on the system (uncontrolled, but known a p r i o r i in the deterministic model).Matrices A(t), B(t) are of dimensions (nxn)
,
(nxr) and assumed to be known.P l a n n i n g p e r i o d ( h o r i z o n ) T is supposed to be fixed. Thus in (1): t = 0,1,
...,
T-1.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 o f t h e s y s t e m
i s g i v e n .
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 a s
where f ( t ) = { f, ( t ) .
. . . ,
f m ( t )1
; G ( t ) and D ( t ) a r e o f d i m e n s i o n s (mxn),
( m x r) a n d a r e g i v e 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
w h e r e a ( t ) ( t = O ,
. . .
, T ),
b ( t ) ( t = O ,. . .
,T-1) a r e known n- a n d m - v e c t o r s ;( * r e ) d e n o t e s t h e i n n e r p r o d u c t .
D e f i n i t i o n s . The v e c t o r s e q u e n c e u = (u ( 0 )
, . . .
, u (T-1 ) ) i s a c o n t r o l ( p r o g r a m ) of 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 O , x ( t ),
. . .
, x ( T )1 ,
which c o r r e s p o n d s t o c o n t r o l u from ( 1 1 , ( 2 ),
i s t h e s y s t e m ' s t r a j e c t o r y . The p r o c e s s { u , x ) , which s a t i g f i e s a l l t h e c o n s t r a i n t s o f t h e p r o b l e m ( i . e . , ( 1 ) - ( 4 ) ) , i s f e a s i b l e . The f e a - s i b l e p r o c e s s { u * , x * ) , m a x i m i z i n g ( 5 ) , i s o p t i m a l .Hence, t h e 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 as f o l l o w s .
P r o b l e m 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 ( 1 ) w i t h t h e i n i t i a l s t a t e ( 2 ) a n d t h e con- s t r a i n t s ( 3 )
-
( 4 ),
w h i c h maximize t h e p e r f o r m a n c e i n d e x ( 5 ) .The c h o i c e o f t h e c a n o n i c a l f o r m o f DLP i s t o some e x t e n t
a r b i t r a r y and 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 o f P r o b l e m 1 . I n p a r t i c u l a r , 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 on 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 be 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 p e r f o r m a n c e i n d e x may b e d e - f i n e d o n l y , f o r e x a m p l e , 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 , e t c .
However, s u c h m o d i f i c a t i o n s may b e e i t h e r r e d u c e d t o t h e
c a n o n i c a l P r o b l e m 1 , o r it i s p o s s i b l e t o u s e f o r them t h e r e s u l t s , s t a t e d b e l o w f o r P r o b l e m 1 [ 4 ] .
3. Discussion
F i r s t o f a l l , i t s h o u l d b e n o t e d t h a t i f T = 1 , t h e n P r o b l e m 1 becomes a c o n v e n t i o n a l LP p r o b l e m .
P r o b l e m 1 i t s e l f c a n a l s o b e c o n s i d e r e d a s a c e r t a i n " l a r g e "
LP p r o b l e m w i t h c o n s t r a i n t s on v a r i a b l e s i n t h e f o r n o f e q u a l i t i e s ( I ) , ( 2 ) a n d i n e q u a l i t i e s ( 3 ) , ( 4 ) . I n t h i s c a s e t h e o p t i m a l c o n - t r o l P r o b l e m 1 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 m a t r i x ( T a b l e 1 ) . B u t i n t h e m a j o r i t y o f c a s e s dynamic LP p r o b l e m s a r e f o r m u l a t e d now d i r e c t l y i n s t a t i c LP l a n g u a g e , a s
f o r e x a m p l e P r o b l e m 2 ( T a b l e 2 ) :
P r o b l e m 2. F i n d v e c t o r s Cx* ( I )
, . . .
, x * ( T ) 1 , w h i c h m a x i m i z es u b j e c t t o
L e t u s 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 P r o b l e m 1 a s a n 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 . One c a n o b t a i n f r o m ( 1 ) :
where
@(t,t+l) = I , I
-
identity matrix.
Using (3) it is also possible to get directly the constraints (3) imposed on control variables. As a result, we shall get the following LP problem with a block-triangular matrix (Table 3).
