Linear programming over an infinite horizon

Linear programming over an infinite horizon

Citation for published version (APA):

Evers, J. J. M. (1973). Linear programming over an infinite horizon. Technische Hogeschool Eindhoven.

https://doi.org/10.6100/IR88742

DOI:

10.6100/IR88742

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Published: 01/01/1973

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AN INFINITE HORIZON

PROEFSCHRIFT

ter verkrijging van de ~raad van doctor in de

technische wetenschappen aan de Technische Hogeschool Eindhoven, op gezag van de rector magnificus, prof.dr.ir,G.Vossers, voor een commissie aangewezen door het college van dekanen in het openbaar te verdedigen op vrijdag 29 juni 1973 te 16.00 uur.

DOOR

Joseph Johannes Maria Evers geboren te Rillegom

prof.dr.J.F.Benders en dr.ir.M.L.J.Hautus.

CHAJ;'TER 2 CHAPTER 3 CHAPTER 4 CHAPTER 5 CHAPTER 6 CHAPTER 7 CHAPTER 8 CHAPTER 9 an infinite horizon.

Mathematical formulation of the linear programming system.

Directedness, feasibility, and regularity.

Partial objective functions.

Inferiority.

Optimality.

Parametrie properties.

Paths of equilibrium.

Semi equilibrium paths.

CHAPTER !0: Equivalent linear programming problems over a finite horizon.

lEFERENCES LIST OF SYMBOLS SUBJECT INDEX 24 38 59 66 85 l i l 124 I 33 169 184 185 187

1. LINEAR PROGRAMMING IN GROWTH MODELSOVER AN INFINITE HORIZON

1.1 Introduction.

Practically all applications of the linear programming theory to growth roodels in the economy (2:265 ), (9:254 ), have in common that they limit themselves to a program over a finite horizon. Into many models, however, the fixatien of a horizon introduces

a certain arbitrariness (4:1~5) which can be avoided by

farmu-lating the problem over an infinite horizon. Then, mathematical-ly speaking, a linear programming problem arises in an iufinite dimensional space. This study presents a mathematica! analysis of such a problem, which results in a general solving procedure. In this analysis we assume a partienlar characteristic with re-gard to the structure of the linear programming problem.

In this chapter four growth ~odels will be ~iscussed in macro

economical terms, in order to arrive at a first formulation of the problem in an economical context, and to show that practic-ally every realistic linear growth problêm can be formulated into a model possessing the specific structure presumed in the mathematica! treatment. As such, the models discussed are of secondary importance. The chapter will be concluded with a brief reference to the most important results.

Since chapter 2 presents a formal mathematica! definition of the problem, and since its mathematica! elaboration is nat connected with the growth roodels outlined above, those exclusively inter-ested in the mathematica! treatment can skip this chapter with-out objection.

1.2 Growth model I.

We consider an economy with m

1goods and n production processes.

The production processes are specified as follows:

a) The production processes are executed in a sequence of

periods of equal duration. The input received at the

b) The production processes are linear: for every period, the input and output of each production process is proportional to the level of activity at which i t is executed during the period.

c) For each production process, the proportion between the

quant-ity of input and output and the level of activquant-ity is constant for all periods.

We shall reprasent the activity levels at which the production

processes are executed during the periods t

=

0,1, .•• ~y a

sequence of non~negative n-dimensional vectçrs x(O),x(1), .•.

in which x.(t) is the activity level for the j-th production J

process, during·the t-th period. The suppositions b) and c) im-ply that, for each sequence of activity levels x(O),x(1), ..• the corresponding quantities of input and output may be -expressed as follows:

Ex ( t) t > 0

(1.2.1)

Ax(t) t > 0

B and A being non-negative m₁xn-matrices of input and output

coefficients; i.e.: b .. x. (t) and a .. x. (t) are the quantities of ~J J lJ J

input and output resp. of the i-th goods if the j-th production process during the t-th period is executed at an activity level

x

3• (t). According to supposition a) the input b .. x. (t) is absorbed . 1 J J at the beginning of period t and the output a .. x. (t) is available

lJ J at the end of this period.

With regard to the transfer of goods between the various product-ion processas we further introduce the following suppositproduct-ions:

d) The transfer of goods between the processes, takes place

time-lessly during the changing of periods.

e) Surpluses are allowed.

f) Surpluses occurring during one change of period are not

If no good~ from without are conveyed towards or from this econo-my, then the latter three suppositions imply that with a given vector of initial activity levels x(O), a sequence of activity

levels x(l),x(2), ••• is feasible if and only if the following

inequalities are satisfied:

Bx (I) ; Ax(O) x ( t ) ! O , t > ;. J (!.2.2) Bx(t+J)-Ax(t) ; 0, t >

We cbserve that the conditions e) and f) appear more limiting than in reality they are. For the preservation of surpluses

during a period as such, can. again be formulated a~ a special

production process.

To the system described thus far, we shall add a number of ele-ments, whereas the conditions a) to f) will remain valid.

Firstly ons may assume that goods from without are made available for the system. In this connection, for instanee the factor labeur might, in a certain manner, be taken into account. We assume, that

this availability of goods has a content of a.tf, for each change

of period (t-I ,t). In which assumption

a

> 0 represents a growth

factor and f a non-negative m-dimensional vector. Under this

ex-tension system (1.2.2) takes the following form:

Bx ( I ) ,::; a.f+Ax{O)

Bx(t+l)-Ax(t) < at+lf, _{t ~} (1.2.3)

x(t) ,;; 0 t ,;;

We now turn our attention towards the consumption aspect. We dis-tinguish two kinds of consumption: autonomous consumption and endogensous consumption. Autonorneus consumption may be interpreted as those quantities of goeds which are necessary to meet the

We assume that, far each change of period (t-l,t), the autonomous

consumption is ~tg; herein

B

> 0 represents a grawth factor and g

a nan-negative m

1-simensional vector. Under this addition, system

(1.2,3) takes the farm of:

Bx(1) < af-Sg+Ax(O)

I

=

Bx(t+l)-Ax(t)

..

<

a

t+If

-

13t+1 g, t >

_I>

IJ

x(t) >

0 .

t >

-( l • 2. 4)

Thus, given the initial activity levels x(O), a sequence of

activ-ity levels x(l),x(2), ••.

is

feasible if and anly if (1.2.4) is

satisfied.

With regard to the process af endogeneaus consumption we intro-duce the following suppositions comparable with a), b) and c):

g) The process of free consumption takes place in a sequence of

periods of the same duration as that of the production pro-cesses.

h) The process of free consumption may consist of various

sub-processes. For each subprocess the input of goods is proport-ional to its level of activity.

i) For each subprocess the proportion of input and the level of

consumption activity remains the same for all periods.

The latter three suppositions are elaborated in a similar manner

as the suppositions a), b) and c). Assuming that there are k

sub-processes for endogeneaus consumption, and repreaenting the act-ivity levels of these subprocesses by a sequence of non-negative k-dimensional veetors xc(l),xc(2), .•• , then, with the help of a non-negative m

1xk-matrix Be of consumption coefficients, the

cor-responding quantity of input may be expressed:

t ~ l . (1.2.5)

conditions a) to i), with a given x(O), a sequence of activity levels (x(l),xc(l)),(x(2),xc(2)), •.. is feasible if and only if

Bx (1 ) + B c x c ( 1 ) < a.f-Sg+Ax(O) = Bx(t+l)+bcxc(t+l)~Ax(t) a.t+lf-St+lg I <

t

(1.2.6)

..

