A universal theorem of the alternative
David Bartl1
Abstract. We present a particular theorem of the alternative for finite systems
of linear inequalities. The theorem is universal in the sense that other classical
theorems of the alternative (Motzkin's Theorem and Tucker's Theorem) are
implicit in it; the theorem itself is an extension of Farkas' Lemma. The presented
result also generalizes and unifies both Dax's new theorem the alternative
[Dax, A. (1993). Annals of Operations Research, 46, 11­60] and Rohn's
residual existence theorem for linear equations [Rohn, J. (2010). Optimization
Letters, 4, 287­292]. The universal theorem of the alternative is established
by using Farkas' Lemma in the setting of a vector space over a linearly ordered
(commutative or skew) field.
Keywords: Theorems of the alternative, systems of linear inequalities,
Farkas' Lemma, linearly ordered vector spaces, linearly ordered fields.
JEL Classification: C65
1 Introduction
In this note, we present a universal theorem of the alternative which unifies both Dax's new
theorem of the alternative (Dax, 1993) and Rohn's residual existence theorem for linear equations
(Rohn, 2010). Moreover, the universal theorem contains Motzkin's Theorem of the alternative
(Motzkin, 1934) and Tucker's Theorem of the alternative (Tucker, 1956) implicitly in it.
The theorem is established by using Farkas' Lemma (Farkas, 1902), which has proved to be a
result of importance in optimization and also in economics (Vohra, 2006). Short proofs of
Farkas' Lemma include those given by Dax (1997), Broyden (1998; Roos and Terlaky, 1999),
Komornik (1998), Scowcroft (2006, pp. 3535­3536, in the Introduction) and Bartl (2012a). See
Fujimoto et al. (2018) for another proof; see also the references therein for further discussion
of earlier proofs. Considering natural numbers m,n,v G N, recall the original results in the
setting of the finite­dimensional vector space EW
, where JV G N is a natural number.
Farkas' Lemma (Farkas, 1902). Let A £ RmxN
be a matrix and let cT
£ RlxN
be a row vector. It holds
Vx E l " : Ax < 0 => cT
x < 0
if and only if
3uT
6 M l x m
, u T
> 0T
: cT
= vfA.
Dax's new theorem of the alternative (Dax, 1993, Sections 5.1 and 5.4). Let cT
6 E l x W
be a row vector, let
B±,..., Bn £ M l x W
be one-row matrices, and let w1 ( ..., w n £ E be positive weights. It holds
V x E l " : cT
x<w1\B1x\+- + wn\Bnx\
if and only if
3v± £ R, —wt <vt< w 1 ( 3vn £ E, —wn <vn< wn: cT
= v1B1 + —I­ vnBn.
Rohn's residual existence theorem for linear equations (Rohn, 2010, Theorem 2). Let cT
£ RlxN
be a row
vector, let Z £ W?xN
be a matrix, and let [bj,..., by} £ M l x p
be any collection of row vectors. It holds
Vx £ M w
: cT
x < max{b[Zx,..., blZx]
if and only if
3vlt ...,vv £ E, vx, ...,vv >Q,v1-\ 1­ vv = 1: cT
= (v^J H 1­ vvbl)Z.
1
Silesian University in Opava, School of Business Administration in Karviná, Department of Informatics and
Mathematics, Univerzitní náměstí 1934/3, 733 40 Karviná, Czech Republic, bartl@opf.slu.cz
­25 ­
Motzkin's Theorem (Motzkin, 1934; Motzkin, 1952, Theorem D6, p. 60). Let A £ RmxN
and B £ RvxN
be
matrices. It holds
^ £ R " : Ax < 0 A Bx < 0
if and only if
3AT
6 M l x m
, AT
> 0T
, Bff £ M l x v
, ff > 0T
, nt + ••• + nv = 1: A1
A + ffB = oT
,
where oT
£ M l x W
is the row vector consisting of N zeros and ...,\iv £ E are the components of j«T
.
