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The convolution of finite systems of linear inequalities
THE CONVOLUTION OF FINITE SYSTEMS OF LINEAH INEQUALITIES* S. N.
CHERNIKOV
Sverdlovsk (Received
4
May
1964)
‘THE present paper deals with the problem of the elimination of unknowns from systems of linear inequalities. Fourier [II has already considered the elimination of unknowns as a natural method of solving such systems. Up to recent times, however. it was used as a basis only for theoretical purposes (see for instance Dines [21 and Kuhn [31); in particular, in [31 it was employed to establish the conditions of consistency of systems of linear inequalities. With this aim an algorithm is suggested there for rule for the combinaeliminating unknowns, which reduces to a definite tion of the inequalities of the system considered, using non-negative numbers as multipliers. In using it, however, for numerical purposes (as also in using other similar algorithms for these purposes) the very large number of combinations needed to carry it out is a great drawback. The content of the present paper is determined to a large extent by searches for the best method of combining the inequalities of an arbitrary linear system, in which the number of constituent combinations is a minimum with a view to eliminating the unknowns. These searches lead to the problem, which is more general than that of the combination of inequalities, of combining an arbitrary system of linear inequalities, such that the new system, composed of the inequality-combinations obtained, has in some subspace of the space of the combined system a set of solutions which coincide with the projection of a set of solutions of the latter into it. This question is considered in the present paDer for systems of linear inequalities in an arbitrary real linear space I!,. If fj(5) are real
linear
(i.e.
additive
(i =
and homogeneous)
-___ *
Zh.
vychisl.
Mat.
1, . * -9 m)
(1) functions,
--mat.
Fiz.
5.
1, 1
3 -
20,
1965.
given on L,
2
S.N.
Chernikov
then a system of the form +G
(1 =
in which tj are real parameters a system of linear inequalities
or simply on L.
fj(z)
1, .... m), real
numbers,
(2) is here called
If U is a subspace of L, V one of its direct complements in L, C(u) the cone of non-negative solutions (ui, . . . , u,,,) of the equation a&(x)
+ . . . + hAa
=
0
on U, n E U and c(i) =
(c&O, . . ., c#))
(i =
1, . . ., k)
is some system of generating elements of the cone C(U), then for any set of values of the parameters tj, j = 1, . . . , m, the set of solutions of the system m
2
tn Cjci'fj(X)
j=i
+2
Cj(%fj < 0
(i=
1, . ..) k)
j=i
in the subspace V, coincides with the projection into it of the set of solutions of system (2) (see Theorem 2.3). We shall call this system a u-convolution of the system (2). In the case of a zero cone C(U) let us agree to consider that the U-convolution of system (2) is empty. In this paper some properties of U-convolutions of system (2) are established and, in particular, the fact that any U’-convolution of an arbitrary Uconvolution coincides with some (17+ U’)-convolution (see Sections 1 3). In conjunction with the defined rule for reducing an arbitrary (U + U’)-convolution to the (II+ U’)-convolution with the least number of inequalities contained in it (see Section 3) this property is used as a basis for the algorithm for a successive reduced convolution of systems of linear inequalities of the form (2) on the space L = An. In the case where tj are free parameters, the algorithm reduces to the best variant of the algorithm for the successive elimination of the unknowns from such systems. The algorithm is used (see Section 4) to find a general formula for non-negative solutions of an arbitrary system of homogeneous linear equations and also to obtain a general formula for the solution of an arbitrary simultaneous system of linear inequalities of the form ajiXi
where all
+
.. . +
ajnXn
Uji and Uj are real
-
aj
4
numbers.
0
(j= 1, .... m),
Finite
systems
of
linear
3
inequalities
The basic results of the paper were published without proof in [41.
1. Fundamental convolutions of a system of linear inequalities We shall call any positive linear combination of one of its arbitrary (non-empty) subsystems, which is identically equal to zero on U, a Uinterlacing of the system of functions (1). The linear combination is here said to be positive if all its coefficients are non-negative and at least one of them is positive. A &interlacing Uj,Ofj,(S) + . . . +
4.+,fj,,,t5)
with positive
coefficients
uj,O, k=l,
...
, s+l,
we
shall call a fundamenta1 U-interlacing, if s of the s + 1 functions entering into it are linearly independent of U; it is clear that with this condition any s of the s + 1 functions fjk(x) are linearly independent of CT.We shall call the vector (ul, . . . , u,,,) with Ujk = Ujko; k = 1, . . . . s + 1, and the remaining Uj zero, a fundamental element of the cone C(U). It is not difficult to see that if the cone C(U) &ntains non-zero elements then it necessarily contains fundamental elements also. This is inferred from the following well-known proposition. Theorem
1.1
If a vector is expressed linearly with non-negative coefficients by means of some system of vectors whose rank is diiferent from zero, then it can be expressed with non-negative coefficients through at ieast one of its linearly independent subsystems. We can easily convince ourselves also that the cone C(u) can contain not more than a finite number of essentially different fundamental elements. Two elements of the cone C(r/, are not considered to be essentially different if they differ from one another only by a positive numerical factor. Two fundamental elements are essentially different only when they differ in the numbers of their non-zero coordinates. Let C be some maximumsystem of essentially elements
cti) =
(uiti), . . . , Us),
i = 1, . . . , h,
different,
fundamental
of the cone C(m. We
shall then call the system i j=i
Uj"'fj(5)
+
i
j=i
Ujti)tj <
0
(i = 1, . . . , h)
(3)
4
S.N. Chernikov
a fundamental U-convolution of system (2). If the cone C(U) is zero we shall consider that the fundamental U-convolution of system (2) is empty. For fixed U the fundamental U-convolution of system (2) is obviously unique except for the positive numerical factors which occur in its inequalities; therefore no distinction will be drawn below between fundamental U-convolutions of system (2) which correspond to one and the same U.
It can easily be seen that the definition essentially tion given here does not differ finition of [51. Theorem
1.2
The system (2) with a L has a solution in only those for which its System (2) with sn empty for any values occurring
U t
In the proof call
of a fundamental U-convolufrom the corresponding de-
non-empty fundamental U-convolution for some L for those values of the parameters tj and fundamental U-convolution has a solution in L. fundamental U-convolution has a solution in U in its parameters.
of the theorem the following is used.
proposition
(which we shall
Minkowskii' s theorem)
Theorem
1.3
If
all the solutions of the system of linear inequalities fj(X) GO, . . . . m, on L satisfy the linear inequality f(x) GO on L (in this case it is said that it is its consequence) non-negative numbers pl,. . . , pm exist, for which the identical relation = 1,
j
f(z) = i Pjfj tz) -
j=i
is valid
for
x E L.
is For systems of linear inequalities on L = Rn this proposition an arbitrary space proved by Minkowskii (see [Sl); it is transferred to L by fairly obvious considerations (the principle is given in L-51). Let V be some direct
Proof.
shall
show that
system tj
=
tj"
(3)
if
z” =
with values has a solution
of
complement of the subspace
zz” + u” (u” E u, u” E v) tj =
tj’, j =
1, . . ., m,
u’ + V0 (zz’ E U).
U in L.
is a solution
of
the system (2)
Suppose that this
We with
is not so.
systems
Finite
Then system (2)
with
then the inequality
tj = t GO
of
tj’ = will
linear
considered therefore
has no solution
tj” + fj (uO)
be a consequence
fj(X) + tj’t < 09
t=
of
in U. But
the system
j = 1, . . ., m,
on the space U + R’ of pairs in view of Minkowskii’s
5
inequalities
[z,t]
(zEU,
tEai),
and
theorem some relation
i
Pj (lj (z) +
tj’t)
j=l
which is identical with respect Pj>O, j = 1, . . . . m.
to x E li and t E
R’ is valid
for
Since system (2) for tj = tj ‘, j = 1, . . . , m, has no solutions in U, the rank of the system of functions fj(X) + tj ‘t on U + R’ is greater by one than the rank 1 of the system of functions fj(x). Using Theorem 1.1 we obtain from the preceding relation a new relation, identical with respect to n E II and t E R1, viz.
t =
2
4j,
(fj, (z) + tjA’t)
(1’d 1)
k=i
with positive But then
qjk and linearly
independent
functions
fjk(x)
+ tjk’t.
I‘+1
C !ljkfjk tx) k=l
is a fundamental U-interlacing . ..* m. Since the inequality
obviously corresponds to (2) with tj = tj’ has no is obvious that it has a proves the first part of
of the system of functions
fj(x),
j
= 1,
this, the fundamental U-convolution of system since it solutions in I;. But this is impossible obtained solution u” E U. The contradiction the theorem.
From these considerations it follows also that if system (2) has no solutions in U for some values of the parameters tj, at least one fundamental U-interlacing of the system of functions (I) exists. If the cone
S.N.
6
Chernikov
C(V) is zero it follows any values theorem.
of
from this that system (2) has solutions in U for of the parameters tj, which proves the second part of the
In view of the assumption, formulated the theorem, we have the inequality
is an arbitrary
of the proof
i=l,.*.,?72.
+tj'
fj(U'+UO)
at the beginning
U-interlacing
of
the system of
functions
j=i
(l),
from this
we obtain
j=i
j=i and also
2
p{fj
(2’) +
j=t
2
pjtj'
<
0
(2” = 24’+ V”).
j=i
In view of the arbitrariness of the choice this means that the inequality m
m
j=i
j=i
of x0 and tj,
j
=
1, . . . , m,
is a consequence of system (31, considered on L i Rm (R” is the space of vectors (tl, . . . . t,,,) 1. Using Minkowskii’ s theorem here we find that the m linear forTI 2 is a positive linear combination of those linear j=l forms of the parameters tj which occur in the inequalities of system (3). pj’tj
Hence it follows that any maximum system of essentially different fundamental elements of the cone C(U) is a system of its generating elements. Since it is obviously irreducible, in addition to this, it is found to be a basis for C(u). On the other hand it is easy to see that each basis for CC.!4 cannot contain elements which differ essentially from the fundamental elements of C(I/). Consequently we have
Finite
systems
of
linear
inequalities
7
Theorem 1.4 Maximum systems of essentially different non-zero cone C(U), and only those, are its
fundamental bases.
elements
of
a
Calling the sum of the numbers of its non-zero coordinates the index of an arbitrary element of the cone CC0 we note an obvious consequence of this assertion: Theorem
2.5
The system of generating elements of a non-zero cone C(U) is its basis when and only when it does not contain any element with an index which includes the index of at least one of its other elements.