_'

t > = x(t),xc(t) > 0

'

t >

_J

= is satisfied.

We complete the introduetion of engogeneaus consumption by ad-dition of a linear utility function. We shall,assume that the utility of each sequence of levels of consumption actlvities xc(l),xc(2), •.. ,xc(T), ••. can he expressedas fellows:

(1.2.7)

where:

- p is a non-negative k-dimensional vector and

- n

a positive discount factor, in which the appreciation of a

succeeding period is expressed in relation 'to the preceding

period. So, generally

u

will be smaller than one.

Starting from the supposition that the economy described above does no cease to exist, the following problem arises: How many periods will the utility function (1.2.7) of a feasible se-quence of activity levels (x(l),xc(l)),(x(2),xc(2)), •.. have to cover for an adequate valuation of such a sequence to be obtain-ed? Clearly, each fixatien of the number of periods covered by the valuation, or differently put, fixatien of the horizon, is bound to introduce a certain arbitrariness. This arbitrariness can be efficiently avoided by effectuating the valuation over an infinite number of periods. This results in the following expression:

""

\' _{~., n} t _p'x c

(t) (1.2.8)

Now, a particular difficulty may present itself. For it is very

well possible that, fora number of~feasible sequences ef

activ-ity levels (x(l),xc(l)),(x(2),xc(2)),.,, the limit (1.2.8) does

not exist. To illustrate this possibility, we consider the case

that there is no autonomous consumption (i.e.: g = 0) and that,

for certain x(O), system (1.2.6) possesses a solution of the form

x(t) :

..

Ct t 1

l

"'

_{t > 1.} 2

J

= xc(t) := a. t x (1.2.9)

For such a fe~sible sequence, we have:

J-(a.11)T+J 2

(a1T) I-(mr) p'x T ;; (1.2.10)

2

If p'x ~ 0 and if ClTI ~ I, then it appears that the sequence of

numbers defined by (1.2. 10) has no upper bound. In that case, the expression (1.2.8) cannot reprasent a sensible utility funct-ion. This difficulty can be eliminated by choosing the positive

coefficient 11 so small that a11 < I.

Apart from the complication as sketched above, the optimization aspect in this economy can be roughly formulated as follows: Given the initia! intensities x(O), find a sequence of activity

levels (x(I),xc(l)),(x(2),xc(2)), . . . , which satisfies (1.2.6) and

for which the limit (1.2.8) attBins a maximum value.

Finally, we shall give the model consisting of the inequalities (I .2.6) and the utility function (1.2.7) and (1.2.8) a more ge-nera! form.

Firstly, we write the left hand side of (1.2.6) in the less spe-cified form:

t ~ I ,

se-quence of m

1-dimensional veetors for which non-negative veetors

f and f are supposed to exist such that -f~f(t)~f, t > I .

Thus, (1.2.6) takes the farm:

;;;i pf(l)+Ax(O),

With the help of two non-negative m

1x(n+k)-matrices:

B

:= (B,Bc) }

A

:= (A,O)

this system can be written:

Bx{ l) < p f ( 1 ) +A x ( 0) Bx(t+I)-Ax(t) < pt+lf(t+l) =

x<

t

>

.

>

o.

:}

'

t > = t >

.

( 1 • 2. I I )

(

I

(1.2.12) (I • 2. 13)

where

x(O),x(l), ...

is a sequence of (n+k)-dimensional vectors,

which correspond with the activity levels (x(O),xc(O)),(x(l),xc(1)), •••

The utility functions (1.2.7) and (1.2.8) will be written in the less specified farm:

T

I

'IT

tp (

t ) I

x (

t ) • T ~ I • ( l • 2. 14) t=l

""

I

'IT

t-p

<

t

>

·x<

t

> •

(1.2.15) t=l

in which ~(1),~(2), •.• is a sequence of m

1-dimensional veetors

for which non-negative veetors

~

and

~

are supposed to exist

such that:

t > I. (1.2.16)

Since negative components are permitted in the veetors

~(1),~(2), ••• we shall use the more general term objective funct-ions, for the expressions (1.2.14) and (1.2.15).

Growth model I, consisting of the inequalities (1.2. 13) and the objective functions (1.2.14) and (1.2.15), will be the point of departure for the growth models now to be discussed.

1.3 Growth model II. ·

We now turn our attention towards durable goods, or briefly, durables. In comparison with the goods of model I, further to be called non-durables, the durable goods possess some characteris-tic properties specified as follows:

a) The durables which are necessary for a production process,

will be adopted at the beginning of a period and will become free at the end of the period.

b) The quantity of durables which are used in a production

pro-cess, is proportional to the activity level of this process.

c) The proportion between the durables used in a productfen

pro-cess, and the level of activity of the propro-cess, is the same for all periods.

The formation and the process of obsolescence of all sorts of durables will be specified as follows:

d) All durables are formed timelessly outoff the non-durables

on the moments of period change.

e) The quantity of non-durables used for the formation of

f) The proportions of the quantities mentioned at e) are constant for all periods.

g) All durables have a finite durability. The curve of

obsolescen-ce is the same for all periods. Moreover, durables cannot change in type.

h) There is no exogeneoua supply of remaval of capacity goods.

The transfer of durables is supposed to be of the same nature as specified in 1.2-d,e,f.

First we elaborate the suppositions 1.3-d,g. For the sake of sim-plicity, we bere assume that the durability of durable goods is three periods at most •. Let L be the number of sorts of durables, then, during each period t, this economy contains the following durables:

- durables formed át the moment of period change (t-J,t); the

quantity of these is expressed by a ~on-negative L-di~ensional

vector z(t;O),

- durables formed at the moment of period change (t-2,t-J);

re-presented by a non-negative L-dimensional v~ctor z(t;l),

- durables formed at the moment of period change (t-3,t-2);

re-presented by a non-negative L-dimensional vector x(t;2). With the help of two non-negative LXL-dimensional matrices y(l) and y(2), for every period t, the actual quantity of durables (viz. 1.3-g) can be expressed by

z(t;O)+y(l)z(t;l)+y(2)z{t;2). (1.3.1)

The suppositions 1.2-e,f and the definition of the veetors

z(t;O),z(t;1),z(t;2), t ~ 0, imply the inequalities:

z(t+l;l) < z(t;O) }

z(t+1;2) ~ z(t;l) ,

For the elaboration of the suppositions 1.3-a to 1.3-f, we start

from the inequalities (!.2.13) of growth model I:

Bx ( 1) < pf(l)+Ax(O)

Bx(t+I)-Ax(t) ~ pt+!f(t+l) (I . 3. 3)

i"(t)~O t~

with m

1 types of non-durables and n1 := n+k processes.

With the help of a Lxn

1-matrix C, the use of durables for every

sequence of -activity levels i"( 1) ,i"(2), .•• , can be expressed by:

ci"(t), t ~ I. (I . 3. 4)

With the help of a negative matrix D the volume of non-durables used for the formation of non-durables can be expressed by:

Dz(t;O), t > I. (1.3.5)

The supposition 1.2-a,d,g and 1.3-a,d imp·ly that for the first period we may jein the expresslons (1.3.1) to (1.3.5) in the fol-lowing inequalities: Bx(I)+Dz(I;O)

-

,,

~ pf(l)+Ax(O)i ci"(l)-z(l;O)y(l)z(l;l)-y(2)z(l;2) < 0

I

z (I ; I ) < z(O;O)

?