Tucker's Theorem (Tucker, 1956, Corollary 2Apart (i)). Let A £ E m x W
and B £ M n x W
£e matrices, where the
matrix B consists of the rows bt,..., bn £ M l x W
. It holds
%x £ M w
: Ax < 0 A Bx < 0 A 6 ^ + ••• + 6n x = - 1
if and only if
3AT
£ M l x m
, AT
> 0T
, 3j«T
£ M l x n
, j«T
> 0T
: XT
A + j«T
B = oT
,
where oT
£ M l x W
w ?Ae row vector consisting ofN zeros.
2 Notation and Farkas' Lemma
Let F be a linearly ordered (commutative or skew) field. Additionally, let V be a linearly ordered
vector space over the linearly ordered field F. The relation of the linear ordering of the
field F and vector space V will be denoted by the symbol < and respectively. Recall that an
element X G F is positive and non-negative if and only if X > 0 and X > 0, respectively. These
two concepts are analogously defined for the elements of the vector space V. Finally, let W be
a vector space over the field F.
For non-negative natural numbers m and n, and for positive natural numbers v1 # ...,vn , let
:W -> F and ...,BjV.\ -> F be linear forms, which make up the linear mappings
A: W -> F m
and Bj-.W ^ Fv
i, respectively, for) = 1, ...,n. (The mappings ^4:14^ -> F m
and Bf. W -> F v
; generalize the concept of the matrices J4 G E M X W
and fl; G RV
;'X W
, which we
could see in the Introduction. Additionally, the linear forms ax, ...,am: W -> F and
/?; V / : 147
-> F correspond to the rows a1 # ..., a m and ...,fcJv^.of the matrix J4 and respectively,
for j = 1, ...,n.)
For any elements X,[J.EF and for any vector u G V, we put t/l/U = /U/l and tu/l = /lit. In
other words, the symbol Y (Greek letter iota) means that the next two entities are to be transposed
and multiplied in the new order.
Now, let m and v be a non-negative and positive, respectively, natural number. As above,
consider the linear mappings A: W -> Fm
and 5: W -> F v
. For any u G K m
and v G K v
, we
stipulate that they consist of the components and vx,..., vv, respectively. Moreover,
we define the linear mappings iuT
A: W -> V and ivT
B: W -> V by twT
.4x = S^iCc^x)^ and
LVT
BX = YX.=i{Bkx)vk, respectively, for every x G The symbol 0 denotes the column
vector consisting of m zeros of the field F. Additionally, the symbol 0T
denotes the row consisting
of m or v zeros of the vector space V. The inequalities Ax < 0 and wT
> 0T
as well as
vT
> 0T
are understood componentwise, that is atx < 0 and ut > 0 for every i = 1,..., m and
also vf c > 0 for every k = 1,..., v, respectively. The symbol e denotes the column vector consisting
of v ones of the field F. We then have ivT
e = iv±l + —I- ivvl = v1 + —Yvv. The
symbol o denotes the zero linear mapping o: W -> V. If m = 0, then iuT
A = o and the inequalities
Ax < 0 and also wT
> 0T
are logically true by convention.
-26-
Finally, let y. W -> V be any linear mapping. (The mapping y. W -> V generalizes the
concept of the row vector c T
G E L X W
, which we could see in the Introduction.)
Considering a non-negative natural number m, we now recall the following generalization
of Farkas' Lemma formulated in the setting of a (possibly infinite-dimensional) vector space
W over a (commutative or skew) linearly ordered field F, see Bartl (2007, Lemma 4.1).
Lemma 1 (Farkas' Lemma). Let F be a linearly ordered (commutative or skew) field, let W be a vector space
over the field F, and let V be a linearly ordered vector space over the linearly ordered field F. Let A:W -> Fm
and Y-W -> V be linear mappings. It then holds
Vx EW: Ax < 0 => yx < 0
if and only if
3u 6 VM
, u T
> 0T
: x = t u T
^ .
See Bartl (2012a) for a very short proof. The original version of Farkas' Lemma presented
in the Introduction is obtained by considering the field F •= E of the real numbers, the finitedimensional
vector space W •= E W
, and the one-dimensional real line V := E 1
.