P r o b l e m 3 . Find the control u*, for which
where the vectors h(t), w(t) and the matrices W(t,-c) depend on the known parameters of Problem 1.
Problems 2 and 3 admit their modifications in the same way a.s Problem 1 (a block diagonal structure with coupling constraints or with both coupling contraints and variables, different types of staircase structure, etc.). They have been studied intensely
[1,2,5-151. But unlike control Problem 1, such formalizations of dynamical problems make no distinction between state and con- trol variables. Therefore this approach makes it difficult to use the ideas and methods of the control theory. This difference will be more significant, when "pure" dynamic problems are considered
(stability and sensitivity of DLP systems, control of the optinal programs, etc.).
The DLP problems in the form of Problem 1 were introduced and studied in [4,16-271
.
4 . D L P M o d e l s
Dynamic linear models, known in the literature, are usually formalized in static LP language, as Problems 2 and 3. To intro- duce DLP models in the form of Problem 1, let us consider, as an example, an ecological system.
E c o Z o g i c a Z S y s t e m s . W e s h a l l c o n s i d e r , a s a n e x a m p l e , t h e p r o b l e m o f o p t i m a l s p e c i e s p o p u l a t i o n u s e w i t h i n a g i v e n p l a n n i n g p e r i o d [ 2 8 , 2 9 , 1 8 ]
.
L e t x i ( t ) b e t h e q u a n t i t y o f b i o l o g i c a l t y p e i a t s t a g e t i = l , . . . , n ) . T h e r e a r e r ways o f u s i n g t h e s e s p e c i e s , w e s h a l l d e n o t e a s u . ( t ) t h e i n t e n s i t y o f way j 1 . . . r ) a t s t a g e t .
3
. L e t t h e n u m b e r s a i j ( t ) d e t e r m i n e t h e q u a n t i t y o f s p e c i e s o f t y p e i , c a u g h t ( r e m o v e d f r o m s p e c i e s ) p e r u n i t i n t e n s i t y o f t h e way j .
I f some s p e c i e s i s a p e s t r e l a t i v e t o t h e o t h e r , s p e c i e s o f n
t y p e i a t s t a g e t w i l l d e c r e a s e by t h e C c i s ( t ) x s ( t ) , w h e r e j = 1
c i s ( t ) d e t e r m i n e s t h e number o f i n 6 i v i d u a l s o f t y p e i , d e v o u r e d by a s i n g l e i n d i v i d u a l o f t y p e s .
T h u s , t h e d y n a m i c e q u a t i o n f o r t h e c h a n g e o f t h e i - t h s p e c i e s q u a n t i t y w i l l b e w r i t t e n a s :
Here a i ( t ) i s t h e c o e f f i c i e n t o f n a t u r a l i n c r e a s e ( a i ( t ) > O )
,
o r m o r t a l i t y ( a i ( t ) < O ) o f t h e s p e c i e s o f t y p e i .I t s h o u l d b e n o t e d t h a t i n ( 9 ) ui ( t ) f o r some j may a l s o
J
d e t e r m i n e t h e q u a n t i t y o f t h e j - t h c h e m i c a l , u s e d a t s t a g e t . The c o n s t r a i n t s h e r e may b e f o r e x a m p l e ,
w h e r e g i k ( t )
-
i s t h e s p e c i f i c r e q u i r e m e n t of s p e c i e s i f o r t h e k - t h r e s o u r c e ; d k i s t h e a v a i l a b e q u a n t i t y o f t h e k - t h r e s o u r c e ;w h e r e
u . ( t )
i s d e t e r m i n e d b y t h e t e c h n o l o g i c a l c o n s t r a i n t s o r t h e s a n i t a r y n o r m s . 3T h e p e r f o r m a n c e i n d e x may b e t h e t o t a l h a r v e s t f o r t h e e n t i r e p l a n n i n g p e r i o d
or a specific structure of the ecosystem, desirable at the terminal stage
where the weight ai(t), Bi(T) coefficient is characterized by the importance of the species of type i.
This simple example illustrates the basic idea for formulating DLP models. In what follows models of this type are only mentioned and not written down in detail.
Economic M o d e l s . Linear programming is closely related to
economic models [I, 21
.