(1.3.6) z(I;Z) < z(O;I)

J

i"(l),z(I;O),z(l;l),z(I;Z) > 0

Bx(t+l)+Dz(t+l;O Cx(t+l}-z(t+l;O)-y(I)z(t+l;l)-y12)z(t+l;2) < 0

=

z(t+l;l)-z(t;O) < 0

=

t ~ z(t+l;2)-z(t; 1) < 0 = x(t),z(t;O),z(t;J),z(t;2) > 0 = (1.3.7)

Thus, given (x(O),z(O;O),z(O;I)), a sequence

(x(t),z(t;O),z(t;l),z(t;2)), t

=

1,2, •.. is feasible if and only

if (1.3.6) and (1.3.7) are satisfied. With the help of the (m

1+L+L+L)x(n1+L+L+L)-matrices

[

'B,

c,

B

·=

0, 0, D, 0 ,

01

-I,-y(l),-y(2) 0, I , 0 0, 0 , I A

r

0, 0, 0, 0, 0, := 0, _{I ' 0,} 0. 0,. I, (1.3.8)

and with the help of the sequence of (m

1+L+L+L)-dimensional

vee-tors defined by

[

f

(t]

}(<) , . : t ~ 1. ( l • 3. 9)

the inequalities (1.3.7) can be written:

13~

(

l ) < p}(I)+Ä:~(O)

11

l!it{t+l )-Ä:~{t) <

₌

p t+lf' (t+l), t ~

Ij .

~(t)

>

₌

0 t ~ ( l • 3. l 0)

where ~(0),~(1), ••• , is a sequenee of (n

1+L+L+L)-dimensional

veetors which corresponds with the sequence

(i(t),z(t;O),z(t;l},z(t;2)}, t • 0,1,2, ••• appearing in (1.3.6)

. ~

and (1.3.7). So, the veetors x(t), t ~ 0 represent quantities

of a different nature. Therefore, we shall use the more general term state vectors.

When we define the sequenee of (n

1+L+L+L)-dimensional veetors

p(1),p(2), ••• by:

p (

t)

,.lfl

t ~ (1.3.11)

p(l),p 2), ..• being the sequence of veetors of the objective

functions (1.2.14) and (1.2.15), then the corresponding

object-ive functions of a sequence }(I),i(2), . . . which satisfies (1.3.19), can be written: T t"'

~

L

TI p(t)'x(t), T ~ 1, t•1 (I • 3. 12)

L

ntp(t)'i(t). (1.3.13) t=l

Clearly, the optimization aspect may be formulated in the same manner as in growth model I.

The system of inequalities (1.3. 10) tagether with the objective functions (1.3. 12) and (1.3. 13) form growth model II. We observe

that this model has the same form as growth model I. Further we

observe that the definitions (!.3.8) and (1.3.9) imply the fol-lowing:

i)

Each row vector ];' .. of matrix

'f!

which contains one or more

1..

vector

~-

of matrix

k

and with non-negative components

'V L • 'lt 'lt

fi(t) of the veetors l:(l),t(2), •••

j) Matiix

k

contains no negative components.

1.4 Growth model III.

Now we shall add import and export facilities. These are speci-fied as follows:

a) The processes of import and export are executed in the same

sequence of periods as the production processes. The

quanti-ty of goo~ to be imported and exported is determined at the

beginning of a period; the actual import and export takes place timelessly at the end of the period.

b) The quantity of goods which are used for the effectuation

of import and export (for instanee transport capacity) is

taken up at the begin~ing of a period, and is proportional

to the quantity of imported and expor~ed goods.

c) Import and export take place at fïxed prices. Import prices

are not lower than export prices.

d) The reserve of payments at the end of a .period is composed

of reserve of payments at the end of the ~receeding period,

multiplied by an interest factor, increaied by the value of the export and decreased by the value of the import at the end of the period.

e) The reserve of pay.ments cannot be negative.

f) The proportion of the quantities mentioned at b), the import

and export prices and the interest factor appearing in d) are the same for all periods.

We take growth model I I as the point of departure:

1;~(1) _{< p (l)+Ax(O)} f' 'V'V

-l;~(t+l)-l:~(t)

< pt+lf'(t+l)

.

_{t > (!.4.1)}

=

=

I

~(t)

> 0 t > = = '

.

where:

- i'(O),i:(l), ..• and f'(I),f'(2), . . . a-re sequences of n

2 and m2

di-mensional veetors (m

2 := ~

1

+L+L+L and n2

:=

n1+L+L+L),

-land~ are m

2xn2-matrices which possess the properties 1.3-i

and 1. 3-j.

For the sake of simplicity, we shall here assume that all types of goods can be imported and exported; the quantities will be

de-noted by the sequences

~f

m

2-dimensional veetors xi(O),xi(l), •..

and xe(O),xe(l), ••• , x1(t) being the quantity of imported goods

at the end of period t and xe(t) the export quantity.

Supposition 1.4-b implies that the quantity of goods used for the effectua·tion of import and export can be expressed by:

i i 0

l

c

x ( t ) , t > (1.4.2) cexe(t), t > 0

J

Supposition 1.4-a implies that (!.4.1) and (1.4.2) can be com-bined into the system:

'VI, _{Bx(I)+C x (l)+C x (I)} i i e e _{~ pt(I)+Ax(O)+x (0)-x (O)} .,_ "'"' i e

"'"' Bx(t+l)+C x (t+l)+C x (t+J)-Ax(t)-x (t) i i e e "'"' i (t) < pt+l'f(t+l), t ~

"' i e

x(t) ,x (t) ,x (t) ~ 0 _{' t ~ 0}

(I • 4. 3)

The suppositions 1.4-c to 1.4-f are elaborated as fellows. The reserves of payment at the end of the periods t • 0,1,2, ..• can be represented by a sequence of non-negative numbers r(O),r(l), . . .

Denoting :he import an~ export by non-negative m

2-dimensional

veetors q1 and qe, and the interest factor by a number a> I,

i i e, e r(l) ~ ar(O)-q 'x (O)+q x (0) . . i i e e r(t+l)-ar(t)+q 'x (t)-q 'x (t) ~ 0, i e . r(t) ,x (t) ,x (t) ~ 0

1

J.

(I • 4. 4)

Now, we may conclude that in such an economy, given the initia!

quantities

(~(O),r(O),xi(O),x

8

(0)),

a sequence

'V e "' i e

(x(l),r(J), (J),x (l)),(x(2),r(2),x (2),x (2)), ..• is feasible

if and only if simultaneously the inequalities (!.4.3) and (1.4.4) are satisfied.

[a

.

0' B ·

-"'.-

,1,

(1.4.5)

and witb the help of a sequence of (m

2+1)-dimensional veetors .{_(1),{(2), . . . defined by:

f1'

0

c

t

>J ,

.{Ct)

:=

l

t ,; 1 ' (1.4.6)

the systems (1.4.3) and (1.4.4) may be eombined in the system:

~~( l) ~ p .{_ (1 ) +A~ ( 0)

.I

~~(t+l)-~~(t)

_;::

t+lf(t+l) (1.4.7) .p 'V ' t ₌ > l ~

IJ

~(t) i; 0

'

t = > where ~(0),~(1), .•• is a sequenee of (n 2+!+m2+m2)-dimensional

veetors which corresponds with the sequence

"' e 'V · i e

(x(O),r(O), (o),x (O)),(x(l),r(l),x (l),x (!)), ...