3 A universal theorem of the alternative
Considering non-negative natural numbers m and n and also positive natural numbers v1#..., vn,
we present a new universal theorem of the alternative.
Theorem 2 (universal theorem of the alternative). Let F be a linearly ordered (commutative or skew) field, let
W be a vector space over the field F, and let V be a linearly ordered vector space over the linearly ordered
field F. Let A:W -> Fm
be a linear mapping, let Bt: W -> FVl
, ..., Bn: W -> FVn
be linear mappings consisting
of the linearforms BLT, ...,B1VL: W -> F, ..., BNL,... ,BNVN: W -> F, respectively, letwt,..., wn £ V be non-negative
weights, and let Y'-W -> Vbe a linear mapping. It then holds
VxEW: AX < 0 => YX
^ iw
i max{^u x,...,i51 V i x} + —I- iwn max{/?n l x, ...,BnVnx)
if and only if
3u 6 7 m
, u T
> 0T
,
£ VVl
, v\ > 0T
, tv^e = w1 ( ..., 3 v n £ VVn
, v\ > 0T
, iv£e = wn : y = LV7A + t V i ^ H 1- iv^Bn .
Proof The implication
Ax < 0 => < ^ max{/S1:Lx, ...,/S1 V i x} + —I- iwn max{/?n l x, ...,BnVnx)
holds for every x £ W if and only if the implication
Ax < 0 A f^x < A ••• A Bnx < eyn => yx < t W i y ! + —I- iwnyn
holds for every x EW and for every y1 ( ... , y n £ F, which equivalently means that
+ eya
yx + iwxyx + —I- iwnyn 4 0.
\Bnx + eyn < 0)
By Farkas' Lemma 1, with the vector space W replaced by W x F n
, it equivalently holds that
(u
v
y i E vm
x x ...xr», I V I >
v
n. .,
= (y t W i ... t w n ) .
It is thus equivalent to say that there exist a non-negative u EVm
and a non-negative vx eVVl
, ...,a non-negative
vn £ VVn
such that ivjei = LW1, w\ei = iwn, which equivalently means that ivje = w1, iv\e = wn, and
also y = iuT
A + ivjB-t + —I- iVnBn, which concludes the proof. •
T
I
w 1 ei
-27-
4 Special cases of the universal theorem of the alternative
We discuss some special cases of the universal theorem of the alternative (Theorem 2) in this
section.
4.1 Farkas' Lemma
In the previous section, we used Farkas' Lemma 1 to prove Theorem 2. At the same time,
Farkas' Lemma 1 is also a special case of Theorem 2 when n = 0. In particular, if n = 0, then
the empty sum iw-^ max{j?n x, ...,(31Vix) + —I- iwn max{/?n l x, ...,(3nVnx} = 0, the zero of the
vector space V, by convention. The remaining terms '3v;- G Vv
i, vj > 0T
, ivje = wf and
'ivJBf vanish in Theorem 2 if n = 0, whence the conclusion is easy to see.
4.2 Dax's new theorem of the alternative
The version of Dax's new theorem of the alternative which was given in the Introduction can
be found in Dax (1993, Sections 5.1 and 5.4). A generalized version can be found in Dax
(1990). Here, we obtain the following generalization of Dax's new theorem of the alternative,
where m and n are non-negative natural numbers, as a special case of Theorem 2.
Theorem 3 (Dax's new theorem of the alternative). Let F be a linearly ordered (commutative or skew)field,let
W be a vector space over the field F, and let V be a linearly ordered vector space over the linearly ordered
field F. Let A:W -> Fm
be a linear mapping, let B^.W -> F, ..., Bn:W -> F be linear forms, let w1 ( ..., w n 6 V
be non-negative weights, and let y.W -> V be a linear mapping. It then holds
Vx £ W: Ax < 0 => yx < i w j f ^ x l H 1- iwn\Bnx\
if and only if
3u 6 Vm
, u T
> 0T
,
3v± £ V, —w1 < V-L < w 1 ( 3vn 6 V, —wn < vn < wn : y = I U 1
A + iv1B1 H 1- ivNBN .