In fact, transformation of static LP to dynamic ones are stimulated in great degree by transition from static input-output models to dynamic ones. Dynamic types of in- put-output economic models were considered, for example, in [30, 311. As the DLP Problem 1, a multisector dynamic economic model was formulated and investigated in [32].E n e r g y S y s t e m s . Many models of short- and long-range devel-
opment of energy systems are formulated as dynamical linear prob- lems [33-361. In [33] the energy model was stated as a DLP prob- lem with time lags.
L a r g e O r g a n i z a t i o n S y s t e m s . Many problems in large organi- zation systems such as, manpower p l a n n i n g o r e d u c a t i o n a l s y s t e m s can be viewed as important applications of DLP. Some dynamic models of such kind were considered, for example, in [37].
I n d u s t r i a l S y s t e m s . Many of the short- and long-range plan- ning problems in industry, as well as the production scheduling problems are reduced to DLP. (See, for example, [10,381).
R e g i o n a l a n d Urban P r o b l e m s . The extensive field of appli- cations of DLP is given by regional and urban planning problems.
(See, for example [39] (agricultural model), [40-421 (water re- sources) , [ 4 3 , 4 8 ] (transportation systems)
.
)Reflecting on these short examples and references it should be noted, that many practical problems of control and optimization in energy, water, ecological, regional, and urban systems may be stated in the form of DLP. Therefore the work on the survey of the existing DLP models and on the design of new ones is of essential interest.
5 . T y p e s o f DLP P r o b l e m s
In the preceding section the DLP problem of a general type was considered. Besides the general one, it is useful to single out dynamic transportation and distribution problems [18,43], integer DLP, convex dynamic programming [19,44]. To illustrate this aspect of the problem, let us consider simple transportation problems of DLP.
T h e Dynamic T r a n s p o r t a t i o n P r o b l e m . We shall consider a transportation network with some homogeneous goods of i-th pro- duction and j; th consumption points (plants)
.
Let ai (t) be the production volume of the i-th point (i=l,...,
n) at stage t(t=O,1,
...,
T-1) and b.(t) be the demand value of the j-th point (j=l,...,
m) at stage t. 3It is assumed that each production or consumption point has an opportunity to store goods. We shall denote by y.(t), zi(t) the quantity of stock goods at the i-th consumption point and j-th 3
production point dt stage t; by ci(t), d.(t) the storage expendi- tures of a unit of goods; u (t) will be the quantity of goods 3
ij
transported from the i-th to the j-th point at stage t; c (t) ij is the transportation cost of a unit of goods.
Then the dynamics of the change of stocks will be determined by the equation
n
y. (t+l) = y . (t)
+
C uij (t)-
bj(t) t y . ( O ) = y 03 3 i= 1 I j
m
zi (t+l 1 = z. (t)
-
E uij (t)+
ai (t),
z. (0) = z 01 1
j=1 i
with the constraints
So, the problem is formulated as follows. To find a transportation plan {u*
.
(t) ) (control u* = {u* (t) ) ) such that it maximizes the1 3 i j
total expenditures
subject to constraints (1 0) and (1 1 )
.
It should be noted that with T = 1 and y . (0) = 0, zi(0) = 0 the problem becomes a conventional LP problem of the transportation 3 type.
6 . T h e o r y o f D L P
The theory of DLP is connected with two main problems:
(i) determination of optimal program;
(ii) realization of this optimal program.
D e t e r m i n a t i o n o f O p t i m a 2 P r o g r a m . This side of the theory is linked with duality relations and optimality conditions for Problem 1 , which are the base for building of numerical methods of DLP.
Analysis of the Lagrange function of Problem 1 reveals the following dual DLP problem [ 1 6,4,17]
.
P r o b l e m I D . To find the dual control X = {X(T-I),
...
X(Q))and the associated dual trajectory p = { p ( ~ ) , ...,p (0)) satisfying the co-state ,{dual) equation
with the boundary condition
subject to the constraints
and minimizing the performance index
Here p(t) E En; X (t) E Em; A (t) - > 0 are Lagrange multipliers for constraints (1 )
-
(4).
The dual Problem I D is a control-type problem as is the pri- mal one 1P. Here the variable X tt) is a dual control and p (t) is a co-state or a dual state at stage t. We- have reversed time in
the dual Problem 13: t = T-1 ,...,1,9.