2. Arbitrary convolutions of a system of linear inequalities Comparing Theorem 1.4 with the definition of an arbitrary U-convolution of system (2) (see the introduction) we obtain the following state-
ment: each U-convolution of system (2) contains mental U-convolution of the system (2).
as a subsystem a funda-
Calling the index of the element Cci) corresponding to it the index of the arbitrary inequality of the U-convolution of system (2) (see the definition of a U-convolution), we see, by virtue of Theorem 1.5, that a U-convolution of system (2) does not cease to be its U-convolution if we eliminate from it any inequality with an index, which includes the index of at least one of the remaining inequalities; by successive elimination of such inequalities from it, it is converted obviously into a fundsmenta U-convolution. If the cone C(u) is non-zero, then from the statement noted here it follows that all U-convolutions of system (2) for fixed U have one and the ssme set of solutions in L for one and the same choice of values entering into its parameters. Therefore from Theorem 1.2 there follows: Theorem
2.1
If the cone C(II) is non-zero the system (2) and its arbitrary U-convolution have solutions in L with the same values entering into their
8
S. N.
Cherni
kov
parameters tje If the cone C(u) is zero the system (2) II for any values of the parameters entering into it. Calling Coral lary
each U-convolution
of system
(2)
complete
if
has a solution
U = L,
in
we obtain
2.2
The system fj(X)
(i =
--aj
1, . . ., m),
where al, . . . . a, are real numbers, is consistent when at least one of its complete convolutions is consistent or empty. If the system considered is consistent then each of its complete convolutions is consistent or empty. Suppose, as before, that II is a subspace of L. We shall denote by V one of its direct complements in L. We shall call a finite system S of linear inequalities on L a UV-combination of the system (2) if it satisfies the following conditions: 1. Each inequality of the system S is a positive linear combination of inequalities of system (2) such that the combination of functions (1) entering into it is identically equal to zero on Lr; For each set of values of the parameters tj the system S in V coincides with the projection solutions of system (2).
2.
Theorem
the set of solutions of into V of the set of
2.3
If the cone C(II) is non-zero a UV-combination of system (2) space 0 in L, and, conversely, is a UV-combination of system V = V” of the subspace U in L, system (2).
then each U-convolution of system (2) is for any direct complement V of the subif a system of linear inequalities on L (2) for at least one direct complement it coincides with some U-convolution of
Proof. Let V be an arbitrary direct L and x = u” be a solution, contained
complement of the subspace U in in V, of some U-convolution S of system (21, taken for some values tj = tj”, j = 1, . . . , m, for which it is consistent. Then for values tj = tj’ = tj” + fj(VO) the system S will have a solution in II. If system (2) and system S are considered as system of linear inequalities on the space II, then by virtue of the first part of Theorem 2.1 it follows that the system
Finite
systems
of
linear
9
inequalities
(f = I, . . ., m)
fj(X) +tj’
has at least one solution u” in u. But then the system (2) for = 1, . . . , m. has a solution u” + vO. Since system (2) is not i with those values of the parameters tj, for which the system s consistent. and since each non-empty convolution of system (2) the first condition of the definition of a /TV-combination, the assertion of the theorem follows from this. Now let
T be some W-combination
an arbitrary
i 4jfj(x) j=1 If
x0=
T for
sistent, is also
U-interlacing
u” + u0 (~0 E U,
arbitrary
values
of
u” E V)
system (2)
is insatisfies first
and
of the system of functions is an arbitrary
tj”, j = 1, . . ., m,
tj =
with V = p
tj = tj”, consistent
solution
(1).
of system
with which it
is con-
then in view of the definition of W-combinations its solution u”. But then in virtue of the same definition system (2) has a
solution
for values
ZJ’+ u0 (u’ E U) fj(Z-3’
+
U”) +
tj" <
0
of
tj
=
tj”
But then
(f = I, . . ., m).
Hence we obtain m
m
2 Qjfj (5’) + 2 j=l
Qjtj’
<
0.
j=i
Since x0 is an arbitrary solution of system T with arbitrarily chosen values of the parameters tj, with which it is consistent, this mesns that the inequality
$
4jfj
j==--i
(Xl+
$QJtj G 0
j-i
is a consequence of the system T, considered as a system of linear inequalities on the space L i Rm (P is the space of vectors (tl, . . ., t,)). Hence, as with the proof of Theorem 1.4, we see that the vectors of the coefficients of the linear combinations, which determine system T from system (21, generate the whole cone CC0 of the system of functions (1). Consequently system T is a U-convolution of system (2). Note.
If
v0EV(U/V
= L)
is a solution
of the U-convolution
of
10
S. N. Chernikov
system (2) for of the system
tj = tjo#
i = 1, . ..,
IIIon
fj(u) + tj’ < 0 for
tj'
= tjo
+fj(Uo)o
finding some solution
(j = i, . . . , m)
we obtain some solution
(2) for
tj = tj"
follows
from Theorem 2.3.
u. E U
no = u. + uo of system
Cj = 1, . . . , ltl). The existence of the element ua here
Calling the maximumsubspace R of L, on which all the functions (1) take only zero values, the kernel of system (21, we note in concluding this paragraph the proposition which now follows. Theorem
2.4
If a U-convolution of system (2) with respect to U @ H is non-empty, z*- -r-I. I~J LLulBis beioa the rank of system (2; by not less than the dimension_ ality
of the direct
complement R’ of the intersection
The rank of system (1) of functions fj(Z) to be the rank of system (2).
R fl LJ
in U.
which enter into it is said
Proof. From th e de f inition of a U-convolution of system (2) we can easily see that its kernel R’ contains the subspace R and II. Since the rank of the U-convolution considered obviously coincides with the dimensionality of the direct complement Q’ of its kernel R’ in L, and the rank of system (2) with the dimensionality of the direct complement in L of its kernel R, on using the direct expansion
L4 =R’.+Q’=
(R+R’-+R)
iQ’=R+
(R’iR+Q’),
where i is some direct complement of the subspace R + U in R’, we obtain the fact that the rank which interests us is lower than the rank of system (2) by no less than the dimensionality of the subspace R” 4 E. But this means that it is lower than the rank of system (2) by no less than the dimensionality of space R, which was to be proved.
3. Repeated convolutions Let U and II’ be two subspaces of (2) is not empty then we shall call sidered on L) a repeated convolution repeated (R U’)-convolution of the
If the U-convolution S of system the U’-convolution of system S (conof system (2) or more accurately a latter; if system S is empty or its
L.
Finite
systems of
linear
11
inequalities
U’-convolution is empty, we shall consider that the repeated (II; U’lconvolution of system (2) is empty. Similarly we define a repeated CR U’; U’$-convolution of system (2) etc. Theorem
3.1
Any repeated (U, U’)-convolution (U + U’)-convolution of the latter.
of
system (2)
coincides
with some
Proof. Let S be some U-convolution of system (2) and S’ some U’-convolution of system S. We shall consider initially the case where both of them are not empty. Let U’ be some direct complement of the intersection U n II’ in II’, and V some direct complement of the subspace I/” = U + U’ = U -k UC in L. It is easily seen that the system S’ is a Y-convolution of system S. In view of the first part of Theorem 2.3 the set of solutions of system S in the subspace M(U, t), t = (tr, . . . . t,), U” -i- V, for any choice of values for the parameters tj, in system (2) is the projection of the set M(t) of solutions of the latter into this subspace. If the system S is considered as a system on the subspace II* i V, in view of Theorem 2.3 the set of solutions of system S’ subspace V coincides with the projections of the set M(U, t) into this subspace and so with the projection into it of the set M(t). Since the system S’ obviously satisfies the first condition of the definition of aUV-combination of system (2) with U = U", it follows from this that it is a II” V-combination of the latter. But then in virtue of the second part of Theorem 2.3 the system S’ must be a U’Lcombination of system (2). Thus in the case considered the theorem is proved to be true. Now let at least one of the two systems S and S’ be empty. If the system S is empty then it can be established without difficulty that system (2) has an empty 07 + U’l-convolution. If system S’ is empty and the system S not empty, then in view of the second part of Theorem 2.1 tha system S has a solution in II’ and so also in U + U’ for any values of the parameters tj entering into it. But then, in view of the first part of the same theorem, system (21 has solutions in U + U’ for all values of the parameters tj entering into it, (here the systems (2) and S are considered as systems on U + U’J. Hence it follows that the system 0, j = 1, . . . , m, has a solution in U + II’. It is clear that with this condition the system of functions (1) cannot have (U + II’)interlacings. Consequently in the case considered the (U + U’)-convolution of system (2) is empty. This proves the theorem. fj(Xl
<
In view of the statement formulated from Theorem 3.1 there follows
at the beginning
of Section
2,
12
S.N.
Chernikov
Corollary 3.2
From the repeated (.!k IJ’)-convolution of system (2) we can choose any inequality which contains a non-fundamental (U + U’)-interlacing of functions (1) (the index of such an inequality includes the index of at least one of the remaining inequalities) and which does not violate its property of being a (U + U’)-convolution of system (2). This possibility is used in the algorithm given below for reduced convolution, which gives successively fundamental U-convolutions of the system fj(X)
+tj=lZjiXi+...+ajnXn+tj
(i
=
1,. . ., m)
(4)
with respect to the subspaces U=
ui, u = uj + uz, . . u = ui + . . . -I-Uk .(
(k G n),
where VI, . . . , U, are subspaces generated in Rn, respectively, by the coordinate vectors el = (1.0, ..., 0). e2 = (0. 1,0, . . . , W, . . . , e, = (0,
. . . ) 0.1).
For the basis of the algorithm the following tions are needed.
supplementary proposi-
Lemma 3.3
If in some (Ui, + . . . + U;)-convolution
of system (4) the coeffi-
cients for some unknownxl, with 2 f iI, . . . , ik, are non-negative (nonpositive), a linear combination of k columns of the matrix of system (4) with numbers i = il, . . . , ik exists, which in the sum with the l-th column of this matrix gives a vector column with non-negative (nonpositive) coordinates; t33nVerSSly if for some numbers i,, . . . . ik and . . . . ik, such a combination exists then in each (UiI + . . . + l/i)1 #iI, convolution of system (4) the coefficients of xl are non-negative (nonpositive). In fact, with the condition of the first solutions of the system aiiut + . . . + adh
=
-Uj
the inequality
0
part of this proposition
(i = it, . . ., ik), (I=
l,...,m)
all
Finite
systems
of
linear
and so the first part of it is obtained by direct Minkowskii’s theorem. Its second part is obvious. From the proposition
13
inequalities
application
here of
just proved there follows
Coral lary 3.4 If a column of coefficients
of xl in a (Ui, + . . . + Uik)-COnVOlUtiOn
of system (4) (I = i,, . .., ik) contains elements with opposite signs, each linear combination of k-columns of the matrix of system (4), which have the numbers i,, . . . , ik, give in the sum with the l-th column of this matrix a vector-column with coordinates of opposite sign. We now prove a basic lemma. Lemma 3.5 Let (II, + . . . + Uk)-convolution of system (4) be non-empty. Let A1 be the column of coefficients of x1 in system (4) and AZ, I = 2, . . . , k, be a column of coefficients of XI in some (111 + . . . + UI_r)-convolution of system (4). If s of the k columns A I, . . . , Ak (s
Let us consider the system aiiui + . . . + amium =
0
(i = 1, . . k), .)
(5)
defining the cone C. If s = k, then in view of Corollary 3.4 its rank is the same as the number s and therefore the proposition to be proved is obtained without difficulty by means of Theorem 1.1. We shall therefore assume that s < k. In this case at least one of the columns Al, . . . , Ak contains no elements of opposite signs. Let k,, . . . , k 2 be the numbers (in increasing order) of all the columns of such a kind. By means of Lemma3.3 we can easily see that without violating the set of solutions of system (5) we can replace successively the k I-th, kl_,-th, . . . , kl-th equations in it by equations with coefficients without change of sign; in connection with Lemma3.3 this is arrived at by adding to each of the substitute equations some linear combination of the equations preceding it.