Proof. In Theorem 2, consider v± = ••• = vn = 2 with p1± = BT, B12 = Pm = Bn, fin2 = —BN. Observe
that the absolute value \BJX\ = max{B; x, —Bjx} = max{/S;1x, Bj2x] for every x 6 W for j = 1,..., n. Moreover
we have for any Vj 6 V that —Wj < Vj < w, if and only if Vj = v ; 1 - Vj2 for some non-negative Vjlt vj2 6 V such
that v ; 1 + Vj2 = Wj fory = 1,..., n. Finally, notice that IVJBJ = t(v; 1 - VJ2)BJ = iv^B^ + LVJ2BJ2 fory = 1,..., n.
The equivalence is easy to see now. •
The original version due to Dax (1993, Sections 5.1 and 5.4), presented in the Introduction,
is obtained by considering the field F ••= E of the real numbers, the finite-dimensional vector
space W •= E W
, the one-dimensional real line V •= E 1
, and by letting m := 0.
4.3 Rohn's residual existence theorem for linear equations
Rohn's residual existence theorem for linear equations (Rohn, 2010, Theorem 2), which was
presented in the Introduction, says in words that the system of linear equations c T
= f T
Z has a
solution f T
G E L X P
in the convex hull of the set {b\,..., by} if and only if cT
x < max{bJZx,...,
byZx} for every x G RN
. We obtain the following generalization of Rohn's residual existence
theorem for linear equations, where m and visa non-negative and positive, respectively, natural
number, as a special case of Theorem 2.
Theorem 4 (Rohn's residual existence theorem for linear equations). Let F be a linearly ordered (commutative
or skew) field, let W be a vector space over the field F, and let V be a linearly ordered vector space over the
linearly ordered field F. Let A: W -» Fm
be a linear mapping, let Bt,..., BV:W -» F be linearforms, letwEV be
a non-negative weight, and let y.W -> V be a linear mapping. It then holds
-28 -
Vx EW: AX < 0 => yx < iwmaxffti, ...,Bvx]
if and only if
3u £ 7 m
, u T
> 0T
, 3^,... ,vv EV, vt,..., vv > 0, v1 + —\-vv=w. Y = fw7
-^ + + —I"t v
n/?n •
Proof. Consider Theorem 2 with n = 1. •
The original version of Rohn's residual existence theorem for linear equations (Rohn, 2010,
Theorem 2), presented in the Introduction, is obtained by considering the field F ••= E of the
real numbers, the finite-dimensional vector space W ••= E w
, the one-dimensional real line
V •= E 1
, by letting m •= 0 and n •= 1, and by taking the weight w := 1. Then, given the row
vector c T
G E l x W
, the matrix Z G E p x W
, and the row vectors b\, ...,by G E l x p
, consider the
mapping y: E w
-> E 1
and the linear forms ...,/?v : E w
-> E defined by y.x •-> cT
;c and
/?fc: x •-> &^ZA: for /C = 1,..., v, respectively, for every ;c G E w
.
More generally, given the row vector c T
G E l x W
, the matrix Z G E p x W
, and the set
{fcT
,..., bl} c E l x p
, consider yet a set {aT
,..., a^} c E l x p
. Then, by Theorem 4, the system
of linear equations c T
= fT
Z has a solution f T
G E l x p
of the form f T
= utCL[ + —I- umcFm +
+ vxb\ + —I- vvby for some ux, ...,um > 0 and for some vx, ...,vv > 0 such that vx + — h
+ vv = 1, that is in the Minkowski sum (the convex hull of {b\,..., by} plus the convex conical
hull of {a[, ...,0^}) if and only if a[Zx, ....a^Zx < 0 implies cT
x < max{bJZx, ...,byZx}
for every x G E w
. See Bartl (2012b, Theorem 5) for further generalization in this direction.