T h e o r e m I . ( T h e DLP " G l o b a l r r D u a l i t y T h e o r e m ) . The solva- bility of either of the 1P or ID problems implies the solvability of the other, with
If the performance index for any of the pair of dual problems
' 1P or 1D is not bounded (from above in 1P, from below in ID), then the other problem has no solution.
Let us introduce Hamilton functions
and
for the primal and dual problems respectively.
T h e o r e m 2 . (Y'he DLP " L o c a Z " D u a Z i t y T h e o r e m ) . The solutions of the primal {u*,x*] and dual {X*,p*) problems are optimal if and only if the values of Hamiltonians coincide:
Thus, the solution of the pair of dynamic dual problems can be reduced to analysis of a pair of static linear programs
max H~ (p(t+l) ru(t))
min HD( ~ ( t )
,
1 (t)linked by the state (1 1 , ( 2 ) and co-state (1 2)
,
(1 3) equations.In particular, it can be shown, that there exist the following optimality conditions for problems 1P and ID [4,17].
T h e o r e m 3 . (Maximum P r i n c i p l e o f P r i m a l ProbZem I P I . The control u* is optimal for Problem 1P if and only if there exists a solution {X*,p*) to dual Problem ID, such that for any t = 0,1,
. . .
,T-1max ~ ~ ( p * ( t + l ) .u(t)) = HD(p*(t+1).u
*
(t)u ( .t)
where the maximization is carried out with respect to all u(t) satisfying constraints (3), ( 4 ) , and X ( t * ) is the optimal dual variable for LP problem (1 5)
.
T h e o r e m 4 . ( M i n i m u m P r i n c i p Z e f o r D u a l P r o b l e m I D ) . The con- trol A* is the optimal for the problem 1D if and only if there
exists a solution { u * , * l to the primal problem 1P such that for any t = 0,1,
...,
T-1min H (x*(t) .A(t)) = HD(xr(t) .A*(t))
,
A (t) D
where the minimization is carried out with respect to all A(t), satisfying the constraints (1 4)
,
' and u* (t) is the optimal dual variable for the LP problem (16).The foregoing optimality conditions define a decomposition principle for solving the pair of dual problems 1P and ID. These conditions permit replacing the solution of the rT and mT - dimen- sional dynamic problems with variables u . (t) and A . (t) (i=l,..
. ,
r;3 3
j=l,
...,
m; t=O,1,...,
T - 1 ) by the successive solution of T static LP problems (15), (16), containing r and m variables respectively and linked by the state equations (I), (12) with the boundary con- ditions ( 2 ) , (13).T h e C o n t r o l o f t h e O p t i m a l P r o g r a m . Unlike static LP the realization of optimal solution in dynamic problems has no less importance than its determination. One should mention here the
questions of realization of the optimal solution as a program (i.e., in dependence o f the numbers of stage: u*(t) (t=O,...,T-1)) or
as a feedback control (i.e., in dependence on the current value of states: u* (t) = u* (t, x* (t) ) ( t = O T - 1 ) ; stability and
sensitivity of the optimal system, connection of optimal solutions for long- and short-range models, etc. These problems are waiting for their solution. We shall mention only some of them here.
(i) It is often necessary to determine in what way the performance index and/or the optimal control will behave when the parameters of the problem are changing (for example, "prices" s(t), b (t)
,
"resources" f(t),
"demand" s (t) ),
(parametric DLP).
Solu-tion methods in this case can be developed on the basis of static parametric LP [ 1 , 2 ] . A general approach to parametric problems of linear and quadratic programming is given in [45].
(ii) In computing the optimal program, especially for the large T I it is very important to know, how the inaccuracy in know- ledge of matrices A(t), B(t) coefficients and other parameters of the system influences the stability of the optimal program and the quality of control (sensitivity problem).
( i i i ) Assume t h a t a n a g g r e g a t e d DLP problem f o r a l a r g e p l a n - n i n g h o r i z o n T i s s o l v e d . How c a n t h e i n f o r m a t i o n a b o u t t h e o p t i - mal d u a l p r o c e s s { p * ( t ) , X * ( t ) ) o f t h e a g g r e g a t e model b e o f any u s e t o t h e o p e r a t i v e s o l u t i o n ( f o r e a c h c u r r e n t s t a t e o f t h e s y s t e m ) o f more d e t a i l e d b u t h a v i n g a s h o r t e r p l a n n i n g h o r i z o n DLP problem?
