Directory UMM :Data Elmu:jurnal:O:Operations Research Letters:Vol28.Issue1.2001:
Operations Research Letters 28 (2001) 9–19
www.elsevier.com/locate/dsw
Two simple proofs for analyticity of the central path
in linear programming
Margareta Halicka
Department of Applied Mathematics, Faculty of Mathematics and Physics, Comenius University, Mlynska dolina,
842 15 Bratislava, Slovakia
Received 2 December 1998; received in revised form 1 January 2000
Abstract
Several papers have appeared recently establishing the analyticity of the central path at the boundary point for both
linear programming (LP) and linear complementarity problems (LCP). While the proofs for LP are long, proceeding from
limiting properties of the corresponding derivatives, the proofs for LCP are very simple, consisting of an application of
the implicit function theorem to a certain system of equations. Inspired by the approach for LCP, this paper gives two
simple ways of proving the analyticity of the central path for LP. One follows the idea for LCP, the other is based on a
c 2001 Elsevier Science B.V. All rights reserved.
proper partition of the system de ning the central path.
MSC: primary: 90C05; secondary: 90C33
Keywords: Linear programming; Interior point methods; (weighted) Central path; Limiting behavior; Analyticity
1. Introduction
Consider the following pair of dual linear programming (LP) problems:
(P)
min{cT x | Ax = b; x¿0};
(D)
max{bT y | AT y + s = c; s¿0};
where A is an m × n matrix, rank(A) = m ¡ n; c; x; s ∈ Rn ; b; y ∈ Rm . We restrict our attention to problems
satisfying the assumption
(AS)
there exist
x ¿ 0; s ¿ 0
such that
Ax = b;
AT y + s = c:
E-mail address: halicka@fmph.uniba.sk (M. Halicka).
c 2001 Elsevier Science B.V. All rights reserved.
0167-6377/01/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 0 0 ) 0 0 0 6 5 - 1
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M. Halicka / Operations Research Letters 28 (2001) 9–19
In the context of interior point methods for solving (P) and (D), it is important to study the properties of certain interior point paths. We recall here the de nition and some properties of these paths, called
!-weighted central paths.
For any ! ¿ 0; ! ∈ Rn , de ne the following parameterized system
Ax = b;
x¿0;
AT y + s = c;
s¿0;
Xs = !;
(1)
where X := diag(x) and ∈ R; ¿0. Note that for = 0 system (1) gives necessary and sucient conditions
for optimality of both (P) and (D). Under assumption (AS), for any ¿ 0 there exists the unique solution
(x(); y(); s()) of (1) such that x() ¿ 0; s() ¿ 0. The set of points {x(); y(); s()}¿0 is called an
(!-weighted) central path. Due to the rank condition for A, we have a one-to-one correspondence between y
and s in (1). This enables us to omit y() from the de nition of an (!-weighted) central path.
The usual central path is the !-weighted central path corresponding to the weight vector ! = (1; : : : ; 1)T .
In this paper we consider the (!-weighted) central path corresponding to the arbitrarily chosen weight vector
! ¿ 0. We study the properties of this central path under the natural parameterization given in (1). That is,
we are interested in the properties of (x(); s()) as a function of ¿ 0.
It is easy to see that the central path has nice analytical properties. Actually, the function
Ax − b
G(x; y; s; ) = AT y + s − c
Xs − !
is real analytic and its Jacobian with respect to (x; y; s) is nonsingular at those points where all components
of both x and s are non-zero. Moreover, the central path satis es
G(x(); y(); s(); ) = 0
(2)
for each ¿ 0. Thus, by the implicit function theorem, the central path is analytic in for ¿ 0. That is,
it is in nitely di erentiable and the Taylor series of (x(); s()) for any 0 ¿ 0 converges to (x(); s()) at
a neighborhood of 0 . (More analytical properties of the central path for ¿ 0 were studied in [17,15].)
The limiting property of the !-weighted central path is that there exists a nite limit of (x(); s()) as
↓ 0, and this limit value forms a strictly complementary solution of (P) and (D). For the proof of this
assertion see [9,6,8]. Therefore, we can extend the domain of the central path (as a function of ) to the
closed interval [0; ∞) by
(x(0); s(0)) := lim (x(); s()):
↓0
(3)
Now, Eq. (2) holds even for = 0, but the study of the analytical properties of the central path becomes
more complicated since some information contained in system (1) vanishes at = 0. In fact, system (1) can
have many solutions at = 0, and the Jacobian, G ′ (x; y; s; 0), at these solutions is singular. Consequently, the
implicit function theorem does not apply to Eq. (2) at = 0. Thus e ort was concentrated on the analysis
of limiting properties of the kth derivatives, (x(k) (); s(k) ()), as ↓ 0. The existence of such limits was
established in [1,16] for k = 1 and, in [4] for any k¿1.