14
S.N. Chernikov
Since, in accordance with the assumption, the (U, + . . . + Ilk)-convolution of system (4) is not empty, the system (5) has non-zero, nonnegative solutions and therefore in each of its so transformed equations zero coefficients must be met with “column” numbers which are common for all the equations considered. It is clear that the arbitrary, non-negative solution (Us, . . . . u,,,) of system (5) can have non-zero coordinates can have non-zero coordinates only at places corresponding to such numbers. Let us replace by zeros all coefficients of system (5) with other column numbers; we shall denote the system thus obtained by P. In the system P the equations with numbers k,, . . . . kl) contain no terms different from zero and therefore its rank does not exceed the number s of its other equations, Using Theorem 1.1 here we easily obtain the statement to be proved. Turning now to tion we give the numbers al, . . ., pair of elements - aqzp. and 0PZ9
the description of the algorithm of reduced convolufollowing definition. Let A be some system of m real a,,, and .Z’ a system of m unknowns ~1, . . , , z,. To each ap > 0 and ag < 0 from A we relate the linear form to each zero element a, from A the form ts; we shall
say that the aggregate of all such forms arrived at is obtained as a result of an A-deformation of system Z. If numbers of opposite sign are met in the system A we shall call the A-deformation an interlacing deformat ion.
THE ALGORITHM OF REDUCED CONVOLUTION 1. Let A, be some non-zero column of coefficients of system (4). for instance a column of coefficients of the unknown xl. Performing the Aldeformation of the left-hand sides of the inequalities of system (4) we obtain a system &, which is its tir-convolution (obviously fundamental). With each inequality of system S1 we shall associate its index. It coincides obviously with the set of numbers entering into its parameters or, what is the same, with the set of numbers of those inequalities by whose combination it was obtained. 2. Let the system Sk, which is a fundamental (II, + . . . + Ilk)-convolution of system (41, be already obtained, and let Aktl be some non-zero column of its coefficients, for instance the column of coefficients of Xk+r. If the Aktr-deformation is not an interlacing deformation we obtain the system Sk+l by performing the Ak+l-deformation of the lefthand sides of the inequalities of the system Sk. If it is an interlacing deformation and s of the k + 1 deformations Ai, i = 1, 2, . . . , k + 1 are interlacing deformations, then to obtain the System Sk+l we proceed as follows:
Finite
systems
of
linear
15
inequalities
(a) We subject the left-hand sides of the inequalities of system Sk to an Ak+l-deformation but do not combine with it the left-hand sides of those inequalities of system Sk, the union of whose indices contains more than s + 1 different elements; (b) From the system so obtained we eliminate one after the other each inequality whose index includes the index of at least one of the inequalities remaining in it. In both cases the system Sk+., is a fundamental (UI + . . . + uk+i)-convolution of system (4). This can be seen without difficulty from propositions 3.2 and 3.5 above. Also as above, with each inequality of the system obtained is associated its index. Extending this process and considering as empty any convolution of an empty convolution, we obtain with a finite number of steps (equal to the rank of system (4)) a fundamental (Rn =U, + . . . +U,). 1. If Ai-deformations are carried out here without limitations in the calculation of the indices of the inequalities obtained, the algorithm considered is transformed into the algorithm of successive convolution of paper [51. Notes.
2. Since, for the performance of each Ai-deformation, in the process considered here for the successive elimination of unknowns from system (4) a non-zero column is taken each time, then in view of Theorem 2.4 the number of steps, leading to the elimination of all the unknowns from system (4) with a non-empty Rn-convolution, coincides with its rank. 3. The algorithm of reduced convolution of system (4) can be compared with the following process of recurrence construction of systems of linear forms from the parameters t 1, . . . , t, which enter into it. Let Ti be a system of linear forms t 1, . . . , t,, and let the system Tk of linear forms with unknowns tl, . . . . t,,, be already constructed. We divide the system Tk arbitrarily
into three subsystems Tk’, Tk2, Tk3
(one of them may be empty) and compare the aggregate Tk2 + Tk3 of such sums of the forms of the system Tk2 with the forms of the system Tk3, in each of which there are not more than s + 1 unknowns tj, if s of the system Ti’
+ Ti3,
i = 1, . . . , k, are non-empty and which differ
from one
another by the unknowns in them (system Ti2 + Ti3 is empty if one of the systems Ti2, Ti3 iS empty). For Tk+l we take the union Tkl U (Tk2+Tk3). It is clear that the number of inequalities
in the arbitrary
16
S.N.
Chernikov
fundamental (Ur + . . . + Ilk)-COnVOlUtiOn of the arbitrary system (4) ‘with fixed m does not exceed the number p,,, of forms in system Tk+r with the greatest number of forms. Thus the number pm gives an evaluation of the possible growth of system (4) with successive elimination of its unknowns (e.g. for m ;r 10, pm = 36). In view of Note 2 this value with fixed m does not depend on the number of unknowns in system (4). 4. Suppose that in the matrix of system (4) we select the first k columns. Let us consider the system a#1 + ‘.* +
4jkZk +
tj <
(j=
0
k columns, e.g.
i, . . . . m).
If 5 qtj
< 0
(i=
1 r**.,S)
1-t
is the system obtained after eliminating from it, by the method put forward here, all the unknownsx1, . . . , Xk, the system i
d:() (ar.
. . . +
k+ta+1+
4jnGI
+
tf)
<
0
(i =
i (. . . , s)
f--i
obviously coincides with the fundamental (u, + . . . + Ilk)-convolution of system (4). This property of the method makes its use very much easier in practical calculations (in the case of system (4) with large matrices). To conclude the present paragraph we shall consider the application of the algorithm of reduced convolution to finding solutions of system (4) for fixed values of the parameters ti. Example.
We are given the system X2 X3 + X4 3 < 0, 52 - x3 - X4 + 1 G 0, -xi + 2x2 + x3 x4 2 < 0, -xi - x2 + 2x3 + X4 - 2 < 0, 3x1 + X2 - 3x3 - 2x4 + 1 < 0, -2xi - X2 + X3 + X4 + 0 6 0. Xl
+
2xi -
Its Ur-convolution
has the form
3X2+0X3+0X4-5 0X2+ X3+2X4-5 X2- x3+3x4-660 3X2+ x3-3x4-3 -3Xx + 3X3 + .g&- 3
-2x2 + 0x3 + 0x4 + 1 G 0 7X~+OX3-5X~---_ < 0 -2x2+3x3+ x4-5 < 0 - Xx-3353X4+2
(2; 6), (3;5), (4;5), (5;6).
Finite
systems
of
By the side of these inequalities
linear
17
inequalities
we have written their indices.
We now form the fundamental (Ur + U3)-convolution 3sz+o54-5 -2Xz+OX4+ 7X2-524-
The fundamental
X2+554-ll
4 0 (1;3), 1<
0 (2;6),
5 <
0 (3;5),
-xz+ox4-
(1;4;6), 3 <
0 (4;5;6).
(Ui j- Us + U4)- and (Vi + US+ U4+ UZ) -convolu-
tions take the form 3x2-5
-7 < 0 (1;2;3;6),
(1;3), 0 (2;6), 0 (4;5;6).
-8
<
0 (1;3;4;5;6).
Since the last of these systems is consistent, the original system is also consistent.
in view of Theorem 2.1
Substituting one of the solutions of the (Ur + U3 + U1)-convolutions, for instance x2 = 1, in the (U, + Us)-convolution, we obtain x4 = 1. Substituting n2 = x4 = 1 in the Ill-convolution we obtain x3 = 0. Finally, substituting x2 = x4 = 1 and x3 = 0 in the original system we find the solution (0, 1, 0, 1) of the latter.
4. The general solution of a system of linear inequalities The algorithm given in the previous paragraph can be used to find a general formula for non-negative solutions of a system of homogeneous linear equations. In fact, let aiiui + ~*~+amium=O
(i=i,...,n)
(6)
be some system of such a kind. It is clear that the set ,Mof its nonnegative solutions can be considered as a cone C(U) with U = Rn for the system of inequalities
fjtx)
+ tj =
UjiXi +
"'
+
UjnXn
+
tj <
0
(j = 1, .*., m)
(7)
with unknowns x1, . . . , z, and Parameters tr, . . . , t,. But then in view of Theorem 1.4 and the definition of a fundamental U-convolution (see Section 1) we have the following theorem.
18
S.N.
Theorem
Chernikov
4. I
If
is a fundamental Rn-convolution of system (7).
then
(z@), . . . , zz,,@)),
k = 1, . . . , I,
is a maximumsystem of essentially different fundamental elements of the cone C(Rn) of non-negative solutions of system (6) and therefore the formula
hi , . . . r&J = f: pk(Up; . . . , u;,‘(q)
(8)
(
k=l
where pl, . . . . p 1 are non-negative parameters, gives all the non-negative solutions of system (6). If the fundamental Rn-convolution of system (7) is empty the zero solution is a unique non-negative solution of system (6). Note Coral lary 4.2 The numbers of the non-zero coordinates of the positive (non-zero, non-negative) solutions of system (61, which has the greatest number of non-zero coordinates, coincide with the numbers of the inequalities of system (7), which enter into the indices of the inequalities of the fundamental Rn-convolution of system (7). For positive values of all the parameters pl, . . . , pl the formula (8) gives positive solutions of system (6) with the greatest number of non-zero coordinates. Notes. 1. The question of finding all the non-negative solutions the arbitrary system of linear equations alful -I- - + amfum = ai
reduces to obtaining equations
(i = 1, . . . . n)
formula (8) for any system of homogeneous linear
alful + *.*+ amwm - uuzm+l = 0
and selecting from it solutions ordinate u,,,+~. 2. If all the coefficients all the solutions
of
(i = 1, . . . , n)
of this system with a positive
of system (6) are rational,
(UP, . . . , u,,,V,
k = 1, . . . , I,
final co-
then obviously
which determine formula (8)
Finite
systems
of
linear
19
inequalities
are also rational. It is not difficult to see that in this case formula (8) with arbitrary, non-negative, rational values of the parameters pk. k = 1, . . . . 2, will give all the non-negative solutions of system (6) with rational coordinates. Multiplying the solutions (ulck), . . . , urn(k)I by suitably selected positive integers we can convert them into positive integral solutions of system (6). Retaining the former notation for the latter, we obtain for?xUla (8) (for rational non-negative pk), adapted to find all the positive integral solutions of system (6) (with rational coefficients). On examining the question of non-negative rational and, in particular, the question of positive integral solutions of an arbitrary system of linear equations with rational coefficients and free terms we must be guided by Note 1. illustrate
To
Theorem 4.1 let us consider the following
example.