( i v ) How c a n t h e l o c a l s y n t h e s i s of t h e s y s t e m , i . e . , t h e c o n t r o l o f t h e form
6u* ( t ) = ~ ( t . 1 bx* ( t ) 1 ( t = O , l f . - . , T - 1 )
f o r s m a l l d e v i a t i o n s of s t a t e s b x * ( t ) from t h e o p t i m a l t r a j e c t o r y x* ( t ) be c a r r i e d o u t ?
7 . E c o n o m i c I n t e r p r e t a t i o n
A s t a n d a r d economic i n t e r p r e t a t i o n c a n b e g i v e n t o t h e p a i r o f d u a l problems 1P and 1 D and r e l a t i o n s between them [ 1 7 , 1 8 1 , a n a l o g o u s t o t h o s e o f t h e s t a t i c LP p r o b l e m s [ 1 , 2 1 .
8 . DLP M e t h o d s
W e 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 p r o b l e m s .
T h e D L P ini its M e t h o d s . T h e s e methods a r e t h e d e v e l o p m e n t o f 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 . Now two main a p p r o a c h e s b e g i n t o b e r e v e a l e d e n a b l i n g u s t o b u i l d DLP f i n i t e methods.
The f i r s t a p p r o a c h i s b a s e d on d e c o m p o s i t i o n m e t h o d s o f LP [ 1 , 4 6 , 4 7 ] , e s p e c i a l l y on Dantzig-Wolfe d e c o m p o s i t i o n [ 1 , 4 6 1 . F o r P r o b l e m s 2 and 3 t h i s t e c h n i q u e was u s e d i n [ l o - 1 2 , 1 5 1 , f o r Prob-
lem 1 i n [ 1 9 , 2 1 , 2 7 1 . I t s h o u l d be n o t e d h e r e t h a t o r i g i n a l l y t h e Dantzig-Wolfe d e c o m p o s i t i o n method was d e v e l o p e d f o r LP p r o b - l e m s w i t h b l o c k - a n g u l a r s t r u c t u r e s u c h a s i n Problem 3 [ 6 1 .
The s e c o n d a p p r o a c h i s b a s e d on t h e f a c t o r i z a t i o n o f c o n s t r a i n t m a t r i x and u s e d f o r P r o b l e m s 2 and 3 i n [13-151 and f o r P r o b l e m 1
i n [ 2 6 ] .
I t e r a t i v e M e t h o d s . The application of the LP finite methods to the dynamic problems causes certain difficulties especially for the large planning horizon T. This can be explained by the fact that in these methods the approach of an approximate point to the optimum is fulfilled over the vertices of the feasible polyhedral set (in some space). But the number of vertices of such a set for the dynamic problems increases exponentially with T I so does the volume of calculation.
The iterative LP methods seems to by-pass these difficulties.
They are also characterized by low demands to the.computerls memory, the simplicity of the computation flowchart, low sensi-
tivity to the disturbances.
We shall differentiate the following iterative methods.
P e n a l t y F u n c t i o n s . This is one of the most universal and simple technique of optimization. But its direct use of
the DLP problem is hampered by relatively low convergence rates in the vicinity of solution. The idea of extrapolation of decision was suggested in [ 2 2 ] which remarkably improves the effectiveness of the method for static LP and is developed for DLP in [ 2 3 1 .
G e n e r a Z i z e d G r a d i e n t s M e t h o d s . The other group of methods is based on finding the extremum of function
$(Alp) = max L(u,x;X,p) x , u ~ O
$(u,x) = min L(u,x;X,p) p,X>O -
where ~ ( u , x ; X , p ) is the Lagrange function of Problem 1P (Problem ID).
I t can be shown that minimization of $(Alp) is equivalent to solution of the dual Problem ID, while the maximization of $(u,x) is equivalent to the solution of primal Problem 1P. But functions
$ and $ nondifferentiable by nature, so the generalized gradient technique [ 4 8 ] is needed. Application of the generalized gradients for DLP reduces the solution of large DLP Problem 1P to successive solutions of small LP problems (1 5)
,
( 16).