When we define the sequenee of (n

~

(t) 0 ~(t) := 0 0 t > 1.

=

(I. 4. 8)

p(J),p(2), ...

being the sequence of veetors of the objective functions (1.3.12) and (1.3.13), then the corresponding

object-ive functions of a sequence ~(1),~(2), ..• which satisfies (1.4.7),

can be written

.,..

~ t

I

1T ~(t)'~(t), T ~ I, t=l (I • 4. 9)

""

I

ortp(t)'~(t)

t '"l '\; (1.4.10)

System (1.4.7) tagether with the objective functions (1.4.9) and (1.4.10) form growth model III. This model has the same form of growth models I and II.

Clearly, the definitions (1.4.5) and (1.4.8) and property 1.3-j imply that growth model III possesses the following property:

g) Each column vector ~.j of matrix~ which contains one or more

negative components, corresponds with a non-negative column

vector ~.j of matrix~· and with non-positive components

p.(t) of the veetors p(l),p(2), . . .

-vJ '\; '\;

!.5 Growth model IV.

Now, the suppositions 1.2-c, 1.3-c,f and 1.4-g will be weakened by introduetion of the possibility of cyclic change, as for in-stance caused by the influence of the seasons. For the sake of simplicity, we here suppose that the cycle consists of two phases: the first and the second half of the year. We put model III as the point of departure.

The state veetors of model III will be represented by a sequenee of pairs of non-negative.n

3-dimensional veetors

l 2 l 2 I

(x (O),x (O)),(x (l),x (!)), .•. , where x (t) represents the

first half of t-th year and x2(t) the seeond. Now model III ean

be written: B(l)x1(!) B(2)x2(J) -A(l)x1(!) ; pf1(t)+A(2)x2(0); ;;; pf2(1) B(l)x 1 (t+l)-A(l)x 2 (t) ;; pt+lf 1 (t+l)l .'t > B( 2 )x 2 (t+I)-A(I)xl(t+l)

~

1f 2 (t+l)J x1{t),x2(t)

~

0 ' t ; (I • 5. I)

where, A(l), A( 2 ), B(l) and B(Z) are m

3xm3-matriees and

I 2 1 2

f (l),f (l),f (2),f (2), . . . a sequenee of m

3-dimensional veetors.

When we define the (m

3+m3)x(n3+n3)-matriees

~

B ( l) _{• 0 ]}

!

,.~

A

(2)l

B

: =

-A (I) B ( 2) . 0 (1.5.2)

'

-

'

and the sequence of (m

3+m3)-dimensional veetors

~

l(t)J

i_(t) :e 2 f ( t) t ~ I, (1.5.3)

then system (1.5.1) can be written

Bx (I) < pi_( 1 )+Ax(O)

:}

=

Bx(t+l)-Ax(t) < pt+li_(t+l)

'

t > (1.5.4)

=

=

!. ( t)

..

>

o,

t >

=

where ~(0),~(1), ••• is a sequence of (n

3+n3)-dimensional veetors

which corresponds with a sequence

(x

1

(0),x2(0)),(x

1

(1),~

2

(I)),

...

satisfying (1.5.1).

Let (p (l).p (l)),(p (2),p (2)), ••• be the veetors of correspond-1 2 I 2

ing objective functions (1.4.9) and (1.4. 10). When wedefine

[ pl(t)l .F_(t) :- 2 p (t)

-·'

t ~ I, (I. 5. 5)

then the corresponding objective function for sequences

~(1),~(2), .•• satisfying (1.5.4), can be written:

T _ 1ftp_(t)'x(t) t=l co

I

1Tt.F_(t) 'x(t) t=l T ~ 1, (1.5.6) (1.5.7)

Thus, we find again that growth model IV, consisting of (1.5.4), (1.5.6), and (1.5.7), bas the same form as the growth roodels I to III. Moreover, the definitions (1.5.2) and (1.5.5) and proper-ty 1.4-g imply:

Each column vector a . of matrix A which contains one or more -.J

negative components, corresponds with a non-negative column vec-tor b . of matrixBand with non-positive components .F_.(t) of

- . J J

the veetors .F_(1),P.(2), ..•

1.6 The linear programming problem over an infinite horizon. It appears that all growth models I to IV give rise to a linear programming problem consisting of the inequalities:

Bx (I) ~ p f ( I ) +A x ( 0)

Bx (t + I ) -A x ( t ) ~ p t+l f(t+l), t ;

l

( I • 6. I )

x(t) ~ 0 , t ;

and an objective function

L

'ITtp(t)'x(t) ,

t•l

(I • 6. 2)

to be maximalized over the n-dimensional veetors x(l),x(2), .•• ,

which satisfy' (1.6.1). Hereln:

- A and B are mxn-matrices.

- f(l),f(2), . . . , is a sequenee of m-dimensional veetors for whieh

m-dimensional veetors f and

I

exist, such that:

;; f(t) ;;;

I

t > 1 ' (1.6.3)

p(l),p(2), . . . , is a sequence of n-dimensional veetors for which

n-dimensional veetors

E

and

p

exist, sueh that:

-E

< p(t) < P ' t > (I • 6. 4)

- p and'IT are positive coeffieients.

- x(O) is a given initial n-dimensional vector, which is

non-negative.

With respect to the matrices A and B and the sequenees of vee-tors f(l),f(2), ••. and p(l),p(2), . . . , it is found that, in these growth models, at least one of the following conditions is satis-fied:

a)

b)

Each row vector b. of matrix B wtich contains orle or more negative

~.

components, corresponds with a non-negative row vector ai. of matrix A, and with non-negative components f.(t) of the

vee-~ .

tors f(l),f(2), .••

Each column vector a . of matrix A which contains one or more • J

negative components, corresponds with a non-negative column vector b . of matrix B, and with non-positive components

• J

p.(t) of the vectors.p(l),p(2), .•• J

In this study we investigate a linear programming problem vhich possesses the structure as sketched above, In this investigation another linear programming problem arises naturally. This problem consists of the linear inequalities:

B'u(t)-A'u(t+l)

u(t) > 0 ' t >

and the objective function

x(O)'A' u(!)+

L

ptf(t)'u(t)

t=l

t ~

(I • 6. 5)

(1.6.6)

to be minimalized over the m-dimensional veetors u(l),u(2), •. ,, which satisfy (1.6.5).

The quantities appearing in this problem are the same as that of the first problem.

It will be shown later, that the coherence between both problems is of the same nature as the coherence between two linear

pro-gramming pro~lems over a finite horizon, which are dual with

respect to each other. This offers the possibility ) to

interpret a sequence of veetors u(l),u(2),., ., which satisfies (1.6.4) and for which (1.6.5) attains its minimal value, as a sequence of prices, i.e.: ui(t) represents the price of the i-th goods at the moment of the period change (t-1 ,t). That means that, in the context of growth roodels I to IV, the expression

b' .u(t)-a' .u(t+l)-7!tp(t) (1.6.7)

• J • J

intensity in the t-th period. Thus, in this manner, we may inter-pret a sequence of veetors u(l),u(2), ••. , which satisfies (1.6.4) as a sequence of prices such that in none of the periods any

process ~ields netto benefits. The expression (1.6.6) might be

taken as the value of the exogeneous goods

x(O)'A'+pf(l),p 2 f(2),p3f(3), ••• , at these prices.

