On the problems of linear programming with integral coefficients

U.S.S.R.

Comput.Maths.Math.Phys

Printed

in Great

0041-5553/84 $lO.OO+O.CO 01986 Pergamon Press Ltd.

.,Vo1.24,No.6,pp.18-23,1984

Britain

ON THE PROBLEMS OF LINEAR PROGRAMMING WITH INTEGRAL COEFFICIENTS*

A.A. VATOLIN

Linear programming, and also systems of linear equations and inequalities all of whose coefficients are specified in the form of ranges of their possible values, are studied.

1. We consider the problems of linear programming mar ((c,2): Az=b, whose coefficients (A=[ajt],.,, given not by specific values c'
, bm17,

b=[b,,

but by ranges

Q&p, Q=[qJl..,

p=

(1.p.) rd”),

(1)

. ,p,l’, c=[G,.

[p,,

. ,c.l’)

are

all

of their possible values: b’
A’GA=ZA”,

~0,

Q’
p’dp
Here C'=[c,',...,c,']', c”=[c~“,.. .,c,,“]~, A’=[a,‘],,., A”=[o,“],,. , etc., are given matrices and b’
. . , a,,, b,, a,,,

, bm,q,,,

>pi. c,,

, cnl= [L,

, tl

be canpiled from the elements of the matrices [A, b], [Q, p] and the vector c, written in a given order, [A,b] and [Q,p]being entered along the rows. We denote by T, the range of T,= (t,:an'
, w~,I,,,EI‘ ,,,, (c,.T)h-8.As=b,

as an 1.~. with integral

coefficients,

Qs~p,s>O)+~).

and the set

M(~~,~~,KP)=(z:~,I*,ETI,,...,xI~C,ETI~,~~I=~, QxO) as a set of solutions

of the problem A.r=b.

Q.&/I.

xx.

(2)

To clarify, for example, the meaning of the set M(re,Qa,Ko) we use a games interpretation. If we regard the choice of the values t.<=Tpf (for x,=3) as the strategy of the first player, as that of the second player (first, the coefficient and the choice of th,=Th (for x,=U) trr ,’ and so on, so that in choosing the coefficient tki the first player t*, is chosen, then uses information about the coefficients 1, chosen, by the second player, where j
19

of the problem L(r,Q,K) for the case where or=V of Dantzig and Wulf /l/. The analcgues for all i, and the limitations of (1) have the form @rGp, s>O ('inexact l.p.'), were studied coefficients in /5. 6/, where other forms of specification of the variations of undetermined were used. 2. We start by considering the set M(T,,S&,K~). We introduce the following notation: a,,..,=-b,, &,+l=-b,“, I N Qh..+l=-Ph, qr,“+,=-p* , [ad n, =+I- [A,

For brevity,

+I,

a$+,=-b,‘,

in the form

M--(z=ty,,...,y"l': XlL~~k,...., Aay=O, IdjGm. The numbers (j-i) (n+i)+ldkGj(n+i)}.

it by 14.

II qr.“+l=-PA’ (j=i, 2,. . , m,k=l,2 ,_.., l),Ab= Qp=[el,. n+t-IQ, -PI, y=[yr, . , yn+,l’~~“+‘.

Then set M can be represented

We fix set. N(j)=(k:

we denote

Q,y
y>O,

(3)

X”tk,=T*“,

y,+,=l}.

of the coefficients We put

tcj-,,(n+,,+,=a,r....,t-b=-aj, >,n+o,

“+, form the

{i: k,=N(j))={i,,

. . .,i.+,),i,< .&), y=1, 2,. . , nj.

Thus,

the collection

We determine rules:

of sets

the coefficients

of the matrix

x,=3,

ai.==ap’, ii,=afi”,

(j-i) (n+l)+i=k,,

x,=V,

The following assertion linear inequalities. The equation

M=M',

n,, j=N,

j=N={j:

N'(j)+;@).

x.=3,

(n+l)+i=k,

x.=V.

gives M in the form of a set of solutions

z=[y,,....