Exanrple. To find a general formula for non-negative solutions system ui +
2u2 -
-ui
-
u3 -
u2 +
ui -
u2
Ul -
u4 +
2%
+
u3
u2 -
-
+
3u5 -
u4 +
2U6 =
0,
U6 =
0,
u5 -
2u4 -
3u5 +
U6 =
0,
u4 -
2u5 +
U6 =
0.
u3 +
of the
In accordance with Theorem 4.1 the question reduces to finding a fundamental R*-convolution of the system Xi + 2X1 -
Its
X3 +
X4 +
t1 <
0,
x3
x4
f2
0,
0x2
OX3 +
+
x3
+
-
-xi
+
2X2 $
-51
-
X2 +
-
X4 +
ts <
0,
X4 +
t4 <
0,
-2xi
-Xz+X3+X4+ts
3x3 --4+t5<0,
has the form
OX4 +
ti +
t3 <
0,
2x4
ti
tr
0,
+
+
=G
-2x2 7x2
X2 -
X3 +
3X4 +
%
t6 <
0,
-2x2
3x2 +
X3 -
3x4 +
t2 +
2t3 <
0,
-x2
X4 +
t2 +
2t4 <
0.
-3X2
+
3X3 +
+
G
X9 -
x2
-
+
2X3 +
3x1+
fundamental Ill-convolution 3x2 +
We
X2 x2
+ +
+ -
Ox3 + 0x3
-
OX4 + 5X4 +
3x3 + 3x3 -
X4 + X4 +
t2 +
ts 4
0,
St3 +
t5 4
0,
t5 <
0,
3t6 <
0.
3t4 + 2t5 +
also form the fundamental (I-J, + UjJ-convolution 3x2 + 0x4 + ti + ts < 0, -2X2 7X2 -
+
OX4 + 5x4 +
t2 + 3ts +
t6 < t5 <
0, 0.
X2 + --3X2
+
5X4 +
054 +
%
+
t4 +
3t4 + 3ts +
t6 <
0,
3t6 <
0.
20
S.N.
The fundamental
Chernikov
(Ui + Us + Uk + UZ)-convolu-
(Vi -I- Us i- Uk)- and
tions take the form 3s~
and
+
ti
+
t3
t2 +
<
0,
-232
+
ts <
0,
--3x2 +
3t4 +
3t5 +
3te &
0,
2ti
3t2 +
2t3 +
3ta ss
0,
3f4 +
3t5 +
3tf3 d
ti +
+
t3 +
0.
In accordance with Theorem 4.1 the general formula for non-negative solutions of the system of equations considered takes the form (Ui, U2,
U3, Uk, U5, U6) =
pi (2,
3, 2, 0, 0, 3)
+
pz(l,
0, 1,3,3,3)
*
We now turn to the question of the general solution of a system of linear inequalities on Rn. Let us consider first the case of the system of homogeneous linear inequalities aj,rxlt < 0
+*. +
UjiXi +
(j = 1, . . . , m)
(9)
It is clear that it can be replaced by the equivalent system of equations UjiXi
+
---
+
ajnxn
-Uj
=
(i =
1, . ..) m)
(10)
with non-negative parameters uj. Considering the rank r of system (9) to be different from zero, we apply to system (101 Gauss’s algorithm for the elimination of unknowns. Without loss of generality we can consider that the formulae obtained for the unknowns and the equations for the parameters take the form
C$iUi +
58 = bsiui +
. . . +
*** +
Ur+i
CriUr +
=
bsnun 0
(s= (i =
1, . . . . n), 1,
. . . , n
-
(W r).
(12)
Let (8) be an expression for the arbitrary non-negative solution of system (121, defined by Theorem 4.1. Substituting it in (111, we obtain the formulae
which determine all the solutions of system (9). In these Xr+i, i = 1, ..,, n - r, are arbitrary parameters and pk, k = 1, . . . , 1, are arbitrary
Finite
non-negative
systems of
Zinear
inequalities
21
parameters.
The case of
the arbitrary
system
aim + -- + ajnxn - Uj < 0 reduces to the case considered, tained from the set of solutions
(I=
since the set of its of the system
ajixi + -** + UjnXn - UjXn+l < 0 of homogeneous inequalities
1, . .
with
xn+i =
m)
.)
(43)
solutions
(j = 1, . .
.)
is ob-
m)
1.
To conclude this paragraph we shall consider the question of finding all the edges of tine cone K of solutions of the system (9) with r =n cm. In this case formulae (11) and equations (12) respectively take the forms = beit+ + -.a + bsnun
XS
(s = 1, . . ., n)
(Ii*)
and C1tW
It all
is easily
+
+CniUn+Un+i=O
seen that
essentially
x(k) =
***
if
different
(s&k), . . . , z,th))
determined hausted
by formula
by the rays reduced CjlXi +
(Xl*), {hz@)},
+
Xj +
all
We can easily the vertices
equal
solutions
are the corresponding
of
to finding “’
(u&~), . . . , u,#Q),
fundamental
Theorem 4.1 the question pletely
uch) =
(i'l, .... m-n).
then all h > 0.
tn+j
<
0
+
k = 1, . . . , I, system (12.)
are and
with coordinates
the edges of the cone K are exall
tj <
in accordance
with
the edges of the cone K is
a fundamental
Cj, m-nxm--n
vectors
Therefore
finding
of
(12*)
RmMn-convolution 0
(j=1,2,
com-
of the system
(j= 1, 2, .... n), .... m-n).
reduce to the question just considered that of finding of a polyhedral set, defined by system (13) with rank
to the number of its
unknowns.
5. A numerical algorithmic scheme for reduced convolution The present section deals basically with the numerical scheme for the reduced convolution algorithm given in Section 3. If we denote by
22
S.N.
Chernikov
the matrix, in which to the left of the vertical line we put the matrix A0 of system (4) and to the right a unit matrix E” of order m, the scheme which interests us may be carried out by the following procedure. 1. We take some non-zero column of the matrix A”, for instance its first column Alo. Carrying out the A, O-deformation (see Section 3) of the system of rows of the matrix Ho, we obtain a system of rows of the matrix H’, situated beneath the matrix Ho on the extension of its columns, in which the part to the left of the vertical line is the matrix of the fundamental Ill-convolution of system (4). We shall denote the second part of the matrix H’ by i?l. If all the elements of the column Alo are positive or negative, we consider that the matrix H1 is empty; in this case the Ill-convolution of system (4) is empty and therefore the process ends. If the matrix HI is non-empty (all the elements of its first column will be zeros) the process is continued. 2. Suppose that the matrix Hk is already obtained and arranged under the matrix Hk-’ by the extension of its columns, the left-hand part Ak of which is the matrix of the fundamental (ul + . . . + Ilk)-convolution of system (4); let us denote the right-hand part of the matrix Hk by Ek. To each row of the matrix Hk, k = 1, 2, . . . , we shall assign an index which is the set of the numbers of the columns of the matrix Ek, which contain the positive elements of this row. We take a non-zero column of the matrix Ak,
e.g.
its (k + I)-th
matrix Ak are zero).
column Ai+1 (the first
k columns of the
If the A& -deformation of the system of rows of
the matrix Hk is not an interlacing
deformation, we obtain a system of
rows of the matrix Hk+' (arranged beneath the matrix Hk on the extension of its columns), by carrying out this deformation. The process terminates with an empty matrix Hk+' . If the At+, -deformation considered is an interlacing deformation and s of the k + 1 deformation &i-’ , i = 1t 2 7 ---, k+ 1, are interlacing deformations, then in the matrix
Hk+' there are rows of the matrix
column Ll;+i
Hk with zero elements in the
and row combinations, obtained by carrying out the given
*kk+i -deformation,
which satisfy
the following
requirements;
(1) Their indices do not exceed the number s + 1;
Finite
systems
of
linear
inequalities
23
(21 The index for none of them includes the index of any other. The first
requirement is satisfied
the Af+1 -defo~ation
automatically
if in carrying out
we do not combine those pairs of rows of the
matrix HI. the union of whose indices contains more than s + 1 elements. The second requirement is obviously satisfied if we also do not combine pairs of rows of the matrix Hk the union of whose indices contain the index of some third row of the matrix Hk. With such a limitation rows which are unnecessary for the composition of the fundamental (U1 + _. , + Uk+l)-convolution of system (4) will obviously not enter into the matrix Hkii. On the other hand it can easily be seen that with this limitation none of the rows which are necessary for the formation of this convolution
occur.
In fact in carrying out the
the third row considered is either simply transferred (if
it has a zero element in the column
A:+,)
Ai+i -deformation to the matrix Hktl
or must be combined with
that of the two rows of the pair taken, whose (k t l)-th
element (its
element found in the column L$+~) is opposite in sign to its
(k + l)-th
element. It is clear that the index obtained with this row combination will be included in the index of the row combination for the pair originally taken, and therefore in accordance with requirement (2) the latter row combination will not appear in the matrix Hktl. It is also clear that it must not be included in the matrix Hktl end in the first case where the third row is considered it is transferred without alteration to the matrix Hktr. The numerical scheme for the convolution of system (4) for fixed values of the parameters tj. e.g. for tj = aj, j = 1, 2, . . . , m, is essentially reduced to the procedure which has been considered. In this case we must introduce into the matrix Ho to the left of the vertical line the (n + l)-th column of the elements al, . . . , a,. If the matrix Ho, so broadened, is subjected to the transformation under consideration the (n + l)-th column of the matrices obtained will be a colt of free terms corresponding to the f~d~ental convolutions of system (4) with tj = oj. As above, the matrices of the coefficients of these convolutions consist of the first n-columns of the matrices obtained, Since the numerical values of the positive elements of the matrices Ek do not interest us (we are interested only in the indices of the rows of the matrices Hkl, when the successive transformations are carried out in the given process they can be replaced by unity. The given numerical scheme enables us to obtain fundamental U-convolutions of system (4) in the case where 17= U1 + . . . + Ilk, k = 1, 2, . . . .
24
S.N.
Chernikov
It can easily be seen that obtaining a fundamental U-convolution of system (4) for arbitrary U c Rn reduces to this case. We can show that the more general question of obtaining a fundamental &convolution of system (2) in an arbitrary real linear space f, with arbitrary 1.1c L reduces to it also. In fact, let U be a finite dimensional subspact: of L of and rrl, . . . . ur one of its bases. Then the fundamental U-convolution system (2) is obviously obtained if the given numerical procedure is applied to the system
JAfj(u’) +
“’
+
Srfj(U')
+
tj
<
0
(j= 1, 2, ...)m)
and in the complete
fundamental convolution obtained tj is replaced by If the subspace U is of infinite dimensions (naturally this is impossible if 1 = R9 then denoting by R the kernel of system (2). we obtain the direct expansion tj
+
_fjtx)s
x
E
L.
u=
(UflR)
--j-IT.
Here II’ is a finite-dimensional subspace of L. Since the fundamental Uconvolution of system (2) obviously coincides with the fundamental II'convolution of the latter, its calculation reduces to the case under consideration. by H.F. Cleaves
Translated
REFERENCES 1.
FOURIER, J. Oeuvres II. Gauthier-Villars, Paris, 1890.
2.
DINES, L.L. Systems of linear inequalities. Ann. 191 - 199, 1918 - 1919.
3.
KUHN, H. W. Solvability and consistency for linear equations equalities. Am. math. Mon., 63, 4, 217 - 232, 1956.
4.
CHERNIKOV, S.N. The convolution of finite systems of linear inequalities. Dokl. Akad. Nauk SSSR, 152. 5, 1075 - 1078, 1963.
5.
CHERNIKOV, S.N. The convolution of systems of linear inequalities. Dokl. Akad. Nauk SSSR, 131, 3, 518 - 521, 1960.
8.
MINKOWSKI,H. Geometrie der
Zahlen,
Leipzig.