A further interpretalon of the conditlans a) and b) can be given as follows. It will be shown (3.2) that condition a) implies that

all sequences x(l),x(2), ••. , satisfying (1.6.1) forsome initia!

vector x(O), also satisfying: t+l

Ax(t)+p f(t+l) ; 0 (1.6.8)

This means that for every period t > I the expression

Ax(t)+pt+lf(t+1) may be interpreted=as the quantity of goods

available for the process~s at the beginning of period t+l. In

that respect we can say that in such an economy all goods are transferred from a preceeding period to the succeeding period. For that reason we call a system (1.6.1) directed, if it satis-fies condition a).

In a similar way it appears that condition b) implies that all sequences u(I);u(2), •.• satisfying (1.6.5), also satisfy:

(1.6.9)

Since, for all period& t > I, the value of the expression

t

=

b' .u(t)-n p.(t) is non~negative, and since

• J J

be taken as the

the vector u(t) may beginning of period prices of the goods at the

t

t, the expression b' .u(t)-n p.(t) may be interpreted as the

• J J

costs per unit of activity level for the j-th process in periód t. Moreover, since (1.6.6) may be interpreted as the netto costs, one might say that the costs b' .u(t)-ntp.(t) are always

account-• J . J

able at the end of period t. Therefore, we shall call a system (\.6.5) directed, if it satisfies condition b).

With respect to the optimization aspect of the two linear pro-gramming problems described above, a sequence of n-dimensional veetors i(l),i(2), ... , will be called an optimal salution of

the first problem if this sequence satisfies (1.6.1) and i f no

sequence of n-dimensional veetors x(l),x(2), .•. exists, which

satisfies (1.6.1) as wellas

T t

~ 1T p (t) 'x(t) ~

r

T 1r t p(t)'x(t)+e: T ~ T ,

*

t= I t"'l

fot some positive number e: and some period

r*

~ 1.

In a similar manner, a sequence of m-dimensional veetors 0(1),0(2),, .. will be called an optimal salution of the second problem if this sequence satisfies (1.6.5) and if no sequence

of m-dimensi~nal veetors u(l),u(2),,,. exists, which satisfies

(1.6.5) ~s wellas T x(O)'A'u(l)+

L

ptf(t)'u(t) < t=l ' • T x(O)'A'û(l)+

L

ptf(t)'G(t)-e:,

T

~ t"'l

forsome positive number e: and some period

r*

~ 1.

The most important questions which are dealt with this study are the following:

*

T '

- When do exist sequences of n-dimensional veetors x(J),x(2), ...

s~tisfying (!.6 1 ) . and when do exist sequences of m-dimensional

veetors u{l),u(2), ... satisfying (1.6.5)?

- When do optimal solutions exist for the first and the second

problem?

- What is the symptotic behavier for t ~ oo of optimal salutlans

x(l),x(2), ... ;x(t), •.• and u(l),u(2), ..• ,u(t), •.. of the first and the second linear programming problem resp.?

1.7 Summary of the most important results.

Presumed that at least one of the conditions 1.6-a or I .6-b 1s

satisfied, i t appears that the linear programming problP~'

con-sisting of (1.6.1), (1.6.2) and of (1.6.5), (1.6.6) are

sens-ibie only if pn < J. In that case, we found that, under certain

conditions which are somewhat strenger than the assumption thát both problems possess feasible solutions, the problems both possess optimal solutions.

It appears that the coherence between both problems is of a similar nature as the coherence between two linear programming problems in a finite dimensional space which are dual with res-pect to each other.

The most advanced results are obtained when the sequences of veetors f(J),f(2), ..• and p(J),p(2), •.. , are svpposed to be

constant ever sincesome ~eriod K ~ J. Then, under some

addit-ional conditions, i t can be shown that all optimal solutions

x(J),x(2), .•. and û(J),û(2), ••• of th~ first and the secend

problem converge to certain fixed veetors ~ and ~ in the

fol-lowing manner:

lim

t+oo

"'

u

This property offers the possibility to construct a linear pro-gramming problem over a finite horizon from which all optimal solutions of the original infinite horizon can be found.

2. HATHEHATICAL FORMULATION OF THE LINEAR PROGRAMMING SYSTEM.

2.1 Introduction.

First, we introduce a number of general concepts and notations. With the help of these implements the formulation of the linear programming system is given, which forms the central theme of this study. Finally, some concepts will be introduced with respect to the structure of this linear programming problem.

2.2 1 - and 1₀₀-space.

The(real) 1₁- ~nd 1₀₀-spaces are particular specimens of the so

called liJ-spaces ( 5:103 ). They are defined as follows.

The 1

1-space is a veetorspace consisting of the sequences of (real)

' }00

numbers 1.si ₁ for which

00

The norm of an element x:={si}7 E 1

1 is defined by

u xn ₁ : = E

I

s .

I

i= l l.

( 2. 2. I )

(2.2.2)

The 1₀₀-space is a veetorspace consisting of the sequences of (real)

numbers {til7 for which

supiCI<oo.

_.

~

l.

The norm of an element x:=tti}7 E 100 is defined by

nxn :=supiS.I· 00 • ~ l. (2.2.3) (2.2.4) The 1

1- and 100-norm may be introduced in an similar way for

The 1

₁

~ and l

₀₀

~spaces considered by us possess a special structure which can be described in the following manner.

Consider the sequences of veetors {x(t)}7 in a k~dimensional real

vectór space Rk. The set of such sequences may be taken as an

oo~dimensional

vector space and denoted by lk. This leads to the following formal definition:

(2.2.5)

k

Now we wish to introduce the 1

₁

~space to be defined as the set

of veetors in lk for which

k L: L: jx.(t)j < oo, t=l i=l ~ (2.2.6) wi th the norm: k llx11 ₁:= L: L: lxi(t)j. t= 1 i= I (2.2.7) k

In a similar way we define the l₀₀~space as the set of veetors in

lk for which

sup maxjx.(t)j

_.

<'oo

~

t ~

The norm of this space is defined by

llxll₀₀:=sup maxlx.(t)j.

t i ~

(2.2.8)

(2.2.9)

It appears that every

lk~

I or

l~~space

may be considered as an 1

₁

~ or l₀₀~space resp. For

U,;r}7 defined by

if x:={x(t)}7 E

1~,

then the sequence

~r=k(t~J)+i:=xi(t), i=1,2, ••. ,k, (2.2.10)

is a vector of 1₁ with the same norm. The other way round, with

the help of an opposite process every vector of 1

1 may be

A similar relation may he constructed between 1

00 and

r!.

This means that all properties of the 1

1- and 1~-space simply

can be transferred to the lk_ and lk-spaces resp. _I

.oo

We shall also use the 1

1- and

1~-norm

for veetors x E lk in another way, namely:

S T T k

llxll ' : = l: l:: lx.(t)!, r;s;J.

1 _{t=S i=l l.} (2. 2. l l)

(2.2.12)

Finally, we define the positive cone of lk, lk and lk by: _l

00

t~ I } , (2.2.13)

A well known property of the 1

1-space is that the positive cone

defined in this manner does not have an interlor point.

However, the interior of lk _~+ is not empty .and is defined by

i= I ,2, •. , k , t ; l '

l

for some E:>O

_{J .}

(2.2.16)

2.3 The a-transform of x E lk.

It is easy to see that for every positive scalar a, the ex-pression

t 00

x a:

=

{a x ( t ) }

represents a one-to-one mapping of lk onto itself. This transfor-mation which will be used frequently, will be indicated by the

term

~-transformation(*).