4.4 Motzkin's Theorem of the alternative
Motzkin's Theorem of the alternative (Motzkin, 1934; Motzkin, 1952, Theorem D6 [in
Chap. Ill, § 13, par. 73], p. 60; cf. Tucker, 1956, Corollary 2A part (i)) is used to establish
optimality conditions in non-linear optimization, see, e.g., Mangasarian (1994), Birbil et al.
(2007). Notice that, depending upon the approach, Farkas' Lemma can be used to establish
optimality conditions in non-linear optimization directly, see, e.g., Franklin (2002). Below, we
obtain the following generalization of Motzkin's Theorem of the alternative, where m and v is
a non-negative and positive, respectively, natural number and 0 denotes the zero vector of the
space V, as a special case of Theorem 2; it is a special case of Theorem 4 actually.
Theorem 5 (Motzkin's Theorem of the alternative). Let F be a linearly ordered (commutative or skew) field,
let W be a vector space over the field F, and let V be a linearly ordered vector space over the linearly ordered
field F. Let A: W -> Fm
be a linear mapping, let ...,/?v : W -> F be linear forms, and let w £ V be a nonnegative
weight. It then holds
tx EW: Ax < 0 A iwmaxffti, ...,/3vx} < 0
if and only if
3u £ Vm
, u T
> 0T
, 3vt,..., vv £ V, vt,..., vv > 0, v1 + —\-vv = w. iuT
A + ivxpx H 1- ivn /?n = o .
Proof. There is no x EW such that Ax < 0 and iwmaxfjSjX, ...,/?vx} < 0 if and only if Ax < 0 implies
0 < twmax{/S1x, ...,f>vx} for every x EW. Now, conclude the proof by considering Theorem 4 with y = o, the
zero linear mapping o: W -> V. m
Remark 1. It holds m a x ^ x , . . . , /3vx) < 0 if and only if Bx < 0, cf. Motzkin's Theorem in the Introduction.
By identifying the vector space V with the one-dimensional line F1
, i.e. by taking V •= F1
,
and by considering the weight w := 1 and also Remark 1, we obtain the generalization of
Motzkin's Theorem due to Bartl (2007, Theorem 5.1).
-29-
To obtain the original formulation of Motzkin's Theorem (Motzkin, 1934; Motzkin, 1952,
Theorem D6, p. 60; see Birbil et al., 2007, Lemma 2.2, for another equivalent formulation),
presented in the Introduction, consider the field F ••= E of the real numbers, the finite-dimensional
vector space W ••= E w
, the one-dimensional real line V •= E 1
, the weight w := 1, and
Remark 1.
4.5 Tucker's Theorem of the alternative
Tucker's Theorem of the alternative (Tucker, 1956, Corollary 2A part (ii)) is dual to Motzkin's
Theorem of the alternative; compare both theorems in the Introduction to see this. The following
generalization of Tucker's Theorem of the alternative, where m and n are non-negative
natural numbers and 0 denotes the zero vector of the space V, is a special case of Theorem 2.
Theorem 6 (Tucker's Theorem of the alternative). Let F be a linearly ordered (commutative or skew) field, let
W be a vector space over the field F, and let V be a linearly ordered vector space over the linearly ordered
field F. Let A: W -> Fm
be a linear mapping, let B:W -> Fn
be a linear mapping consisting of the linear forms
Bx,..., Bn: W -> F, and let w £ Vn
be a column vector of non-negative weights w1 ( ..., w n 6 V. It then holds
tx 6 W: Ax < 0 A Bx < 0 A LWtBtx + ••• + iwnBnx < 0
if and only if
3u 6 Vm
, u T
> 0T
, 3v 6 Vn
, vT
> wT
: iifA + ivT
B = o .
Proof. There is no x 6 W such that Ax < 0 and Bx < 0 and also iw^B^x + —I- iwnBnx < 0 if and only if
Vx 6 W: (Q)X<Q => 0 4iw1B1x + - + iwnBnx.
By Theorem 2, with v1 = ••• = vn = 1 and y = o, the zero linear mapping o: W -> V, and also with the linear
mapping A:W -> Fm
replaced with the linear mapping y J : W -* Fm
x Fn
, it equivalently holds that
3u 6 Vm
, u T
> 0T
, 3v 6 Kn
, vT
> 0T
: iifA + iv1
B + iw^ + ••• + iwnBn = o .