Math.
Teubner.
(Ser.
$896.
2),
20,
and in-
CHERNIKOV
Sverdlovsk (Received
4
May
1964)
‘THE present paper deals with the problem of the elimination of unknowns from systems of linear inequalities. Fourier [II has already considered the elimination of unknowns as a natural method of solving such systems. Up to recent times, however. it was used as a basis only for theoretical purposes (see for instance Dines [21 and Kuhn [31); in particular, in [31 it was employed to establish the conditions of consistency of systems of linear inequalities. With this aim an algorithm is suggested there for rule for the combinaeliminating unknowns, which reduces to a definite tion of the inequalities of the system considered, using non-negative numbers as multipliers. In using it, however, for numerical purposes (as also in using other similar algorithms for these purposes) the very large number of combinations needed to carry it out is a great drawback. The content of the present paper is determined to a large extent by searches for the best method of combining the inequalities of an arbitrary linear system, in which the number of constituent combinations is a minimum with a view to eliminating the unknowns. These searches lead to the problem, which is more general than that of the combination of inequalities, of combining an arbitrary system of linear inequalities, such that the new system, composed of the inequality-combinations obtained, has in some subspace of the space of the combined system a set of solutions which coincide with the projection of a set of solutions of the latter into it. This question is considered in the present paDer for systems of linear inequalities in an arbitrary real linear space I!,. If fj(5) are real
linear
(i.e.
additive
(i =
and homogeneous)
-___ *
Zh.
vychisl.
Mat.
1, . * -9 m)
(1) functions,
--mat.
Fiz.
5.
1, 1
3 -
20,
1965.
given on L,
2
S.N.
Chernikov
then a system of the form +G
(1 =
in which tj are real parameters a system of linear inequalities
or simply on L.
fj(z)
1, .... m), real
numbers,
(2) is here called
If U is a subspace of L, V one of its direct complements in L, C(u) the cone of non-negative solutions (ui, . . . , u,,,) of the equation a&(x)
+ . . . + hAa
=
0
on U, n E U and c(i) =
(c&O, . . ., c#))
(i =
1, . . ., k)
is some system of generating elements of the cone C(U), then for any set of values of the parameters tj, j = 1, . . . , m, the set of solutions of the system m
2
tn Cjci'fj(X)
j=i
+2
Cj(%fj < 0
(i=
1, . ..) k)
j=i
in the subspace V, coincides with the projection into it of the set of solutions of system (2) (see Theorem 2.3). We shall call this system a u-convolution of the system (2). In the case of a zero cone C(U) let us agree to consider that the U-convolution of system (2) is empty. In this paper some properties of U-convolutions of system (2) are established and, in particular, the fact that any U’-convolution of an arbitrary Uconvolution coincides with some (17+ U’)-convolution (see Sections 1 3). In conjunction with the defined rule for reducing an arbitrary (U + U’)-convolution to the (II+ U’)-convolution with the least number of inequalities contained in it (see Section 3) this property is used as a basis for the algorithm for a successive reduced convolution of systems of linear inequalities of the form (2) on the space L = An. In the case where tj are free parameters, the algorithm reduces to the best variant of the algorithm for the successive elimination of the unknowns from such systems. The algorithm is used (see Section 4) to find a general formula for non-negative solutions of an arbitrary system of homogeneous linear equations and also to obtain a general formula for the solution of an arbitrary simultaneous system of linear inequalities of the form ajiXi
where all
+
.. . +
ajnXn
Uji and Uj are real
-
aj
4
numbers.
0
(j= 1, .... m),
Finite
systems
of
linear
3
inequalities
The basic results of the paper were published without proof in [41.
1. Fundamental convolutions of a system of linear inequalities We shall call any positive linear combination of one of its arbitrary (non-empty) subsystems, which is identically equal to zero on U, a Uinterlacing of the system of functions (1). The linear combination is here said to be positive if all its coefficients are non-negative and at least one of them is positive. A &interlacing Uj,Ofj,(S) + . . . +
4.+,fj,,,t5)
with positive
coefficients
uj,O, k=l,
...
, s+l,
we
shall call a fundamenta1 U-interlacing, if s of the s + 1 functions entering into it are linearly independent of U; it is clear that with this condition any s of the s + 1 functions fjk(x) are linearly independent of CT.We shall call the vector (ul, . . . , u,,,) with Ujk = Ujko; k = 1, . . . . s + 1, and the remaining Uj zero, a fundamental element of the cone C(U). It is not difficult to see that if the cone C(U) &ntains non-zero elements then it necessarily contains fundamental elements also. This is inferred from the following well-known proposition. Theorem
1.1
If a vector is expressed linearly with non-negative coefficients by means of some system of vectors whose rank is diiferent from zero, then it can be expressed with non-negative coefficients through at ieast one of its linearly independent subsystems. We can easily convince ourselves also that the cone C(u) can contain not more than a finite number of essentially different fundamental elements. Two elements of the cone C(r/, are not considered to be essentially different if they differ from one another only by a positive numerical factor. Two fundamental elements are essentially different only when they differ in the numbers of their non-zero coordinates. Let C be some maximumsystem of essentially elements
cti) =
(uiti), . . . , Us),
i = 1, . . . , h,
different,
fundamental
of the cone C(m. We
shall then call the system i j=i
Uj"'fj(5)
+
i
j=i
Ujti)tj <
0
(i = 1, . . . , h)
(3)
4
S.N. Chernikov
a fundamental U-convolution of system (2). If the cone C(U) is zero we shall consider that the fundamental U-convolution of system (2) is empty. For fixed U the fundamental U-convolution of system (2) is obviously unique except for the positive numerical factors which occur in its inequalities; therefore no distinction will be drawn below between fundamental U-convolutions of system (2) which correspond to one and the same U.
It can easily be seen that the definition essentially tion given here does not differ finition of [51. Theorem
1.2
The system (2) with a L has a solution in only those for which its System (2) with sn empty for any values occurring
U t
In the proof call
of a fundamental U-convolufrom the corresponding de-
non-empty fundamental U-convolution for some L for those values of the parameters tj and fundamental U-convolution has a solution in L. fundamental U-convolution has a solution in U in its parameters.
of the theorem the following is used.
proposition
(which we shall
Minkowskii' s theorem)
Theorem
1.3
If
all the solutions of the system of linear inequalities fj(X) GO, . . . . m, on L satisfy the linear inequality f(x) GO on L (in this case it is said that it is its consequence) non-negative numbers pl,. . . , pm exist, for which the identical relation = 1,
j
f(z) = i Pjfj tz) -
j=i
is valid
for
x E L.
is For systems of linear inequalities on L = Rn this proposition an arbitrary space proved by Minkowskii (see [Sl); it is transferred to L by fairly obvious considerations (the principle is given in L-51). Let V be some direct
Proof.
shall
show that
system tj
=
tj"
(3)
if
z” =
with values has a solution
of
complement of the subspace
zz” + u” (u” E u, u” E v) tj =
tj’, j =
1, . . ., m,
u’ + V0 (zz’ E U).
U in L.
is a solution
of
the system (2)
Suppose that this
We with
is not so.
systems
Finite
Then system (2)
with
then the inequality
tj = t GO
of
tj’ = will
linear
considered therefore
has no solution
tj” + fj (uO)
be a consequence
fj(X) + tj’t < 09
t=
of
in U. But
the system
j = 1, . . ., m,
on the space U + R’ of pairs in view of Minkowskii’s
5
inequalities
[z,t]
(zEU,
tEai),
and
theorem some relation
i
Pj (lj (z) +
tj’t)
j=l
which is identical with respect Pj>O, j = 1, . . . . m.
to x E li and t E
R’ is valid
for
Since system (2) for tj = tj ‘, j = 1, . . . , m, has no solutions in U, the rank of the system of functions fj(X) + tj ‘t on U + R’ is greater by one than the rank 1 of the system of functions fj(x). Using Theorem 1.1 we obtain from the preceding relation a new relation, identical with respect to n E II and t E R1, viz.
t =
2
4j,
(fj, (z) + tjA’t)
(1’d 1)
k=i
with positive But then
qjk and linearly
independent
functions
fjk(x)
+ tjk’t.
I‘+1
C !ljkfjk tx) k=l
is a fundamental U-interlacing . ..* m. Since the inequality
obviously corresponds to (2) with tj = tj’ has no is obvious that it has a proves the first part of
of the system of functions
fj(x),
j
= 1,
this, the fundamental U-convolution of system since it solutions in I;. But this is impossible obtained solution u” E U. The contradiction the theorem.
From these considerations it follows also that if system (2) has no solutions in U for some values of the parameters tj, at least one fundamental U-interlacing of the system of functions (I) exists. If the cone
S.N.
6
Chernikov
C(V) is zero it follows any values theorem.
of
from this that system (2) has solutions in U for of the parameters tj, which proves the second part of the
In view of the assumption, formulated the theorem, we have the inequality
is an arbitrary
of the proof
i=l,.*.,?72.
+tj'
fj(U'+UO)
at the beginning
U-interlacing
of
the system of
functions
j=i
(l),
from this
we obtain
j=i
j=i and also
2
p{fj
(2’) +
j=t
2
pjtj'
<
0
(2” = 24’+ V”).
j=i
In view of the arbitrariness of the choice this means that the inequality m
m
j=i
j=i
of x0 and tj,
j
=
1, . . . , m,
is a consequence of system (31, considered on L i Rm (R” is the space of vectors (tl, . . . . t,,,) 1. Using Minkowskii’ s theorem here we find that the m linear forTI 2 is a positive linear combination of those linear j=l forms of the parameters tj which occur in the inequalities of system (3). pj’tj
Hence it follows that any maximum system of essentially different fundamental elements of the cone C(U) is a system of its generating elements. Since it is obviously irreducible, in addition to this, it is found to be a basis for C(u). On the other hand it is easy to see that each basis for CC.!4 cannot contain elements which differ essentially from the fundamental elements of C(I/). Consequently we have
Finite
systems
of
linear
inequalities
7
Theorem 1.4 Maximum systems of essentially different non-zero cone C(U), and only those, are its
fundamental bases.
elements
of
a
Calling the sum of the numbers of its non-zero coordinates the index of an arbitrary element of the cone CC0 we note an obvious consequence of this assertion: Theorem
2.5
The system of generating elements of a non-zero cone C(U) is its basis when and only when it does not contain any element with an index which includes the index of at least one of its other elements.