The coëfficient of transformation a will ever be positive. The image xa of x E lk generated by this transformation will be called the a-transform of x.

In conneetion with this transformation we introduce the following

concept: a vector x E lk ·is called a-dominated if

(2.3.2)

This is equivalent with the condition that a positive number M exists such that

(2.3.3)

So, in this prospect the sequence of numbers lx(t)H

00, t~l is

dominated by atM, t ; l .

2.4 Linear functionals.

~ith

the help of a vector y E lk, a sequence of numbers can be

joined to every x E lk in the following manner:

T

<y,x>T:=

E

y(t)'x(t),

t= I

( 2. 4. I)

If such a sequence converges, the limit will be denoted by

<y,x>₀₀:=1im <y,x>T. (2.4.2)

T+"'

The expression (2.4. J) and (2.4.2) may be taken as linear

functio-nals on lk. It is well known that for every x E lk and y E lk

1 00

the sequence {<y,x>T};=I converges.

~In the context of this investigation confusion with the well

k

This implies that for every y E 1

00, <y,x>00 is a bounded (and. so

a continuous) linear functional on

1~

and in the same manner,

k

that for every y E 1

1, <y,x>00 is a bounded linear functional on

lk.

00

2.5 Formulation of the linear programming system.

Now we shall give a formal definition of the growth model as

described in §1.2 to §l.S. To this end we consider solutions

x E ln of the system of linear inequalities

+ Bx {I) _{< pf(l)+Ax(O)}

l

;; pt+l f(t+l),t~l ( 2. 5. I) Bx(t,+J)-Ax(t) where

A and B are m x u-matrices,

f:={f(t)}7 E

1:

x(O) E is the initial vector,always supposed to be non-negative,

p is a positive scalar.

In conneetion with the economical back ground we shall term the

numbers 0,1,2, ..• used in the context of .(2.5.1) as periods.

Unless indicated otherwise, the initial vector x(O) is supposed to be a fixed given quantity.

By introduetion of so called slack variables y:={y(t)}7 E

}=•

the system (2.5. I) can be converted into the system of linear

equalities:

Bx (I} +y(J)

=

pf(I}+Ax(O)

I ) ,

(2.5.2)

. t+l

Bxtt+l)-Ax(t)+y(t+l)

=

p f(t+l),

The systems (2.5.1) and (2.5.2) are equivalent and from now on,

Naw we shall consider far every x E ln the linear functionals

+ T

<p ,x> := E fftp(t)'x(t), T~l,

1f T t=l

where p E ln and 1f is a positive scalar.

00

(2.5.3)

System (2.5. I) ar (2.5.2), tagether with the linear functionals

(2.5.3) will be analysed simultaneously with another system

already mentioned in chapter J. In mathematica! respect

these systems are related by a so called duality-relatian. This relation will be pointed out later and in the first instanee i l

-lustrated in §2.7. Adopting the camman nomenclature of the theory

of linear programming in a finite dimensional spaee, we shall call

the system (2.5. I) or (2.5.2) tagether with the linear functionals

(2.5.3) the primal system and the system to be formulated now,

the The whole consisting of the primal and dual system

will be indicated by the term

In the dual system we consider veetors u E lm satisfying

+ t

B'u(t)-A'u(t+l)~w p(t),

or LP-system

(2.5.4)

or formulated as a system of linear equalities, we consider veetors

(u,v) E lm x 1° satisfying.

+ +

B'u(t)-A'u(t+l)-v(t)=wtp(t), (2.5.5)

where A' and B' are the transposed matrices of (2.5. 1), while

p:={p(t)}7 and ff>O eorrespond with (2.5.3).

With a fixed initial vector x(O) we further consider for all

veetors u E 1m.or (u,v) E lm x ln satisfying (1.5.4), (2.5.5)

+ + +

resp. the linear funetionals.

T t

x(O)'A'u(l)+ l: p f(t)'u(t),

t= I

oo m where f:={f(t)}

1 E 100 and p>O are quantities already introduced

in the primal system.

l

Denoting the sequence of veetors {f(l)+PAx(O),f(2), ••. ,f(t), ••. }

0 { 0 }""

by f := f (t) ₁, the linear functionals of (2.5.6) can be written

T t o

:=

E

p f (t)'u(t), (2.5. 7)

t=l

The systems (2.5.4) or (2.5.6), combined with (2.5.7), form the dual system.

We wish to introduce some terms frequently appearing in this in-vestigation.

The LP-system is called:

- primal feasible (P-feasible) when system (2.5.1) possesses a

solution x E 1+0; this salution x or a solution (x,y) E ln x lm. _{+ +}

of (2.5.2) will be called primal feasible,

virtuallyprimal feasible ( easible) when there is a initial

vector x(O) such that the LP-system is P-feasible,

(D-feasible) when system (2.5.4) possesses a

salution u E lm+; this salution u or a solution (u,v) E lm x 1°

+ +

of (2.5.5) will be called dual feasible,

(P- and feasible) when the LP-system is P- and D-feasible,

(P0- and D-feasible} when the LP-system is

The following terms have reference to the existence of special sort of feasible salutians:

The LP-system is called:

- primal regular (P-regular) when (2.5.2) possesses a solution

(x,y) E

1:

x

1:

such that (xl/p'yl/p)

E

1:~

int(l:+); this

solutian is called

m n m n

{u,v) E 1+ x 1+ such that (u

₁₁

~,vl/n) E 1~+ x int(l00+); this

solution is called dual-regular.

0

The notions .virtually primal regular {P -regular), regular and virtually regular may be introduced in a si mi lar manner as the eer-responding noiions with respect to the feasibility.

We further shall call the LP-system (Vtr·t·u:ally)superregular if simultaneously:

- p~< I '

- the LP-system is (virtually) regular,

- the systems

Bx(t+I)-Ax(t)+y(t+l)=(!)tf(t),

. ~

I t B'u(t)-A'u(t+l)-v(t)=(p) p(t),

have solutions (x,y) and (u,v) forsome K~l, such that

(x ,y )Eln x int(lm) and (u , v ) E lm x int(ln ).

u u oo + oo+ p p m+ oo+

The systems (2.5.1) and (2.5.2) will be called

and the systems (2.5.4) and (2.5.5) conditions.

(2.5.8)

The linear functionals (2.5.3) and (2.5.7) will be indicated by the terros primal and dual objective functions resp.

On the set of primal and dual feasible solutions we wish to

install a partial ordering which refers directly to the optimization

aspect of the LP-systems as mentioned in §1.2 and §1.6.

To this end a P-feasible solution x is called with

respect to a P-feasible solution ;~ if a number• E>O and a period

s;1

exist such that

A feasible salution u is called inferior with respect to a

D-feasible salution ~. if a number E>O and a period s~l exist such

that

s.

(2.~5.10)

2.6 Formulation of the problem.

Briefly summarized this investigation deals with the following problems:

a) When is the LP-system feasible?

b) Are the sets of P- and D-feasible solutions bounded in some

respect?

c) Do~s an upper bound exist for the primal partial objective function

and a lower bound for the dual partial objective function?

d) When do a P-feasible

i

and D-feasible u exist, such that the

sequences {<pn,;>T};=l and {<pn,u>T converge and such that

every P-feasible x and every D-feasible u, for which the

sequences {<pn,x>T};=l and

{<f~,u>T};=I

are not convergent,

is inferior with respect to ; , ~ resp.?