Recently, almost simultaneously, several papers have appeared proving the analyticity of the central path
at the boundary [7,12–15]. While two of them deal with LP problems [7,15], the others treat linear complementarity problems (LCP) where the linear programming problem is regarded as a special case. The proofs
for LP by Halicka [7] and Wechs [15] proceed from the investigation of the limits of explicit formulas for
high-order derivatives for ¿ 0 and also establish the geometric growth of the derivatives at the boundary.
M. Halicka / Operations Research Letters 28 (2001) 9–19
11
Although these proofs for LP are constructive and insightful, they are rather technical and quite long. On
the contrary, the proof by Stoer and Wechs [12] (see also [13]) for the monotone LCP is, surprisingly, very
simple. It consists of an application of the implicit function theorem to a system of equations, the existence of
which was deduced from the system of equations de ning the central path. Let us note that the construction
of that system is described in the follow-up paper by Stoer et al. [14].
Inspired by the approach for LCP, this paper gives two simple ways of proving the analyticity of the
central path for LP. The main idea of both proof techniques is common: we rewrite the system of equations
describing the central path, so that the corresponding Jacobian remains nonsingular as ↓ 0. One proof
technique (Section 3) follows the ideas for LCP from [12–14] and allows geometric interpretation. The other
(Section 4) uses a partition of A, rst described in [7]. Here, the system is rewritten in a di erent form and
the resulting Jacobian exhibits some (skew-)symmetric properties. A common conclusion of both these proof
techniques is given in Section 5.
2. Preliminaries
The main tool in developing limiting properties of the central path is the concept of an optimal partition
(B; N ) of (P) and (D). Recall that the optimal partition is a partition of the index set {1; : : : ; n} such that
every strictly complementary optimal solution (x; s) of (P) and (D) has the following property: xi ¿ 0; si = 0
for i ∈ B and xi = 0; si ¿ 0 for i ∈ N . For further details of the optimal partition concept see [2], or [11].
We use the notation xB and xN to refer to the restriction of any vector x ∈ Rn to the coordinate sets B and N ,
respectively. Correspondingly, the submatrices AB and AN are formed by taking the B and N columns of A,
respectively. Thus, we can write A = (AB ; AN ) by reordering the columns in A, if necessary. Finally, by |B|
and |N | we denote the cardinality of the index sets B and N , respectively. In this notation, we can rewrite
Eqs. (1) as
AB xB + AN xN = b;
ATB y + sB = cB ;
ATN y + sN = cN ;
XB sB = !B ;
XN sN = !N :
(4)
Since (x(0); s(0)) is strictly a complementary optimal solution of (P), (D) and (B; N ) is the optimal partition,
we have
xB (0) ¿ 0;
xN (0) = 0;
sB (0) = 0;
sN (0) ¿ 0:
(5)
We now use a technique, as used by Stoer and Wechs [12,13] in the context of LCP’s, and de ne the new
“tilde” variables
s˜B := XB−1 !B ⇒ sB = s˜B ;
x˜N := SN−1 !N ⇒ xN = x˜N :
(6)
Then, instead of (xB (); xN (); sB (); sN ()), we study the corresponding “tilde” path (xB (); x˜N (); s˜B ();
sN ()) for ¿0. It is easy to see that for each ¿0 the “tilde” path is the solution to the system
AB xB + AN x˜N = b;
(7a)
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M. Halicka / Operations Research Letters 28 (2001) 9–19
ATB y + s˜B = cB ;
(7b)
ATN y + sN = cN ;
(7c)
XB s˜B = !B ;
(7d)
X˜ N sN = !N :
(7e)
More precisely, (xB (); x˜N (); s˜B (); sN ()) is even the unique solution to (7) such that xB () ¿ 0; sN () ¿ 0.
Moreover, by (5) and (6), we have s˜B (0) ¿ 0 and x˜N (0) ¿ 0.
The Jacobian of (7) is
AB 0
0
AN
0
0
0
0
IN
ATB
IN
0
0 ;
ATN
J (xB ; y; sN ; x˜N ; s˜B ; ) = 0
S˜
0
0
XB
B 0
0
0
X˜ N
SN
0
and its values along the “tilde” path are denoted by J (), i.e. J () := J (xB (); y(); sN (); x˜N (); s˜B (); )
for ¿0. It is easy to see that J () is nonsingular for ¿ 0, but,
at =
0, its nonsingularity depends on the
AB 0
is nonsingular, which is equivalent
m × |B| matrix AB . In fact, J (0) is nonsingular if and only if
0 ATB
to the two conditions |B| = m; rank(AB ) = m. Note that these conditions are also equivalent to the uniqueness
of the optimal solutions of (P) and (D) (see [5,10]). The case of |B| =
6 m or rank(AB ) 6= m is treated in the
next two sections.