2. Arbitrary convolutions of a system of linear inequalities Comparing Theorem 1.4 with the definition of an arbitrary U-convolution of system (2) (see the introduction) we obtain the following state-
ment: each U-convolution of system (2) contains mental U-convolution of the system (2).
as a subsystem a funda-
Calling the index of the element Cci) corresponding to it the index of the arbitrary inequality of the U-convolution of system (2) (see the definition of a U-convolution), we see, by virtue of Theorem 1.5, that a U-convolution of system (2) does not cease to be its U-convolution if we eliminate from it any inequality with an index, which includes the index of at least one of the remaining inequalities; by successive elimination of such inequalities from it, it is converted obviously into a fundsmenta U-convolution. If the cone C(u) is non-zero, then from the statement noted here it follows that all U-convolutions of system (2) for fixed U have one and the ssme set of solutions in L for one and the same choice of values entering into its parameters. Therefore from Theorem 1.2 there follows: Theorem
2.1
If the cone C(II) is non-zero the system (2) and its arbitrary U-convolution have solutions in L with the same values entering into their
8
S. N.
Cherni
kov
parameters tje If the cone C(u) is zero the system (2) II for any values of the parameters entering into it. Calling Coral lary
each U-convolution
of system
(2)
complete
if
has a solution
U = L,
in
we obtain
2.2
The system fj(X)
(i =
--aj
1, . . ., m),
where al, . . . . a, are real numbers, is consistent when at least one of its complete convolutions is consistent or empty. If the system considered is consistent then each of its complete convolutions is consistent or empty. Suppose, as before, that II is a subspace of L. We shall denote by V one of its direct complements in L. We shall call a finite system S of linear inequalities on L a UV-combination of the system (2) if it satisfies the following conditions: 1. Each inequality of the system S is a positive linear combination of inequalities of system (2) such that the combination of functions (1) entering into it is identically equal to zero on Lr; For each set of values of the parameters tj the system S in V coincides with the projection solutions of system (2).
2.
Theorem
the set of solutions of into V of the set of
2.3
If the cone C(II) is non-zero a UV-combination of system (2) space 0 in L, and, conversely, is a UV-combination of system V = V” of the subspace U in L, system (2).
then each U-convolution of system (2) is for any direct complement V of the subif a system of linear inequalities on L (2) for at least one direct complement it coincides with some U-convolution of
Proof. Let V be an arbitrary direct L and x = u” be a solution, contained
complement of the subspace U in in V, of some U-convolution S of system (21, taken for some values tj = tj”, j = 1, . . . , m, for which it is consistent. Then for values tj = tj’ = tj” + fj(VO) the system S will have a solution in II. If system (2) and system S are considered as system of linear inequalities on the space II, then by virtue of the first part of Theorem 2.1 it follows that the system
Finite
systems
of
linear
9
inequalities
(f = I, . . ., m)
fj(X) +tj’
has at least one solution u” in u. But then the system (2) for = 1, . . . , m. has a solution u” + vO. Since system (2) is not i with those values of the parameters tj, for which the system s consistent. and since each non-empty convolution of system (2) the first condition of the definition of a /TV-combination, the assertion of the theorem follows from this. Now let
T be some W-combination
an arbitrary
i 4jfj(x) j=1 If
x0=
T for
sistent, is also
U-interlacing
u” + u0 (~0 E U,
arbitrary
values
of
u” E V)
system (2)
is insatisfies first
and
of the system of functions is an arbitrary
tj”, j = 1, . . ., m,
tj =
with V = p
tj = tj”, consistent
solution
(1).
of system
with which it
is con-
then in view of the definition of W-combinations its solution u”. But then in virtue of the same definition system (2) has a
solution
for values
ZJ’+ u0 (u’ E U) fj(Z-3’
+
U”) +
tj" <
0
of
tj
=
tj”
But then
(f = I, . . ., m).
Hence we obtain m
m
2 Qjfj (5’) + 2 j=l
Qjtj’
<
0.
j=i
Since x0 is an arbitrary solution of system T with arbitrarily chosen values of the parameters tj, with which it is consistent, this mesns that the inequality
$
4jfj
j==--i
(Xl+
$QJtj G 0
j-i
is a consequence of the system T, considered as a system of linear inequalities on the space L i Rm (P is the space of vectors (tl, . . ., t,)). Hence, as with the proof of Theorem 1.4, we see that the vectors of the coefficients of the linear combinations, which determine system T from system (21, generate the whole cone CC0 of the system of functions (1). Consequently system T is a U-convolution of system (2). Note.
If
v0EV(U/V
= L)
is a solution
of the U-convolution
of
10
S. N. Chernikov
system (2) for of the system
tj = tjo#
i = 1, . ..,
IIIon
fj(u) + tj’ < 0 for
tj'
= tjo
+fj(Uo)o
finding some solution
(j = i, . . . , m)
we obtain some solution
(2) for
tj = tj"
follows
from Theorem 2.3.
u. E U
no = u. + uo of system
Cj = 1, . . . , ltl). The existence of the element ua here
Calling the maximumsubspace R of L, on which all the functions (1) take only zero values, the kernel of system (21, we note in concluding this paragraph the proposition which now follows. Theorem
2.4
If a U-convolution of system (2) with respect to U @ H is non-empty, z*- -r-I. I~J LLulBis beioa the rank of system (2; by not less than the dimension_ ality
of the direct
complement R’ of the intersection
The rank of system (1) of functions fj(Z) to be the rank of system (2).
R fl LJ
in U.
which enter into it is said
Proof. From th e de f inition of a U-convolution of system (2) we can easily see that its kernel R’ contains the subspace R and II. Since the rank of the U-convolution considered obviously coincides with the dimensionality of the direct complement Q’ of its kernel R’ in L, and the rank of system (2) with the dimensionality of the direct complement in L of its kernel R, on using the direct expansion
L4 =R’.+Q’=
(R+R’-+R)
iQ’=R+
(R’iR+Q’),
where i is some direct complement of the subspace R + U in R’, we obtain the fact that the rank which interests us is lower than the rank of system (2) by no less than the dimensionality of the subspace R” 4 E. But this means that it is lower than the rank of system (2) by no less than the dimensionality of space R, which was to be proved.
3. Repeated convolutions Let U and II’ be two subspaces of (2) is not empty then we shall call sidered on L) a repeated convolution repeated (R U’)-convolution of the
If the U-convolution S of system the U’-convolution of system S (conof system (2) or more accurately a latter; if system S is empty or its
L.
Finite
systems of
linear
11
inequalities
U’-convolution is empty, we shall consider that the repeated (II; U’lconvolution of system (2) is empty. Similarly we define a repeated CR U’; U’$-convolution of system (2) etc. Theorem
3.1
Any repeated (U, U’)-convolution (U + U’)-convolution of the latter.
of
system (2)
coincides
with some
Proof. Let S be some U-convolution of system (2) and S’ some U’-convolution of system S. We shall consider initially the case where both of them are not empty. Let U’ be some direct complement of the intersection U n II’ in II’, and V some direct complement of the subspace I/” = U + U’ = U -k UC in L. It is easily seen that the system S’ is a Y-convolution of system S. In view of the first part of Theorem 2.3 the set of solutions of system S in the subspace M(U, t), t = (tr, . . . . t,), U” -i- V, for any choice of values for the parameters tj, in system (2) is the projection of the set M(t) of solutions of the latter into this subspace. If the system S is considered as a system on the subspace II* i V, in view of Theorem 2.3 the set of solutions of system S’ subspace V coincides with the projections of the set M(U, t) into this subspace and so with the projection into it of the set M(t). Since the system S’ obviously satisfies the first condition of the definition of aUV-combination of system (2) with U = U", it follows from this that it is a II” V-combination of the latter. But then in virtue of the second part of Theorem 2.3 the system S’ must be a U’Lcombination of system (2). Thus in the case considered the theorem is proved to be true. Now let at least one of the two systems S and S’ be empty. If the system S is empty then it can be established without difficulty that system (2) has an empty 07 + U’l-convolution. If system S’ is empty and the system S not empty, then in view of the second part of Theorem 2.1 tha system S has a solution in II’ and so also in U + U’ for any values of the parameters tj entering into it. But then, in view of the first part of the same theorem, system (21 has solutions in U + U’ for all values of the parameters tj entering into it, (here the systems (2) and S are considered as systems on U + U’J. Hence it follows that the system 0, j = 1, . . . , m, has a solution in U + II’. It is clear that with this condition the system of functions (1) cannot have (U + II’)interlacings. Consequently in the case considered the (U + U’)-convolution of system (2) is empty. This proves the theorem. fj(Xl
<
In view of the statement formulated from Theorem 3.1 there follows
at the beginning
of Section
2,
12
S.N.
Chernikov
Corollary 3.2
From the repeated (.!k IJ’)-convolution of system (2) we can choose any inequality which contains a non-fundamental (U + U’)-interlacing of functions (1) (the index of such an inequality includes the index of at least one of the remaining inequalities) and which does not violate its property of being a (U + U’)-convolution of system (2). This possibility is used in the algorithm given below for reduced convolution, which gives successively fundamental U-convolutions of the system fj(X)
+tj=lZjiXi+...+ajnXn+tj
(i
=
1,. . ., m)
(4)
with respect to the subspaces U=
ui, u = uj + uz, . . u = ui + . . . -I-Uk .(
(k G n),
where VI, . . . , U, are subspaces generated in Rn, respectively, by the coordinate vectors el = (1.0, ..., 0). e2 = (0. 1,0, . . . , W, . . . , e, = (0,
. . . ) 0.1).
For the basis of the algorithm the following tions are needed.
supplementary proposi-
Lemma 3.3
If in some (Ui, + . . . + U;)-convolution
of system (4) the coeffi-
cients for some unknownxl, with 2 f iI, . . . , ik, are non-negative (nonpositive), a linear combination of k columns of the matrix of system (4) with numbers i = il, . . . , ik exists, which in the sum with the l-th column of this matrix gives a vector column with non-negative (nonpositive) coordinates; t33nVerSSly if for some numbers i,, . . . . ik and . . . . ik, such a combination exists then in each (UiI + . . . + l/i)1 #iI, convolution of system (4) the coefficients of xl are non-negative (nonpositive). In fact, with the condition of the first solutions of the system aiiut + . . . + adh
=
-Uj
the inequality
0
part of this proposition
(i = it, . . ., ik), (I=
l,...,m)
all
Finite
systems
of
linear
and so the first part of it is obtained by direct Minkowskii’s theorem. Its second part is obvious. From the proposition
13
inequalities
application
here of
just proved there follows
Coral lary 3.4 If a column of coefficients
of xl in a (Ui, + . . . + Uik)-COnVOlUtiOn
of system (4) (I = i,, . .., ik) contains elements with opposite signs, each linear combination of k-columns of the matrix of system (4), which have the numbers i,, . . . , ik, give in the sum with the l-th column of this matrix a vector-column with coordinates of opposite sign. We now prove a basic lemma. Lemma 3.5 Let (II, + . . . + Uk)-convolution of system (4) be non-empty. Let A1 be the column of coefficients of x1 in system (4) and AZ, I = 2, . . . , k, be a column of coefficients of XI in some (111 + . . . + UI_r)-convolution of system (4). If s of the k columns A I, . . . , Ak (s
Let us consider the system aiiui + . . . + amium =
0
(i = 1, . . k), .)
(5)
defining the cone C. If s = k, then in view of Corollary 3.4 its rank is the same as the number s and therefore the proposition to be proved is obtained without difficulty by means of Theorem 1.1. We shall therefore assume that s < k. In this case at least one of the columns Al, . . . , Ak contains no elements of opposite signs. Let k,, . . . , k 2 be the numbers (in increasing order) of all the columns of such a kind. By means of Lemma3.3 we can easily see that without violating the set of solutions of system (5) we can replace successively the k I-th, kl_,-th, . . . , kl-th equations in it by equations with coefficients without change of sign; in connection with Lemma3.3 this is arrived at by adding to each of the substitute equations some linear combination of the equations preceding it.