It appears that P- and D-feasible solutions x, u as mentioned under d) exists, when the LP-system is regular and pn<l. With regard to the optimalization aspect of the LP-system, we can

conclude that these feasible salution

i,

u are "better" then

those for which the sequences of partial objective functions do not converge. For this reasen we further restriet ourselves to the subset of P- and D-feasible solutions, for which the sequence of partial objective functions converge, and direct the investigation to the following problems:

e) When does a P-feasible i exist~ for which the linear

functio-nal <pn,x>₀₀ on this subset of P-feasible solutions attains

his supremum? Every P-feasible salution i with this property

will be called a P-optimal solution.

f) When does a D-feasible û exists, for which the linear

functio-o

above, attains his infinum? Every D-feasible salution û with this property will be called a D-optimal solution.

g) Wha~ is the asymptotic behavior of the components x(t) and

û(t) of P- and D-optimal solutions

x

and û for t ~ «>?

We write the problems posed in e) and f) as follows

Bx (I)

Bx(t+I)-Ax(t) ; p t+l f 0 (t+l), t;l

inf <f0 u> IB'u(t)-A'u(t+l) __ > ntp(t),

Elm p' 00

u +

t> f

(2.6.1)

(2. 6. 2)

The problems (2.6.1) and (2.6 •. 2) will be called the primal and dual pr.oblem resp. and the whole, consisting of (2.6.1) and (2.6,2) the linear programming problem (LP-problem).

We dominate the functionals <pn,x>₀₀ and <f~,u>

00

as the prim~

and dual objective function resp.

2.7 tluality.

We illustrate the duality relation between the primal and dual problem already suggested with the help of a LP-problem formu-lated for a finite horizon problem.

To this end, we consider the following programming problem in a finite dimensional euclidean space:

T max

TE

ntp(t)'x(t) {x(t)} 1t=l

I

Bx (I) _{( 2. 7. I)} t+ I o

Bx(tH)-Ax(t);p f (t+l),t=I,2, •. ,T-I

-Ax(T) T+ 1 _f o _(T+I)

where all quantities are the same as used in §2.5.

This problem can be taken as being generated by the cutting down of the primal problem (2.6. 1) at a period T.

Applying the well known duality rules from the theory of linear

programmingin a finite dimensional space on (2.7. I) we encounter

the following problem:

min {u(t)}i+l T+l i: ptf0 (t)'u(t) t= l B'u(t)-A'u(t+l) u(t)~O, p(t),t=1,2, .• ,T t=l,2, .. ,T+I (2.7.2) This problem. toa, may be considered as being generated by the cutting down of the dual problem (2.6.2).

The mathematica! caberenee between the primal and duàl problem of §2.6 will appear to be of the samenature as that between the problems sketched above.

In this context, we remark that the dual system can be written in a similar farm as the primal system:

A'u(t+l)-B'u(t)~-ntp(t)

) u(t)~O t~ I = o T t o <-fp,u> :• i: -p f (t)'u(t),

T

t=l

Consequently the dual problem can be written:

sup <-f0,u> uElm p "" + IA'u(t+I)-B'u(t)<-ntp(t),

I

= (2.7.3) (2.7.4)

By virtue of this simi!arity between the prima! and dual system (problem), further to be indicated by the term symmetry, many

to the dual system (problem).

Finally, we point out a difference which is caused by the appearence

of the ~nitial vector x(O) in the primal system (problem).

This difference bas led to the .introduction of two concepts of feasibility. for the primal system, whereas one type of feasibility suffices for the dual system.

2. 8 a.-transformed primal -and dual systems.

The a.-transformation (§2.3) of lk onto itself suggests a similar transformation for the primal and dual systems, which we want to introduce now.

We shall call the system

Bx (I) t+ I o Bx(t+l)-a.Ax(t);(a.p) f (t+l), t~l T 1T t . <prr/a.'x>T:=

E (-)

p(t)'x(t), t=l a.

The a-transformed primal system, and

T

:= I: (~) tf0 (t) 'u(t),

t=l the a.-transformed dual system.

( 2. 8. I)

(2.8.2)

Clearly, x is a P-feasible salution (of the untransformed LP-system) if and only if xa. satisfies (2.8.1), and u is a D-feasible solution

(of the untransformed LP-system) if and only if ua. satisfies (2.8.2)

Bx( 1)

Bx(t+l)-aAx(t) ;(ap) t+ I f (t+I), o t~l. (2.8.3)

The m-transformed primal problem, and

inf

<f~,u>

00

I

B'u(t)-~A'u(t+l)

> (m1T)tp(t),

t~l,

Elm """ u. ==

u +

(2.8.4)

the m-transformed dual-problem.

A specific property of these problems is:

x is a P-optimal solutiorr(of the untransformed LP-problem) if and only if xm is a P-optimal salution of the a-transformed primal problem. A similar relation holds for the dual problem.

2.9 Concepts with respect to the structure of the LP-systefu. We shall analyse the LP-system under different suppositions. The most important of which are the following:

The LP-system is called: exponential, when f(t)-'t' )

p(t)=~

semi-exponential, when f(t)=1'

~

t

p(t)=~

J

, for some T~l,

i.e. if a period T~l exists such that the sequences {f(t)};

and {p(t)}; are constant,

prima! directed (P-directed), if every row vector b. of B,

l .

non-negative row vector ai. of A, and with non-non-negative components

fi(t), t~l of the sequence {f(t)}7. (see also §1.)

duaL directed (D-directed), if every column a . of A, which • J

possesses a negative component, corresponds with a non-negative column vector b . of B, and with non-positive components p.(t),

• J "" J

t~l of the sequence {p(t)}

1•

We remark that the latter two definitions are symmetrie; for writing the dual feasibility conditions as follows

A'u(t+I)-B'u(t)

~

nt(-p(t)),

it appears that they correspond completely.

Finally, with the help of two examples, we introduce a brief notation for suppositions about the LP-system(problems):

LP-system(P- or D-directed; p(t) +

~.

t + 00 ) : the system is

supposed to be P-or D-directed and the sequence {p(t)}7 to

converge to ~ E Rn

LP-problem((f0 ,p) E F0 x P; P-or D-directed): f0 and Pare

supposed to be veetors of F0, P resp. and such that, for these

3. DIRECTEDNESS, FEASIBILITYa AND REGULARITY.

3.1 Introduction.

In this chapter a number of conditions will be derived with respe< to the feasibility, regularity and boundness of feasible solutions In these derivations the concept of directedness takes a central p 1 ace.

3.2 Theorem.

The L.P.-system is then and only then P-directed if for every

f(t),

t;;;

1:

( ). E n+n+m . f .

each x,y,z R + sat~s y~ng

Bx-Ay-z < f(t), "(3. 2. 1) also satisfies -Ay-z ~ f(t). (3.2.2) Necessary: such that n+n•

suppose that for some t > I, there is a (x,y,z} E R +

-Ay-z

f

f(t)

l .

Bx-Ay-z < f(t)

Then an index i exists for which bi.

l

0 and for which

i

0

or fi (c) < 0. This, however, is impossible in conneetion with

the definition of P-directedness.