3. The rst approach
As mentioned above, Stoer and Wechs proved the analyticity of the central path for the monotone LCP
in [12] and for the sucient LCP in [13]. The main idea of their proof resides in adding some equations to
the original system of equations describing the central path. Then, the enlarged (overdetermined) system of
equations has a Jacobian with full column rank, and the existence of a subsystem with a nonsingular Jacobian
is ensured. Thus, the analyticity follows by an application of the implicit function theorem. The study of this
problem was completed in [14], where the authors describe explicitly how to select a subsystem from the
enlarged system with a nonsingular Jacobian at = 0.
It is well-known that LP problems can be formulated in the LCP format. So, the results (found in [12–
14]) concerning the analyticity of the central path for LCP are also valid for LP. As mentioned above, the
analyticity of the central path was proven directly for LP in [7,15], but the proofs are long. Thus, the value of
[12–14] to LP is that the analyticity of the central path can be proved in a very simple manner, speci cally the
system of equations for the central path may be rewritten, so that the corresponding Jacobian is non-vanishing
at = 0. Obviously, the proof technique and the resulting system described in [12–14] depend on the LCP
format, so it is not clear what the proof and the resulting system would look like in terms of LP. In this
section the idea of the proof by Stoer and Wechs for LCP is applied directly to the LP problem (without the
reformulation of LP to LCP). Moreover, while the procedure of Stoer and Wechs is described in two di erent
steps ( rst adding some equations to the original system, then the selection of the subsystem), here the result
will be obtained in one step.
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M. Halicka / Operations Research Letters 28 (2001) 9–19
Denote
P :=
AB
0
0
ATB
!
;
Q :=
AN
0
0
IB
!
and
q :=
b
cB
!
:
Then (7a) and (7b) can be rewritten as
!
!
x˜N
xB
+ Q
= q:
P
y
s˜B
(8)
Let rank(P) = r ¡ m + |B|. Then there exists a nonsingular (m + |B|) × (m + |B|) matrix W such that
!
W1
and W2 P = 0:
W=
W2
Here W2 is an (m + |B| − r) × (m + |B|) matrix, and we have
W2 w = 0 ⇔ ∃u:
w = Pu:
(9)
To see this, W2 P = 0 implies Col(P) ⊆ Null(W2 ). However, since W is nonsingular, rank(W2 ) = m + |B| − r,
which implies dim(Null(W2 )) = r. Now, since rank(P) = r, we get Col(P) = Null(W2 ), from which (9) follows.
Multiplying (8) by W on the left we obtain
!
!
xB
x˜N
(10a)
+ W1 Q
= W1 q;
W1 P
y
s˜B
W2 Q
x˜N
s˜B
!
= W2 q:
From (10b) at = 0 we get W2 q = 0. Thus instead of (10b) we have
!
x˜N
= 0:
W2 Q
s˜B
(10b)
(11)
We now have that for any ¿0, the “tilde” path is a solution to the system of Eqs. (10a), (11) and (7cde).
Denote JI () the Jacobian of system (10a), (11) and (7cde) with respect to (xB ; y; sN ; x˜N ; s˜B ), evaluated at
(xB (); y(); sN (); x˜N (); s˜B ()) and ¿0. It is easy to see that JI () is nonsingular for ¿ 0, since this
is a property of the Jacobian J () of original system (7). The next theorem summarizes these results and
establishes the nonsingularity of JI (0).
Theorem 1. For any ¿0 the “tilde” path (xB (); x˜N (); y(); s˜B (); sN ()) is a solution to system (10a); (11)
and (7cde) of 2n + m equations; and the nonsingularity of the Jacobian JI () extends to = 0.
Proof. It suces to prove that if JI (0) dh = 0 for some vector dh, then dh = 0.
Let dh = (d xB ; dy; dsN ; d x˜N ; d s˜B ), then the equations of JI (0) dh = 0 are
!
d xB
= 0;
W1 P
dy
W2 Q
d x˜N
d s˜B
!
= 0;
(12a)
(12b)
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M. Halicka / Operations Research Letters 28 (2001) 9–19
ATN dy + dsN = 0;
(12c)
S˜B (0) d xB + XB (0) d s˜B = 0;
(12d)
X˜ N (0) dsN + SN (0) d x˜N = 0:
(12e)
First, we consider (12a). Since W2 P = 0, we have WP
d xB
dy
!
= 0 and due to the nonsingularity of W we
obtain
P
d xB
dy
!
= 0:
(13a)
Further, we consider (12b). According to (9) applied to (12b), there exist u1 and u2 such that
!
!
d x˜N
u1
= 0:
+Q
P
d s˜B
u2
(13b)
Moreover, setting u3 = −ATN u2 , we have
ATN u2 + u3 = 0:
(14)
From (13a) and (13b) we obtain
!