7-l,...,

m(n+i)+(j-1)

if

1

M’-

if

ti(n+i)+(j-1) (n+l)+i=k,,

if

for each

A==[%],,,+,, A= [c%J~,~+~, Q=[qii]++i by the following (j-l) (n+l)+i=k,,

&=qli”,

1.

is obtained

a,,=ar",Cr=ajt',if

q,,=qa’,

Theorem

Z(j, 7), y=l,Z,...,n,

of a certain

system of

where

y-1’: &/GO, Ay>O, Qy
~30,

y.+r=l

(IF-aJJ z ~~W,T,

y&O,

I

holds. Proof. Let us set d=[d,, . . _, d,+,]ER”+‘, D,={d,: d:Gd&d,“}, i=l, 2,. We assume that the coefficients di are fixed for {i, 2,. ..) nfl). i+h. arbitrary y~Y=(yB0: y,+,=l) and shall show that the equivalence

( V d&ED& (d, y) =

g diyi60) i-1

9

. . , n-l-l t

ad let We also fix an

(d, y)x+dr”yr
if

(4)

is valid. Here and below we use the notation n+L (d,y)l=

dry,. r,' r-1,,+

Let (d,y)r+dr"yrGO. Then because y1 is non-negative, we have dTy,
(d, y)cO) -

(d,y)=M y),+dry*G(d, Y)A+ follows from the fact that

(d, y)k+d,‘yAGO.

(5)

If a d+D, exists such that (d, y)GO, then since y1 is non-negative we have (d, y)r+d'yrG(d. The validity of the inverse implication follows from the fact that d,‘ED,. y)GO. Consider the set W,=(y=y:

n,d,,=&,

. , xn+A,=D,n.,,

(6 ~)a},

where s1,....Xn+. is an arbitrary ordered collection of the quantifiers is the permutation of the set {i, &...,n+i}. If, for example, x.+,=V, A=ilI+,, we have

Djm+t*(d,y)lO)}=(y~Y:n,d,,EDj,,.

.,JL~,~=D,~>

3 and V; then using

@,Y),~+,+d:+,~~,,+t~‘%

{is, ....in+.) (4) for

20 Similarly, we apply the quantifier (d,y),n+,+d,r+,y,n+lGO n,,d,.=D,,to the inequality obtained, and for use the equivalence (4) if n,=V, or (5) if n.=3. h=j. Proceeding the *ame way, as a result we obtain II+,

w,=(

y=Y:

r,Z,y,
where

6,,=

/_I

d”

n.=v,

d,:,

x,=3.

I

'I'

in

l~Jc(l, 2. .,n-rl),d,‘=d,‘=di for SJ. We assume Let d’=[dt’ ,..., d:,,], d’=[d,‘,. .d,*+,]. that the coefficients d,’ and i#A. Suppose that Y'is a set of y's which are fixed when da and possibly some linear inequalities (arising as a satisfy the conditions y>o, y,+,=l. we write the system of conditions result of applying the quantifiers 3. see below). (d',y)GO,

d,‘=d?=d(,

(d',Y)~O,

i=J,

y=Y’

(6)

and show that the equivalence (Yd,=Di

(d’. y) ~0,

(dz, y) SO,

y=Y’)

-

((d’,

y) ,+d,“yiGO,

(d’, y)i+dLy&O,

y=Y’)

(7)

holds. Let the left-hand side of the above be satisfied. Since Vd&=D, (d',y)GO, we have (d',y)r+dl”yiO) that (d’, y)l+d,‘y&O. BY analogy with (4), and inverse implication is justified. Thus, by applying the quantifier V&ED* to system (6), we obtain a system of similar form: (d',y)i+d,“yAO.

(d’, y)

d,‘=d,‘=di,

>+d,‘yal

i~J’=l\(h},

y=Y’

Next, we show that (3 dl=Di:

(d’, y) GO, (d’, y) 20,

y= Y’) -

(8)

(d’, y) I+

“+i

dh’y,dO,

(d*, y)I+d>.“ylaO,

c

(d,‘-d,t)y,GO,

y=Y’)

i-1, ICJ Let the left-hand -(d?. y)r. Hence form:

side of (d’,

y)r-(d*,

(8) be true. Then a dlEDl exists such that The last inequality can be rewritten y),
diyr<-

in the

(d',Y)A,&/La eqUiValent

y&O.

(9)

l-l, ICI

Since

y, are non-negative,

we have

(d',y) r+&‘yaG (d’, Y)GO,

(d’, y) h+dL’yl>

(d’, y) 20.