We have ivT
B + + —I- iwnBn = ivT
B + iwT
B = i(v + w)T
B. By considering v = v + w, it is equivalent
to say that iuT
A + ivT
B = o for some non-negative u £ Vm
and for some v 6 Vn
such that vT
> wT
, which
means we are done. •
Remark 2. If the weights w 6 Vn
are positive, then vT
> wT
> 0T
, cf. Tucker's Theorem in the Introduction.
By identifying the vector space V with the one-dimensional line F1
, i.e. by taking V •= F1
,
and by considering the weights wt ••= ••• := wn ••= 1, we obtain the generalization of Tucker's
Theorem due to Bartl (2007, Theorem 5.2).
To obtain the original formulation of Tucker's Theorem (Tucker, 1956, Corollary 2A
part (ii)), presented in the Introduction, consider the field F •= E of the real numbers, the finitedimensional
vector space W ••= E w
, the one-dimensional real line V •= E 1
, and the weights
wt ••= ••• ••= wn ••= 1.
5 Concluding remarks
We presented a new universal theorem of the alternative (Theorem 2). We proved this result
by using Farkas' Lemma 1. We then showed that many other theorems of the alternative
(Farkas' Lemma itself, Dax's new theorem of the alternative, Rohn's residual existence theorem
for linear equations, Motzkin's Theorem of the alternative and Tucker's Theorem of the
alternative) are special cases of the universal theorem of the alternative (Theorem 2).
-30-
Fan (1956) and Chernikov (1968) also considered (finite) systems of linear inequalities and
theorems of the alternative from the algebraic point of view, i.e. in a vector space of arbitrary
dimension. Fan (1956) studies linear inequalities in a vector space over the field of real numbers
E and also over the field of the complex numbers (C. Chernikov (1968) presents a theory
of linear inequalities in a vector space over a linearly ordered commutative field F. In this
paper, we consider any linearly ordered (commutative or skew) field F, a vector space W over
the field F, and a linearly ordered vector space V over the linearly ordered field F.
Let the linear mappings Bt:W -> FVl
, Bn:W -> FVn
and the non-negative weights
w 1 # w n G V be as in Theorem 2. Notice that the mapping p:W -*V defined by
p:x •-> iWl maxjftji, ...,/?1 V i x} + ••• + twn max{/?n l x, ...,BnVnx\
is sublinear, that is p(Ax) = Ap(x) for all positive A G F and for all x G W and also p(x + y) ^
< p(x) + p(y) for all x,y G VK. It seems that Theorem 2 can be used to obtain optimality
conditions for some optimization problems with a non-smooth objective function of a special
form.
By using a discrete variant of Farkas' Lemma (Bartl, 2020), we could achieve analogous
results in the discrete setting of a module over a non-trivial linearly ordered commutative ring.
Finally, by using an extended variant of Farkas' Lemma, which allows the system 'Ax < 0'
to be infinite, we could generalize the results even further.
Acknowledgements
This work was supported by the Czech Science Foundation under grant number GACR
21-03085S, and by the Ministry of Education, Youth and Sports of the Czech Republic within
the Institutional Support for Long-Term Development of a Research Organization in 2021.
References
Bartl, D. (2007). 'Farkas' Lemma, other theorems of the alternative, and linear programming in infinitedimensional
spaces: a purely linear-algebraic approach'. Linear andMultilinear Algebra, 55 (4), pp.
Bartl, D. (2012a). 'A very short algebraic proof of the Farkas lemma'. Mathematical Methods ofOperations
Research, 75 (1), pp. 101-104.
Bartl, D. (2012b). 'Separation theorems for convex polytopes andfinitely-generatedcones derived from
theorems of the alternative'. Linear Algebra and its Applications, 436 (9), pp. 3784-3789.
Bartl, D. (2020). 'A discrete variant of Farkas' lemma and related results'. Optimization, DOI:
10.1080/02331934.2020.1768535.
