Internat. J. h.
Math. Sci.
8O9
Vol. 5 No. 4 (1982) 809-812
RESEARCH NOTES
A NOTE ON GORDAN’S THEOREM OVER CONE DOMAINS
BRAD SKARPNESS
Department of Statistics
Virginia Polytechnical Institute
Blacksburg, Virginia 24061
V.A. SPOSlTO
Department of Statistics
Iowa State University
Ames, Iowa 50011
(Received April 6, 1982 and in revised form September 8, 1982)
ABSTRACT.
This note presents a proof of
Gordan’s Theorem over general closed, conve
cone domains which follows in a natural way appealing to the standard definitions of
closed convex cones and their respective polar cones.
KEV ’WORDS AND H.
Gordan’s Theorem, closed convex cones, polar cones.
1980 THEMATICS SUBJECT CLASSIFICATION CODES.
I.
90C99.
INTRODUCTION
Solvability theorems or theorems of the alternative for problems involving
linear system of equations has played a major role in mathematical programming,
and linear analysis, (Craven [I], Luenberger [2] or Mangasarian [3]).
Mmngasarian [3] presents a rigorous development of the classical theorems
of alternatives, see Chapter 2 in [3].
Many theoretical aspects in mathemati-
cal programming appeal to various theorems of the alternative in establishing
certain optimality conditions and duality properties.
Ben-lsrael [4], and Berman and Ben-lsrael [5] extended the classlcal
theorems of the alternative to problems with linear equations over polyhedral
cone domains.
Thus extending the classical formulations over nonnegative
orthants as originally presented by Gordan [6]
linear system over inequalities (> 0).
i.e., problems involving a
In particular, Ben-Israel proves a
810
B. SKARPNESS AND V.A. SPOSITO
certain theorem, (Theorem 2 in
to establish extensions
[4]), which is utilized
of the classical theorems of the alternative of Gordan, Motzkin, and Slater to
problems formulated with linear equations over polyhedral convex cone domains.
Ben-Israel underscores the fact that Theorem 2 cannot be used to generalize
these results to general (nonpolyhedral) closed convex cones; see page 134 in
[4]; hence, restricting the extension of the theorems of the
problems involving general convex cone domains.
alternative to
Berman and Ben-Israel (cor.
1.5 in [5]), however, were able to establish a generalized Gordan’s Theorem
Mazur’s
over general closed convex cone domain by appealing to
Theorem
(Bourbaki [7]) or the Hahn-Banach Theorem (Schaefer [8]).
Craven (pg 31-33 in [i]), presents several theorems of the alternative
however, a generalization of Gordan’s
over general closed .cone domains
Therefore, the purpose
Theorem is not explicitly given in this development.
of this note is to present a proof of Gordan’s generalized theorem appealing
only to the standard definitions of cones and their respective polar cones;
hence, differing from the proof given by Berman and Ben-Israel in [5].
2.
ALTERNATIVE PROOF
An alternative proof of Gordan’s theorem over arbitrary convex cone
domains is now presented over finite dimensional space.
Consider the
following definitions
Definition I.
have that
C
n
if for any vector
y e C
and
k > 0
we
ky e C
Definition 2.
A cone
Definition 3.
C
En;
E
is a cone in
C
is pointed if
C
{0}
N (-C)
will denote the polar cone of an arbitrary cone
C
in
that is
C*
{y*
E
n
IY
*,
y
>_
0
for all
y
C}
(2.1)
Gordan’s theorem over convex cone domains, Lemma 2 below, is established
appealing to the following lemma:
GORDON’ S THEOREM OVER CONE DOMAINS
Lemma I.
C
Then
Int
(C*)
(Gordan’s Theorem for Arbitrary Cone Domains).
Let
M
Let
C
be a closed convex cone in
E
n
@
iff
is a pointed cone.
Berman and Ben-Israel ([5], Lemma 0).
Proof:
Lemma 2.
nonvacuous
in
811
E
n,
m x n
C
matrix, with
be any given
any arbitrary pointed, closed convex cone
then exactly one of the following systems is consistent;
(i)
Mx
(ii)
M y
0
x e C
for some
or
,
g
Int(-C ), y
# 0
x
m
E
E
(Not (li) implies (1)).
Proof:
Let
{zlz
S
S1
and, reover,
Em}
M y, y
S2
and
E
disjoint convex sets in
S
{zlz
2
g
are convex sets.
n
Int(-C*)},
Since
S
S
and
n
S
then
S
2
are two
2
v (nonzero)
then there exists a hyperplane
such that
v z
_>
v z
for all
2
z
I
e
for all
Sl;
z
2
e S 2, (the closure of S
2)
Hence,
_>
v M y
Assume
C
v
v z
then there exists
v
y*
However, for any given
such that
v z
2
> v M y
Now letting
letting
E
TM
for all
S
z^ 6
z
2
there exists
v M y > v z
then
0
-v M Mv > 0
we have that
z
S
g
2
(2.2)
2
such that
,
z2>0
which violates (2.2).
0
-Mv
y
TM
y e E
for all
2
-*
z
2
kz
e S
k > 0
where
2
Hence, it follows that
hence, v M y
6
C
However,
0
Mv
Therefore,
v
0
C
v
O) hence, (i) holds.
(v
To show now that (ii) implies not (i)
,
Let
such that
since
be such that
y
M y
,
,
,
Mx
0
Int(-C
,
x
M y
,
Int(-C ), and assume there exists
# 0; then necessarily
,
with
Hence, the result follows.
x
,
,
g
e C
(x
,
,
y (Mx)
0
x
A contradiction,
,v
# 0), implies that
e C
x
(M y
,
< 0
B. SKARPNESS AND V.A. SPOSITO
812
3.
CONCLUDING REMARKS
Lemma 2 requires that the cone
by Lemma
,
C
be closed, convex, and pointed; hence,
Clearly, relaxing this requirement could result
Int (-C)
in the inconsistency of both (i) and (ii) in Lepta 2.
ACKNOWLEDGEMENTS
The authors wish to thank the referees for many valuable and helpful
suggest ions.
References
Mathematical Programming and Control Theory.
Chapman and
I.
CRAVEN, B. D.
Hall, 1978.
2.
LUENBERGER, D. G. Introduction to Linear and Nonlinear Programming.
Addlson-Wesley, Reading, Mass. 1973.
3.
MANGASARIAN, O. L.
N.Y. 1969.
4.
BEN-ISRAEL, A. Theorems of the alternative for complex linear inequalities.
Israel Journal of Mathematics, 7, (1969), 129-136.
5.
BERMAN, A. and BEN-ISRAEL, A. More on Linear inequalities with applications
Journal of Mathematical Analysis and Applications, 3,
to matrix theory.
(1971) 482-496.
6.
GORDAN, P.
ker
Coefflcienten.
Nonlinear Progra,,ing.
McGraw-Hill, New York,
Auflsungen linearer Glerchungen mit reelen
Math. Ann., 6, (1873), 23-28.
die
7.
BOURBAKI. N.
Espaces vectorlels topologiques.
8.
SCHAEFER, H.
Topological vector spaces.
Herman & Cie, Paris. 1953.
Macmillan, New York. 1966.