14
S.N. Chernikov
Since, in accordance with the assumption, the (U, + . . . + Ilk)-convolution of system (4) is not empty, the system (5) has non-zero, nonnegative solutions and therefore in each of its so transformed equations zero coefficients must be met with “column” numbers which are common for all the equations considered. It is clear that the arbitrary, non-negative solution (Us, . . . . u,,,) of system (5) can have non-zero coordinates can have non-zero coordinates only at places corresponding to such numbers. Let us replace by zeros all coefficients of system (5) with other column numbers; we shall denote the system thus obtained by P. In the system P the equations with numbers k,, . . . . kl) contain no terms different from zero and therefore its rank does not exceed the number s of its other equations, Using Theorem 1.1 here we easily obtain the statement to be proved. Turning now to tion we give the numbers al, . . ., pair of elements - aqzp. and 0PZ9
the description of the algorithm of reduced convolufollowing definition. Let A be some system of m real a,,, and .Z’ a system of m unknowns ~1, . . , , z,. To each ap > 0 and ag < 0 from A we relate the linear form to each zero element a, from A the form ts; we shall
say that the aggregate of all such forms arrived at is obtained as a result of an A-deformation of system Z. If numbers of opposite sign are met in the system A we shall call the A-deformation an interlacing deformat ion.
THE ALGORITHM OF REDUCED CONVOLUTION 1. Let A, be some non-zero column of coefficients of system (4). for instance a column of coefficients of the unknown xl. Performing the Aldeformation of the left-hand sides of the inequalities of system (4) we obtain a system &, which is its tir-convolution (obviously fundamental). With each inequality of system S1 we shall associate its index. It coincides obviously with the set of numbers entering into its parameters or, what is the same, with the set of numbers of those inequalities by whose combination it was obtained. 2. Let the system Sk, which is a fundamental (II, + . . . + Ilk)-convolution of system (41, be already obtained, and let Aktl be some non-zero column of its coefficients, for instance the column of coefficients of Xk+r. If the Aktr-deformation is not an interlacing deformation we obtain the system Sk+l by performing the Ak+l-deformation of the lefthand sides of the inequalities of the system Sk. If it is an interlacing deformation and s of the k + 1 deformations Ai, i = 1, 2, . . . , k + 1 are interlacing deformations, then to obtain the System Sk+l we proceed as follows:
Finite
systems
of
linear
15
inequalities
(a) We subject the left-hand sides of the inequalities of system Sk to an Ak+l-deformation but do not combine with it the left-hand sides of those inequalities of system Sk, the union of whose indices contains more than s + 1 different elements; (b) From the system so obtained we eliminate one after the other each inequality whose index includes the index of at least one of the inequalities remaining in it. In both cases the system Sk+., is a fundamental (UI + . . . + uk+i)-convolution of system (4). This can be seen without difficulty from propositions 3.2 and 3.5 above. Also as above, with each inequality of the system obtained is associated its index. Extending this process and considering as empty any convolution of an empty convolution, we obtain with a finite number of steps (equal to the rank of system (4)) a fundamental (Rn =U, + . . . +U,). 1. If Ai-deformations are carried out here without limitations in the calculation of the indices of the inequalities obtained, the algorithm considered is transformed into the algorithm of successive convolution of paper [51. Notes.
2. Since, for the performance of each Ai-deformation, in the process considered here for the successive elimination of unknowns from system (4) a non-zero column is taken each time, then in view of Theorem 2.4 the number of steps, leading to the elimination of all the unknowns from system (4) with a non-empty Rn-convolution, coincides with its rank. 3. The algorithm of reduced convolution of system (4) can be compared with the following process of recurrence construction of systems of linear forms from the parameters t 1, . . . , t, which enter into it. Let Ti be a system of linear forms t 1, . . . , t,, and let the system Tk of linear forms with unknowns tl, . . . . t,,, be already constructed. We divide the system Tk arbitrarily
into three subsystems Tk’, Tk2, Tk3
(one of them may be empty) and compare the aggregate Tk2 + Tk3 of such sums of the forms of the system Tk2 with the forms of the system Tk3, in each of which there are not more than s + 1 unknowns tj, if s of the system Ti’
+ Ti3,
i = 1, . . . , k, are non-empty and which differ
from one
another by the unknowns in them (system Ti2 + Ti3 is empty if one of the systems Ti2, Ti3 iS empty). For Tk+l we take the union Tkl U (Tk2+Tk3). It is clear that the number of inequalities
in the arbitrary
16
S.N.
Chernikov
fundamental (Ur + . . . + Ilk)-COnVOlUtiOn of the arbitrary system (4) ‘with fixed m does not exceed the number p,,, of forms in system Tk+r with the greatest number of forms. Thus the number pm gives an evaluation of the possible growth of system (4) with successive elimination of its unknowns (e.g. for m ;r 10, pm = 36). In view of Note 2 this value with fixed m does not depend on the number of unknowns in system (4). 4. Suppose that in the matrix of system (4) we select the first k columns. Let us consider the system a#1 + ‘.* +
4jkZk +
tj <
(j=
0
k columns, e.g.
i, . . . . m).
If 5 qtj
< 0
(i=
1 r**.,S)
1-t
is the system obtained after eliminating from it, by the method put forward here, all the unknownsx1, . . . , Xk, the system i
d:() (ar.
. . . +
k+ta+1+
4jnGI
+
tf)
<
0
(i =
i (. . . , s)
f--i
obviously coincides with the fundamental (u, + . . . + Ilk)-convolution of system (4). This property of the method makes its use very much easier in practical calculations (in the case of system (4) with large matrices). To conclude the present paragraph we shall consider the application of the algorithm of reduced convolution to finding solutions of system (4) for fixed values of the parameters ti. Example.
We are given the system X2 X3 + X4 3 < 0, 52 - x3 - X4 + 1 G 0, -xi + 2x2 + x3 x4 2 < 0, -xi - x2 + 2x3 + X4 - 2 < 0, 3x1 + X2 - 3x3 - 2x4 + 1 < 0, -2xi - X2 + X3 + X4 + 0 6 0. Xl
+
2xi -
Its Ur-convolution
has the form
3X2+0X3+0X4-5 0X2+ X3+2X4-5 X2- x3+3x4-660 3X2+ x3-3x4-3 -3Xx + 3X3 + .g&- 3
-2x2 + 0x3 + 0x4 + 1 G 0 7X~+OX3-5X~---_ < 0 -2x2+3x3+ x4-5 < 0 - Xx-3353X4+2
(2; 6), (3;5), (4;5), (5;6).
Finite
systems
of
By the side of these inequalities
linear
17
inequalities
we have written their indices.
We now form the fundamental (Ur + U3)-convolution 3sz+o54-5 -2Xz+OX4+ 7X2-524-
The fundamental
X2+554-ll
4 0 (1;3), 1<
0 (2;6),
5 <
0 (3;5),
-xz+ox4-
(1;4;6), 3 <
0 (4;5;6).
(Ui j- Us + U4)- and (Vi + US+ U4+ UZ) -convolu-
tions take the form 3x2-5
-7 < 0 (1;2;3;6),
(1;3), 0 (2;6), 0 (4;5;6).
-8
<
0 (1;3;4;5;6).
Since the last of these systems is consistent, the original system is also consistent.
in view of Theorem 2.1
Substituting one of the solutions of the (Ur + U3 + U1)-convolutions, for instance x2 = 1, in the (U, + Us)-convolution, we obtain x4 = 1. Substituting n2 = x4 = 1 in the Ill-convolution we obtain x3 = 0. Finally, substituting x2 = x4 = 1 and x3 = 0 in the original system we find the solution (0, 1, 0, 1) of the latter.
4. The general solution of a system of linear inequalities The algorithm given in the previous paragraph can be used to find a general formula for non-negative solutions of a system of homogeneous linear equations. In fact, let aiiui + ~*~+amium=O
(i=i,...,n)
(6)
be some system of such a kind. It is clear that the set ,Mof its nonnegative solutions can be considered as a cone C(U) with U = Rn for the system of inequalities
fjtx)
+ tj =
UjiXi +
"'
+
UjnXn
+
tj <
0
(j = 1, .*., m)
(7)
with unknowns x1, . . . , z, and Parameters tr, . . . , t,. But then in view of Theorem 1.4 and the definition of a fundamental U-convolution (see Section 1) we have the following theorem.
18
S.N.
Theorem
Chernikov
4. I
If
is a fundamental Rn-convolution of system (7).
then
(z@), . . . , zz,,@)),
k = 1, . . . , I,
is a maximumsystem of essentially different fundamental elements of the cone C(Rn) of non-negative solutions of system (6) and therefore the formula
hi , . . . r&J = f: pk(Up; . . . , u;,‘(q)
(8)
(
k=l
where pl, . . . . p 1 are non-negative parameters, gives all the non-negative solutions of system (6). If the fundamental Rn-convolution of system (7) is empty the zero solution is a unique non-negative solution of system (6). Note Coral lary 4.2 The numbers of the non-zero coordinates of the positive (non-zero, non-negative) solutions of system (61, which has the greatest number of non-zero coordinates, coincide with the numbers of the inequalities of system (7), which enter into the indices of the inequalities of the fundamental Rn-convolution of system (7). For positive values of all the parameters pl, . . . , pl the formula (8) gives positive solutions of system (6) with the greatest number of non-zero coordinates. Notes. 1. The question of finding all the non-negative solutions the arbitrary system of linear equations alful -I- - + amfum = ai
reduces to obtaining equations
(i = 1, . . . . n)
formula (8) for any system of homogeneous linear
alful + *.*+ amwm - uuzm+l = 0
and selecting from it solutions ordinate u,,,+~. 2. If all the coefficients all the solutions
of
(i = 1, . . . , n)
of this system with a positive
of system (6) are rational,
(UP, . . . , u,,,V,
k = 1, . . . , I,
final co-
then obviously
which determine formula (8)
Finite
systems
of
linear
19
inequalities
are also rational. It is not difficult to see that in this case formula (8) with arbitrary, non-negative, rational values of the parameters pk. k = 1, . . . . 2, will give all the non-negative solutions of system (6) with rational coordinates. Multiplying the solutions (ulck), . . . , urn(k)I by suitably selected positive integers we can convert them into positive integral solutions of system (6). Retaining the former notation for the latter, we obtain for?xUla (8) (for rational non-negative pk), adapted to find all the positive integral solutions of system (6) (with rational coefficients). On examining the question of non-negative rational and, in particular, the question of positive integral solutions of an arbitrary system of linear equations with rational coefficients and free terms we must be guided by Note 1. illustrate
To
Theorem 4.1 let us consider the following
example.
Exanrple. To find a general formula for non-negative solutions system ui +
2u2 -
-ui
-
u3 -
u2 +
ui -
u2
Ul -
u4 +
2%
+
u3
u2 -
-
+
3u5 -
u4 +
2U6 =
0,
U6 =
0,
u5 -
2u4 -
3u5 +
U6 =
0,
u4 -
2u5 +
U6 =
0.
u3 +
of the
In accordance with Theorem 4.1 the question reduces to finding a fundamental R*-convolution of the system Xi + 2X1 -
Its
X3 +
X4 +
t1 <
0,
x3
x4
f2
0,
0x2
OX3 +
+
x3
+
-
-xi
+
2X2 $
-51
-
X2 +
-
X4 +
ts <
0,
X4 +
t4 <
0,
-2xi
-Xz+X3+X4+ts
3x3 --4+t5<0,
has the form
OX4 +
ti +
t3 <
0,
2x4
ti
tr
0,
+
+
=G
-2x2 7x2
X2 -
X3 +
3X4 +
%
t6 <
0,
-2x2
3x2 +
X3 -
3x4 +
t2 +
2t3 <
0,
-x2
X4 +
t2 +
2t4 <
0.