Birbil, S. t, Frenk, J. B. G. and Still, G. J. (2007). 'An elementary proof of the Fritz-John and KarushKuhn-Tucker
conditions in nonlinear programming'. European Journal of Operational Research,
180 (1), pp. 479-484.
Broyden, C. G. (1998). 'A simple algebraic proof of Farkas' lemma and related theorems'. Optimization
Methods and Software, 8 (3-4), pp. 185-199.
Chernikov, S. N. (1968). Linear inequalities. Moscow: Nauka. [In Russian. HepHHKOB, C. H. (1968).
JluneuHbie uepaeeucmea. MocKBa: Hayica. (CoBpeMeHHaa ajire6pa).]
Dax, A. (1990). 'A new theorem of the alternative'. Mathematical Programming, 41 (1-3), 297-299.
-31 -
Dax, A. (1993). 'The relationship between theorems of the alternative, least norm problems, steepest
descent directions, and degeneracy: A review'. Annals of Operations Research, 46 (1), pp. 11-60.
Dax, A. (1997). 'An elementary proof of Farkas' lemma'. SIAM Review, 39 (3), pp. 503-507.
Fan, K. (1956). 'On systems of linear inequalities'. In Kuhn, H. W. and Tucker, A. W. (eds.) Linear
Inequalities and Related Systems. Princeton: Princeton University Press, pp. 99-156. (Annals of
Mathematics Studies; no. 38).
Farkas, J. (1902). 'Theorie der einfachen Ungleichungen'. Journalfür die reine und angewandte Mathematik,
124 (1), pp. 1-27.
Franklin, J. N. (2002). Methods of mathematical economics: Linear and nonlinearprogramming, fixedpoint
theorems. Philadelphia: SIAM. ISBN 0-89871-509-1.
Fujimoto, T., Perera, B. B. U. P. and Giorgi, G. (2018). 'A proof of the Farkas-Minkowski theorem by
atandem method'.Metroeconomica, 69 (1), pp. 142-150.
Komornik, V. (1998). 'A simple proof of Farkas' lemma'. The American Mathematical Monthly, 105
(10), pp. 949-950.
Mangasarian, O. L. (1994). Nonlinear programming. Philadelphia: SIAM. ISBN 0-89871-341-2.
Motzkin, T. S. (1934). Beiträge zur Theorie der linearen Ungleichungen. Basel: University of Basel.
Doctoral dissertation. Published also in Jerusalem: Azriel, 1936.
Motzkin. T. S. (1952). Contributions to the theory of linear inequalities. [A translation of Motzkin
(1934) by Fulkerson, D. R] Santa Monica: RAND Corporation, 7 March 1952. RAND Corporation
Translation 22. Reprint in: Cantor, D., Gordon, B. and Rothschild, B. (eds.) Theodore S. Motzkin:
Selected Papers. Boston: Birkhäuser, 1983, pp. 1-80. (Contemporary Mathematicians). ISBN
3-7643-3087-2.
Röhn, J. (2010). 'A residual existence theorem for linear equations'. Optimization Letters, 4 (2), pp.
287-292.
Roos, C. and Terlaky, T. (1999). 'Note on a paper of Broyden'. Operations Research Letters, 25 (4),
pp.183-186.
Scowcroft, P. (2006). 'Nonnegative solvability of linear equations in certain orderedrings'.Transactions
of the American Mathematical Society, 358 (8), pp. 3535-3570.
Tucker, A. W. (1956). 'Dual systems of homogeneous linear relations'. In Kuhn, H. W. and Tucker,
A. W. (eds.) Linear Inequalities and Related Systems. Princeton: Princeton University Press, pp.
99-156. (Annals of Mathematics Studies; no. 38).
Vohra, R. V. (2006). 'The ubiquitous Farkas lemma'. In Alt, F. B., Fu, M. C. and Golden, B. L. (eds.)
Perspectives in Operations Research. Boston: Springer, pp. 199-210. (Operations Research/Computer
Science Interfaces Series; vol. 36). ISBN 0-387-39934-8.
-32-