Next, let the right-hand side of (8) hold. Then d,'y*<-(d',u)~, d,"y&-(d*, y)~, and at the same time by (9) we have (d',y)i>- (d',Y)~. Taking into account the inequalities obtained, and also the non-negativity of yr, it is obvious that (z: &'y&a
-(d', y)AzG-(d',

y),)+@.

Hence it follows that a dl=D, exists such that (d', y)l+dlyk
Cd’, y) ~+&‘yd4

(d’,

y)r+dl”y&O,

d,‘=d,z=d,,

We note

3 are applied that if several quantifiers inequalities of type (9) obtained will be a consequence inequality (9) which arises in applying the quantifier most on the right in this succession. Consider the set n,d,,EDj,,. . , n,+A.,EQ..,,

Wz=(y=Y: It can be transformed

i~/‘=J\{h),

y=Y’.

to system (6) in succession, then the of the inequality y>O, and of that XI, this quantifier being the outer-

(4 Y)=O}.

into W,=(y: n&ED,,, (d’, y)
,n,d,.EDj,,(n,+A,ED,..,,

y)>O, d,'=d:=dt, i=J={l, 2,...,

n+l),

FY)).

Using the same argument as in transforming the set W,, taking into account the equivalences (7) and (8) and discarding the inequalities - sequences of type (9) (see the remark above), we obtain

where

dj:, ii,*= {d(,

n.=V, x.=3;

21

Z(O, y) (r=i, 2,. . no) and n, are identical to set Z(1,r) and number n,: the latter correin (3) has the form Aay=O, QpyGO, ~20, Y.+I---1 (4 y)=O, spond to the case where the system x,=n., k.=j., s=l, 2,.. .,~fl. y)O, y*+,=i and QpyGO the formula obtained for W,, and NOW it remains to apply to each line of system &y=O the formula obtained for W,. The theorem is proved. to each line of system let the coefficients trihave the chosen fixed values and for iGi<,a Let OdczLpGv, _ We determine the coefficients &e, Ass, g,~job of the mX(n+l), mX(n+l), tX(n+l) matrices &,, by the rules 9

i&y"=&"'=E,,, if

&Y-n,, -so -.a a,< =u,i

=tr,,

.&‘)=L,.,

(j-i)(n+l)Ci=kh,

if

m(n+l)+(j-i)

1ChGa, IGhdcz,

(n+l)+t=kh,

(j-I) (n+l)ii=k,

if

if m(n+l)+(j-1)

a
(n+l)+i=k&,

a
ii,,%&, =$8 a&; ,I, 9,x -aR=?j,l. Using notation like that at the beginning of In all other cases Sect.2, for each j, la and x.=V), to find the first player's strategy (i.e., the choice of thi,where such that z satisfies system (2) with the coefficients chosen in accordance with i>a,xi=3) these strategies. The answer to this question is the following assertion which defines a set of the desired strategies in the form of a set of solutions of a certain system of linear inequalities.

and let the coefficients ta, of the matrices Corollary 1. Let OGa
. , p): xp+,t,b+r=I’~R+I.., xvt,v=T~v,A.z’=b,

{th I.ET~. 1(i=a+l,.

1 tlzETA,(i=a+l,. (&-&)yi’
z rczlj,,, or

, p): .&y’GO, r=l.

2..

_, n!,

A,,y’>O, jmV,

As,Q,

and xo+,=

Qz’Gp}=

Q,sy’fO,

y’=[z,‘,

,5,‘,

11’

1

=M(a.8,5’)).

Proof.

1n addition to fixing the coefficients G,,...,t,,, we fix the coefficients the+,, assigning to each of them an arbitrary value from the intervals TIm+,,...,T,,. Then, by Theorem lwehave the following formula for MB (for this we assume tr:=tr’;=ir I, th:=t,;=th,, i=l, 2,. ,a, j=a+l,..., B):

. . ..LB’

Mp=(z:

Y-B+,~++,ET~~+~,

s=[y1,.

In other words, the condition

. ,

for an arbitrary

. , jcv~~=T~~.