Modified Lagrange F u n c t i o n Methods. The i d e a o f t h e method was s u g g e s t e d i n [ 4 9 ] , a l t h o u g h t h e m e t h o d s o f t h i s g r o u p b e g a n t o b e d e v e l o p e d o n l y r e c e n t l y [50-511. T h i s a p p r o a c h c o m b i n e s t h e a c c u r a c y of f i n i t e m e t h o d s w i t h s i m p l i c i t y o f i t e r a t i v e o n e s . The t h e o r y o f t h e method f o r DLP was c o n s i d e r e d i n [ 2 4 1 . One o f
i t s r e a l i z a t i o n s b a s e d o n employment o f t h e Kalman-Bucy f i l t e r t e c h n i q u e [ 5 2 ] was g i v e n i n [ 2 5 ] .
9. Some Extensions
N a t u r a l l y , a l l t h e p r a c t i c a l p r o b l e m s c a n n o t be k e p t w i t h i n t h e framework o f DLP. H e r e w e s h a l l s t a t e t h e f i e l d s o f DLP d e v e l - o p m e n t , w h i c h a r e o f t h e g r e a t e s t i n t e r e s t .
Nonlinear Dynamic Programming. T h i s i s e s s e n t i a l l y t h e o p t i - m a l c o n t r o l t h e o r y o f t h e g e n e r a l t y p e o f d i s c r e t e s y s t e m s w i t h
s u b s t a n t i a l u s e o f n o n l i n e a r programming t e c h n i q u e s . Some a p p r o a c h e s i n t h i s d i r e c t i o n h a v e b e e n c o n s i d e r e d i n [ 4 , 4 4 ] .
Stochastic DLP. W e s h a l l o n l y n o t e [ 5 3 , 5 4 ] h e r e t h e p a p e r s o n m u l t i - s t a g e s t o c h a s t i c programming.
Maxi-min (mini-max) D L P Problems. The s o l u t i o n o f s u c h p r o b - l e m s i s o f c o n s i d e r a b l e p r a c t i c a l i n t e r e s t when g u a r a n t e e d c o n t r o l q u a l i t y i s t o b e o b t a i n e d u n d e r t h e c o n d i t i o n s o f u n c e r t a i n t y , a s w e l l a s f o r s e n s i t i v i t y a n a l y s i s , a n d game p r o b l e m s o f p l a n n i n g .
L e t i n P r o b l e m 1 t h e v a l u e s o f v e c t o r s s ( t ) b e unknown, a n d o n l y t h e r a n g e o f t h e i r v a r i a t i o n s St b e known, w h i c h i s assumed t o be bounded p o l y h e d r o n s .
P r o b l e m 4. F i n d c o n t r o l u* a n d t h e t r a j e c t o r y x* s u b j e c t t o ( 1 ) - ( 3 ) a n d p r o v i d i n g
max min J ( u , s ) = w-
,
u S 1 1
w h e r e s = { s ( t ) E St}, t h e p e r f o r m a n c e i n d e x J 1 i s d e t e r m i n e d f r o m ( 5 ) .
P r o b l e m 5. F i n d c o n t r o l u* a n d t r a j e c t o r y x* s u b j e c t t o ( 1 ) - ( 3 ) a n d p r o v i d i n g
max min
...
max min J 1 = o-u(0) s(3) u(T-1) s(T-1) 2 -
The solution of Problems 4, 5 guarantees the values of the performance index J 1 no worse than oi, if the program control u*
is realized and no worse than o (with o- > o-) if there is a 2 2 - 1
possibility of recalculation of the program for each x(t) (the feedback control u(t) = u(t,x(t)) of a system). The solution of Problems 4, 5, is considered in [ 5 5 ]
.
Above a short survey has been given of the contemporary state- of-the-art in dynamic linear programming, reflecting the author's possibilities and point of view. The development of optimization methods for dynamic problems, i.e., planning and control methods
for large scale problems (which are of such a necessity in our dynamic world), irrespective of the directions they will take, will, undoubtedly, enrich the practice of decision making in complex systems.
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