Sufficient: assume that the LP-system is not P-directed, then the1

is a b .. < 0 such that a.

t

0 or f.L. (t) < 0 for some t > I. This

~ J ~.

implies that there exists a (x,y) E Rn such that for some t > 1

+

bi.x-ai.Y

~

fi(t) )

-ai.Y > fi(t) •

(3.2.3)

From this i t appears that i t is possible to choose a zE R::

zi ~ 0 and sufficiently large the other components, such that

(x,y,z) satisfies (3.2.1) but not (3.2.2).

3.3 Theorem.

The LP-system is then and only then D-directed if for every p(t),t;;: 1:

( ) E m+m+n . .

each u,v,w R + sat~sfy1ng

B'u-A'v+w;;: p(~), (3. 3. I)

also satisfies

B'u+w > p(t). (3.3.2)

Proof.

The theorem follows from theorem 3.2 and from the symmetry between the primal and dual system.

3.4 Remark.

The following two propositions only will be used as auxiliary theorems.

3.5 Proposition.

Let

A

be a diagonal n x n-matrix which is defined for a LP-system

(P- or D-directed) as follows:

in case the LP-system is P-directed:

A: I,

À . . := 1 , if a _{. j}

i,

0

)

JJ

À •• :=

o,

i f a > 0

JJ . j =

then for every ( x, y, z) E ₁₊ n x ₁₊ m x ₁₊ m satisfying

Bx(t)-Ax(t-J)+y(t) = Ytf(t)+z(t), t > K,

=

( 3. 5. I)

forsome y > 0, K ~ 1, and for every monotonous non-increasing

sequence of numbers {8(t)};_ 1, there exists a ({~(t)};_,, {~(t)};) E 1° x lm with ~(K-J) = 8(K-I)x(K-~),such that ~( t) ~(t) Proof. 0(t-J)(Ytf(t)+z(t))

I

8(t)Ax(t)+8(t-l)(I-A)x(t) t~K(3.5.2) 8(t)y(t)

First consider the case that the LP-system is P-directed, so that A:= I .

From theorem 3.3 it then follows that every (x,y,z)

E

1:

x

1:

x

1:

satisfying (3.5.1) forsome y > 0, K ~ I, also satisfies

-Ax(t-1) < Ytf(t)+z(t), t

~

K.

Since the sequence {8(t)};_

1 is monotonous non-increasing, this

implies that

t

-(0(t-l)-8(t))Ax(t-1)~(0(t-l)-8(t))(y f(t)+z(t)),t > K.

(3.5.5) The equalities (3.5.1) imply:

t

By adding (3.5.5) and (3.5.6) we find

'B0(t)x(t)-A0(t-l)x(t-1)+0(t)y(t) (t-l)(ytf(t)+z(t)),t

~

K.

Sincè A:= I we may conclude that there exists a (~,~) E 1 n x l m

satisfying (3.5.2).

Now we consider the case that the LP-system is not P-directed. Then the D-directedness and the definition of A imply that

(3.5.7)

Since {0(t)}~-l is monotonous non-increasing and {x(t)};_

1 is non-negative, we may conc)ude that

BA (0(t)-0(t-l))x(t) ~ 0

j

~

0

t ; K.

-A(I-A)(0(t-1)-0(t))x(t) _

The equality (3.5. I) implies

(3.5.8)

0(t-l)(Bx(t)-Ax(t-l)+y(t))=0(t-I)(Ytf(t)+z(t)),t

~

K.

(3.5.9) Adding (3.5.8) and (3.5.9) it appears that there exists a

({~(t)}~_

₁

, {~(t)}~)

satisfying (3.5.2).

3.6 Propo~;~ition.

If the LP-system is P- or D-directed, then for every

( x,y ) E l n + x 1m + sat1sfy1ng • .

Bx(t)-Ax(t-J)+y(t)

for some y > 0, K ~ I, .and for every ~ > 0, T > K, a

(;,y)

E Rn:m exists such that

(3.6.2) T-l l t

y > 1: (-) y(t)

t=K

a

Proof.

Let (x,y) E x 1: be a solution of (3.6.1) forsome y > 0,

K > l. Defining the sequence {0(t)

1 forsome T > K by

0(t):= I, t = K-J,K, ••• ,T-l

I.

(1{1:):= _0' t > T

=

it follows from proposition 3.5 that a

({~(t)};, {~(t)};)

exists

such thac: = yKf(K)+Ax(K-1) t+l B.t(t+J)-A)6(t)+~(t+l) = Y f(t+J}, t=K,K+I, ••• T-1 (3.6.3) ~(t) x(t), t=K,K+l, ..• T-1 ~ ( t) ; ·Y ( t) t=K,K+I , ••• T-l )6(T), ~(T) ; 0

I T l t T l t (B--A) ~ (-) ~(t)+ ~ (ä) ~(t) ~ T ~ (-) f(t)+-Ax(K-1) y t I a t=K a t=K t=K a a T l t ~ (-) ~(t) t=K

a

T-l I t ~ E (a) x(t) t=K T 1 t. · T-l I t l: (-) ~(t) > ~ (-) y(t) t=l

a

t=l

a

From this the proposition follows immediately.

3.7 Remark.

(3.6.4)

The manner in which the supposition that the LP-system is P- or

D-directed functions appears especially from proposition 3.6.

By this i t is possible to cut down the LP-system to a system with a finite number of periods, which leads to a finite number

of linear inequalities, as constructed in (3.6.2). The latter is

essential to the argumentation of the next theorem.

3.8 Theorem.

"'

For a LP-system (P-or D-directed; f(t) + f, t + 00 ) the following

~roperties hold:

a) If the LP-system is P0-feasible, then the system

(B-.!_A)z <

p =

(3.8.1)

z ;

has a solution.

b) If for some initial vector the LP-syatem has a P-feasible

t~ - m

solution (x,y) such that y(t) ; p y for some y E int (R+) then the system

c)

has a solution.

- m

If for some K~l and y E int(R+) the system

Bx(t)-Ax(t-l)+y(t)

=

(l)tf(t)

1T

x(t-1) ;:; 0

y ( t)

has a solutiDn, then the system

(B-1TA)z < }, (3.8.3)

ha;; a solution.

Proof.

(a) Let (x,y) E ln:m be a P-feasible solution for some x(O). Then

it follows from proposition 3.6 that the system

I - -(B--A)x+y p T

tclA~(O)+ E

f(t)) p t= I T I t T E

Cp-)

x(t) t=l IT-I I t y > -

E (-)

y(t) = Tt=l p

has a solution for every T > I.

Defining

l 1 T

g(T):=~(-Ax(O)+

E

f(t)},

p t•l

system (3.8.4) implies that

T> I,

(3.8.4)

(3.8.5)

g (T) E C, T> I • (3.8.7)

The supposition f(t) +

~.

t +

~

implies

g(T)+f', t + (3.8.8)

From the definition of C it fellows ,174) that this set is

closed. On this ground, (3.8.7) and (3.8.8) imply

~

E C.

In conneetion with the definition (3.8.5) of C we may conclude

that the system (3.8.1) has a solution;

( b) Let (x,y) E ln:m be a P-feasible salution for

~ome x(~)

such

that

1' (3.8.9)

- m

forsome y E int(R+).

From (3.8.4), (3.8.5), (3.8.6) and (3.8.9) i t may be derived

that

Then the convergence (3.8.8) implies

(f'-[~)

E C and there

by the existence of a salution for (3.8.2)

( c) From proposition 3.6 it may be derived that the system

i

T

T-K(~Ax(K-1)+

L

f(t))

t=K

has a salution for every T>K. The proef may be derived in a similar way as the one for b).