As=b,

Qdp,

r>O)-

ynl’: xmsyco, A,,ydO, Q&GO,

X%0

the collection

tbq+z,....tke

Ar’=b, xP+lthB+,ET,B+,,-..,x~t*,ET,~,,

when

ensures

satisfaction

of

Qx’dp,

and only when this collection belongs to kf(a, B,I'). The corollary is proved. Concluding this item of our argument we consider the case where t{=--m, t;‘=+m for a t,'=--m* then Ti= (tkER: t,4t,“)). part of the elements of the ve_ctor T (for example, if To each coefficient of the system which defines the set M' there corresponds ii*,aj& q,I a certain element tl, (for example, the element t,,-'H"+ll+~ .) corresponds to the coefficients E Then we call the number h a rank of this coefficient. Let some coefficients of aji, a# 1. a certain inequality of this system have infinite values. Then we shall set the left-hand side of the inequality under consideration equal to the infinite value of that coefficient which has the highest rank (or to the infinite value with opposite sign, if this coefficient occurs in the inequality with a minus sign). For example, if in the inequality

(U,.-&,)y&O

z IEL(I.7,

22 =a,,,,,=+=J have the highest rank, we set its left-hand side equal the coefficients a,,,,=-= and +a%0 or -m>O in a system which specifies to --m. Then the presence of the inequality +m>O, --aoO:z~M'). The proof is carried out on the lines of the proof of Theorem 1, and is not therefore given here. The corresponding analogue of Corollary 1 can be easily formulated. 3.

Now investigating

problem

L(T, S&K),

by Theorem

1 we obtain

the following.

Corollary 2. Let h
P*=C*"

and

E,;=c,'if

Itlax(e: (F,z),e,r=M'), v+j=r., o.=v vi-i==-rr,o~=8 and

Note. As an 1.~. problem

with integral

mar(&(.z:m,tr,ET,,,._., o,t,,

coefficients,

Then the problem

respectively. the problem

E T,,, (c,z)= 8, AI-b,

Qzcp,

(10)

z>O)-a),

L(r,Q,Kf can be specified. which has a scmewhat different interpretation compared with Theorem 1, problem (10) is also equivalent to a certain problem of 1.~. Let us write an 1.~. problem, and its dual problem: max (t-l: (c,2)>8, Qzdp, min {p: (p, u) Gp, Q’uac,

5>0),

By

(11)

t&O}.

(12)

Let q-mn+m, .T,==T~+,X.. .XT,,dV, Q,= (a,, . . , o,-,), and let p+ u=p. K,={ m,, mu_,) be a certain permutation of the set (nf1,...,~). The 1.~. problem L(r,, a,, K,) with the integral coefficients

.,

Here

max(8: (z:o,~~,=T,~,...,o,,&

ET,,,_-, P--9

(c,z)29,

corresponds to problem (11). Let the operation l as applied latter's replacement by a dual quantifier, that is

z90)+0').

QzGp,

to the quantifier

oi entail the

R, if W-V, tit'= 1 V, if 0,=3. Putting Q,*=(0,',...,$__J. coefficients

we shall describe the 1.~. problem

as a problem dual to the problem respect to L'(T,,Q,',K,).

L(r,, 8,, K,). Then problem

L'(G, Q,',&)

with the integral

L(T,,Q,,K,} is in turn dual with

L(s,,Q,, KS), L*(T,,Q,',K,) can be Corollary 3. From the fact that one of the problems solved there follows the solvability of the other, and the equality of their optimal values. This assertion holds because by Corollary 2 the problems L(7,, Q,. K,), L'(G, Q,', K,) are equivalent to the mutually dual problems of 1-p. 4. We will give two examples of the results obtained. Let us denote by f(T) the optimum value of problem (1). Suppose the operator P is applied to problem (11, taking, on the sets Tr, . . . , T, , the upper bound of the function f(T) over all elements of the vector T with numbers from the set Z,, and the lower bound over all Z,UZ,=(1, Z, and Z, are taken from the condition elements T with numbers from Z,, where We assume that by definition the upper bound of the linear function on an 2,...,n), z,nz,=0. The following is a corollary to Theorem 1. empty set equals --. P does Corollary 4. Suppose that (1, 2,...,mn+m)cZ,.Then the value of the operator not depend on the order of the operators sup and inf in this operator, and it equals sup ((E,z):A'zdb", A”z>b’,

QzGp, ~0).

if the numbers of these elements in vector T belong to Z,.and where C,=c.",&=q,f‘, ji,=p,” I othenrise. I-c,=c, , q,S=q,,“, p,=pj Consider the system of linear equations &=6, all of whose coefficients are given approximately within the bounds, that is we know only that AHJ~=(A:A’~A6A”), 6d,={b:b’G Without Let the signs of the elements of the desired vector z be known beforehand. b
X=(z>O: BA=U,, 3b4l,, Az=b}, a class of solutions of the approximate system Az=b,

~20,

AdJ.,,

that are of equivalen:accuracy: (13)

b=!J,.