-3X2
+
3X3 +
+
G
X9 -
x2
-
+
2X3 +
3x1+
fundamental Ill-convolution 3x2 +
We
X2 x2
+ +
+ -
Ox3 + 0x3
-
OX4 + 5X4 +
3x3 + 3x3 -
X4 + X4 +
t2 +
ts 4
0,
St3 +
t5 4
0,
t5 <
0,
3t6 <
0.
3t4 + 2t5 +
also form the fundamental (I-J, + UjJ-convolution 3x2 + 0x4 + ti + ts < 0, -2X2 7X2 -
+
OX4 + 5x4 +
t2 + 3ts +
t6 < t5 <
0, 0.
X2 + --3X2
+
5X4 +
054 +
%
+
t4 +
3t4 + 3ts +
t6 <
0,
3t6 <
0.
20
S.N.
The fundamental
Chernikov
(Ui + Us + Uk + UZ)-convolu-
(Vi -I- Us i- Uk)- and
tions take the form 3s~
and
+
ti
+
t3
t2 +
<
0,
-232
+
ts <
0,
--3x2 +
3t4 +
3t5 +
3te &
0,
2ti
3t2 +
2t3 +
3ta ss
0,
3f4 +
3t5 +
3tf3 d
ti +
+
t3 +
0.
In accordance with Theorem 4.1 the general formula for non-negative solutions of the system of equations considered takes the form (Ui, U2,
U3, Uk, U5, U6) =
pi (2,
3, 2, 0, 0, 3)
+
pz(l,
0, 1,3,3,3)
*
We now turn to the question of the general solution of a system of linear inequalities on Rn. Let us consider first the case of the system of homogeneous linear inequalities aj,rxlt < 0
+*. +
UjiXi +
(j = 1, . . . , m)
(9)
It is clear that it can be replaced by the equivalent system of equations UjiXi
+
---
+
ajnxn
-Uj
=
(i =
1, . ..) m)
(10)
with non-negative parameters uj. Considering the rank r of system (9) to be different from zero, we apply to system (101 Gauss’s algorithm for the elimination of unknowns. Without loss of generality we can consider that the formulae obtained for the unknowns and the equations for the parameters take the form
C$iUi +
58 = bsiui +
. . . +
*** +
Ur+i
CriUr +
=
bsnun 0
(s= (i =
1, . . . . n), 1,
. . . , n
-
(W r).
(12)
Let (8) be an expression for the arbitrary non-negative solution of system (121, defined by Theorem 4.1. Substituting it in (111, we obtain the formulae
which determine all the solutions of system (9). In these Xr+i, i = 1, ..,, n - r, are arbitrary parameters and pk, k = 1, . . . , 1, are arbitrary
Finite
non-negative
systems of
Zinear
inequalities
21
parameters.
The case of
the arbitrary
system
aim + -- + ajnxn - Uj < 0 reduces to the case considered, tained from the set of solutions
(I=
since the set of its of the system
ajixi + -** + UjnXn - UjXn+l < 0 of homogeneous inequalities
1, . .
with
xn+i =
m)
.)
(43)
solutions
(j = 1, . .
.)
is ob-
m)
1.
To conclude this paragraph we shall consider the question of finding all the edges of tine cone K of solutions of the system (9) with r =n cm. In this case formulae (11) and equations (12) respectively take the forms = beit+ + -.a + bsnun
XS
(s = 1, . . ., n)
(Ii*)
and C1tW
It all
is easily
+
+CniUn+Un+i=O
seen that
essentially
x(k) =
***
if
different
(s&k), . . . , z,th))
determined hausted
by formula
by the rays reduced CjlXi +
(Xl*), {hz@)},
+
Xj +
all
We can easily the vertices
equal
solutions
are the corresponding
of
to finding “’
(u&~), . . . , u,#Q),
fundamental
Theorem 4.1 the question pletely
uch) =
(i'l, .... m-n).
then all h > 0.
tn+j
<
0
+
k = 1, . . . , I, system (12.)
are and
with coordinates
the edges of the cone K are exall
tj <
in accordance
with
the edges of the cone K is
a fundamental
Cj, m-nxm--n
vectors
Therefore
finding
of
(12*)
RmMn-convolution 0
(j=1,2,
com-
of the system
(j= 1, 2, .... n), .... m-n).
reduce to the question just considered that of finding of a polyhedral set, defined by system (13) with rank
to the number of its
unknowns.
5. A numerical algorithmic scheme for reduced convolution The present section deals basically with the numerical scheme for the reduced convolution algorithm given in Section 3. If we denote by
22
S.N.
Chernikov
the matrix, in which to the left of the vertical line we put the matrix A0 of system (4) and to the right a unit matrix E” of order m, the scheme which interests us may be carried out by the following procedure. 1. We take some non-zero column of the matrix A”, for instance its first column Alo. Carrying out the A, O-deformation (see Section 3) of the system of rows of the matrix Ho, we obtain a system of rows of the matrix H’, situated beneath the matrix Ho on the extension of its columns, in which the part to the left of the vertical line is the matrix of the fundamental Ill-convolution of system (4). We shall denote the second part of the matrix H’ by i?l. If all the elements of the column Alo are positive or negative, we consider that the matrix H1 is empty; in this case the Ill-convolution of system (4) is empty and therefore the process ends. If the matrix HI is non-empty (all the elements of its first column will be zeros) the process is continued. 2. Suppose that the matrix Hk is already obtained and arranged under the matrix Hk-’ by the extension of its columns, the left-hand part Ak of which is the matrix of the fundamental (ul + . . . + Ilk)-convolution of system (4); let us denote the right-hand part of the matrix Hk by Ek. To each row of the matrix Hk, k = 1, 2, . . . , we shall assign an index which is the set of the numbers of the columns of the matrix Ek, which contain the positive elements of this row. We take a non-zero column of the matrix Ak,
e.g.
its (k + I)-th
matrix Ak are zero).
column Ai+1 (the first
k columns of the
If the A& -deformation of the system of rows of
the matrix Hk is not an interlacing
deformation, we obtain a system of
rows of the matrix Hk+' (arranged beneath the matrix Hk on the extension of its columns), by carrying out this deformation. The process terminates with an empty matrix Hk+' . If the At+, -deformation considered is an interlacing deformation and s of the k + 1 deformation &i-’ , i = 1t 2 7 ---, k+ 1, are interlacing deformations, then in the matrix
Hk+' there are rows of the matrix
column Ll;+i
Hk with zero elements in the
and row combinations, obtained by carrying out the given
*kk+i -deformation,
which satisfy
the following
requirements;
(1) Their indices do not exceed the number s + 1;
Finite
systems
of
linear
inequalities
23
(21 The index for none of them includes the index of any other. The first
requirement is satisfied
the Af+1 -defo~ation
automatically
if in carrying out
we do not combine those pairs of rows of the
matrix HI. the union of whose indices contains more than s + 1 elements. The second requirement is obviously satisfied if we also do not combine pairs of rows of the matrix Hk the union of whose indices contain the index of some third row of the matrix Hk. With such a limitation rows which are unnecessary for the composition of the fundamental (U1 + _. , + Uk+l)-convolution of system (4) will obviously not enter into the matrix Hkii. On the other hand it can easily be seen that with this limitation none of the rows which are necessary for the formation of this convolution
occur.
In fact in carrying out the
the third row considered is either simply transferred (if
it has a zero element in the column
A:+,)
Ai+i -deformation to the matrix Hktl
or must be combined with
that of the two rows of the pair taken, whose (k t l)-th
element (its
element found in the column L$+~) is opposite in sign to its
(k + l)-th
element. It is clear that the index obtained with this row combination will be included in the index of the row combination for the pair originally taken, and therefore in accordance with requirement (2) the latter row combination will not appear in the matrix Hktl. It is also clear that it must not be included in the matrix Hktl end in the first case where the third row is considered it is transferred without alteration to the matrix Hktr. The numerical scheme for the convolution of system (4) for fixed values of the parameters tj. e.g. for tj = aj, j = 1, 2, . . . , m, is essentially reduced to the procedure which has been considered. In this case we must introduce into the matrix Ho to the left of the vertical line the (n + l)-th column of the elements al, . . . , a,. If the matrix Ho, so broadened, is subjected to the transformation under consideration the (n + l)-th column of the matrices obtained will be a colt of free terms corresponding to the f~d~ental convolutions of system (4) with tj = oj. As above, the matrices of the coefficients of these convolutions consist of the first n-columns of the matrices obtained, Since the numerical values of the positive elements of the matrices Ek do not interest us (we are interested only in the indices of the rows of the matrices Hkl, when the successive transformations are carried out in the given process they can be replaced by unity. The given numerical scheme enables us to obtain fundamental U-convolutions of system (4) in the case where 17= U1 + . . . + Ilk, k = 1, 2, . . . .
24
S.N.
Chernikov
It can easily be seen that obtaining a fundamental U-convolution of system (4) for arbitrary U c Rn reduces to this case. We can show that the more general question of obtaining a fundamental &convolution of system (2) in an arbitrary real linear space f, with arbitrary 1.1c L reduces to it also. In fact, let U be a finite dimensional subspact: of L of and rrl, . . . . ur one of its bases. Then the fundamental U-convolution system (2) is obviously obtained if the given numerical procedure is applied to the system
JAfj(u’) +
“’
+
Srfj(U')
+
tj
<
0
(j= 1, 2, ...)m)
and in the complete
fundamental convolution obtained tj is replaced by If the subspace U is of infinite dimensions (naturally this is impossible if 1 = R9 then denoting by R the kernel of system (2). we obtain the direct expansion tj
+
_fjtx)s
x
E
L.
u=
(UflR)
--j-IT.
Here II’ is a finite-dimensional subspace of L. Since the fundamental Uconvolution of system (2) obviously coincides with the fundamental II'convolution of the latter, its calculation reduces to the case under consideration. by H.F. Cleaves
Translated
REFERENCES 1.
FOURIER, J. Oeuvres II. Gauthier-Villars, Paris, 1890.
2.
DINES, L.L. Systems of linear inequalities. Ann. 191 - 199, 1918 - 1919.
3.
KUHN, H. W. Solvability and consistency for linear equations equalities. Am. math. Mon., 63, 4, 217 - 232, 1956.
4.
CHERNIKOV, S.N. The convolution of finite systems of linear inequalities. Dokl. Akad. Nauk SSSR, 152. 5, 1075 - 1078, 1963.
5.
CHERNIKOV, S.N. The convolution of systems of linear inequalities. Dokl. Akad. Nauk SSSR, 131, 3, 518 - 521, 1960.
8.
MINKOWSKI,H. Geometrie der
Zahlen,
Leipzig.
Math.
Teubner.
(Ser.
$896.
2),
20,
and in-