Here, in accordance with /4/, the problem will consist of finding (13) (a solution which gives a stable approximation to the normal that is of solving the problem z>O), min())z~):zEX),

a normal solution

solution of system &=(?, of system

(, i41

23 where

11 .I1 is the Euclidean norm. The use of Theorem 1 enables us, by transforming X=(120:

to convert ities.

(14) into the problem

The author

expresses

A’xdb”,

of finding a normal

his gratitude

the set X to the form

A”s>b’) solution

to 1.1. Eremin

of a system of linear

inequal-

for his interest.

REFERENCES 1. DANTZIG G.B., Linear programming and extensions, Princeton University Press, 1963 (Russian translation, Progress, Moscow, 1966). (Nestatsion2. EREMIN 1.1. and MAZUROV V.D., Non-stationary processes of linear programming arnye protsessy matematicheskogo programmirovanya), Nauka, Moscow, 1979. 3. EREMIN I.I., MAZUROV V.D. and ASTAF'EV N.N., Non-eigenproblems of linear and convex programming (Nesobstvennye zadachi lineinogo i vypuklogo prograuunirovaniya), Nauka, Moscow, 1983. 4. TIKHONOV A.N., Approximate systems of linear algebraic equations, Zh. vychisl. Mat. mat. Fiz., 20, No.6, 1373-1383, 1980. 5. SOYSTER A.L., Convex programming with set-inclusive constraints and applications to inexact linear programming, Operat. Res., 21, No.5, 1154-1157, 1973. 6. TIMOKBIN S.G. and SRAPKIN A-V., On linear programming when there is inexact data, Ekonomika i matem. metody, 17, No.5, 955-963, 1981.

Translated

U.S.S.R. Comput.Maths.Math.Phys., Printed in Great Britain

Vo1.24,No.6,pp.23-30,1984

by

W.C.

0041-5553/84 $lO.OO+O.oO 01986 Pergamon Press Ltd.

THE METHOD OF SUCCESSIVE APPROXIMATIONS IN A PROBLEM OF THE OPTIMAL CONTROL OF ONE NON-LINEAR PARABOLIC SYSTEM* A.T. LUK'YANOV

and S.YA. SEROVAISKII

The necessary conditions for optimality in the form of variational inequalities for the problem of the optimal control of a parabolic system with a step non-linearity is established. The convergence of the method of successive approximations for realizing the optimality condition is proved. TO solve the problem of optimal control, the necessary conditions of optimality (the principle of maximum, variational inequalities, the condition of stationarity, etc.), are widely used; see for example /l-3/. The conditions lead to boundary value problems which include an equation of state, an adjoint equation, and, properly, an optimality condition. The system obtained can be solved by the method of successive approximations /4/. In the present paper we discuss the problem of the optimal control of a system described by parabolic equations with a step non-linearity. The unique solvability of the equations of state, and the existence of the optimal control are proved. By studying a quasi-adjoint system (see /S/j, we establish the necessary optimality conditions in the form of variational To solve the optimality conditions the method of successive approximations is inequalities. used. The convergence of the method is proved. 1. Statement of the problem. Let Q be an open bounded domain of space R”, with T=const>O, Q=BX(O, T), X=SX(O, T). Consider a system described by the equation +-by(u) with the initial

and boundary

Ipy(u)=u,

a sufficiently

smooth boundary

s,

k t) EQ.

conditions Y(U)1I-0=y0, s=Q,

y(v)=O,

(z,t)Ez

(1.2)

where u is the control specified on the closed and bounded set (I (a set of feasible controls) in the space L,(Q),y(u) is the function of the state of the system, which corresponds to the u, A is the Laplace operator, control p=const>O, and y, is a known function. The problem of optimal control consists of searching for a function u form the set U, which minimizes the functional v+Z[u, y(u); where

’ 24,11,1638-1648,1984 *ZI~.vychisl.Mat.mat.Flz.,