!
d x˜N
u1 + d xB
+Q
=0
P
u2 + dy
d s˜B
for all ∈ R:
(15)
Thus, from (15), (12c) and (14) it follows that
AB (u1 + d xB ) + AN d x˜N = 0;
ATB (u2 + dy) + d s˜B = 0;
ATN (u2 + dy) + (u3 + dsN ) = 0
Since the vectors
!
u1 + d xB
d x˜N
and
for all ∈ R:
d s˜B
u3 + dsN
(16)
!
are orthogonal we have
u1T d s˜B + d x˜TN u3 + (d xBT d s˜B + d x˜TN dsN ) = 0
for all ∈ R
which implies
d xBT d s˜B + d x˜TN dsN = 0:
From (12d) and (12e) we now obtain
d s˜B = −XB (0)−1 S˜B (0) d xB
(17)
(18a)
and
dsN = −X˜ N (0)−1 SN (0) d x˜N :
(18b)
Substituting (18a) and (18b) into (17) yields
(19)
− d xBT XB (0)−1 S˜B (0) d xB − d x˜TN X˜ N (0)−1 SN (0) d x˜N = 0
−1 ˜
−1
˜
and since XB (0) S B (0) and X N (0) SN (0) are positive de nite, we obtain d xB =0 and d x˜N =0. Consequently,
(18a) and (18b) yield d s˜B = 0 and dsN = 0.
M. Halicka / Operations Research Letters 28 (2001) 9–19
15
Finally, by (12c) and dsN = 0, we get ATN dy = 0 and from (13a) we obtain ATB dy = 0. Now, the full row
rank of A yields dy = 0. Thus dh = 0 and the theorem is proved.
The results of this section are closely related to the geometric interpretation of the central path as described
by Adler and Monteiro [1] (see also [3]). By this interpretation the (!-weighted) central path is, at any ¿0,
the (!-weighted) analytic center of a certain set P(), where P(0) is the set of optimal solutions for (P) and
(D). In our case, we de ne P(); ¿0, as the set of variables xB ¿0; y; sN ¿0 satisfying (10a) and (7c)
where x˜N = x˜N () and s˜B = s˜B () are xed. Then it is straightforward to verify that Eqs. (10a), (11) and
(7cde) represent necessary and sucient conditions describing the (!-weighted) analytic center of P() for
all ¿0.
Since the analytic center of P(); ¿0, is de ned uniquely, the above interpretation implies the uniqueness
of the (positive) solutions of (10a), (11) and (7cde) for all ¿0.
4. The second approach
We now present an alternative procedure that yields a di erent algebraic description for the central path
with non-vanishing Jacobian. First we note that, prior to this section, the symbol := was used to de ne the
object on its left-hand side. In this section we also use the symbol =: to de ne the object on the right.
Let rank(AB ) =: r. From the rank conditions on A and AB one has that A can be partitioned in the form
(by reordering the columns in AB or in AN , if necessary)
"
..
AB11
A = [AB . AN ] =:
AB21
|{z}
r
where
R :=
"
AB11
AN 11
AB21
AN 21
#
AB12
AB22
|{z}
..
.
..
.
|B|−r
AN 11
AN 12
AN 21
|{z}
AN 22
|{z}
m−r
#
}r
}m − r;
|N |−m+r
is a nonsingular m × m matrix. Multiplying A by R−1 on the left, we obtain
.
.
A := R−1 A = R−1 [AB .. AN ] =: [A B .. A N ]
and also
.
A = [A B .. A N ] =:
Ir
A B12
0(m−r)×r
A B22
..
.
..
.
0r×(m−r)
Im−r
Note that A B22 = 0, since rank(AB ) = rank(A B ) = r and thus
#
"
"
#
0
A N 12
Ir A B12
;
A N =
:
A B =
0
0
Im−r A N 22
AN 12
:
A N 22
(20)
De ne b := R−1 b and y := RT y. Using the corresponding partition of all vector variables Eqs. (7) can be
written in the form
# " #
"
#
"
b1
x˜N 1
xB1
;
(21)
=
+ AN
AB
x˜N 2
xB2
b2
16
M. Halicka / Operations Research Letters 28 (2001) 9–19
T
A B
"
T
A N
"
y 1
y 2
y 1
y 2
#
+
#
"
+
"
s˜B1
s˜B2
sN 1
sN 2
#
#
=
=
"
"
cB1
cB2
cN 1
cN 2
#
#
;
;
(22)
(23)
XB1 s˜B1 = !B1 ;
(24)
XB2 s˜B2 = !B2 ;
(25)
X˜ N 1 sN 1 = !N 1 ;
(26)
X˜ N 2 sN 2 = !N 2 :
(27)
Substituting (20) into (21) – (23) yields
xB1 + A B12 xB2 + A N 12 x˜N 2 = b1 ;
(28)
x˜N 1 + A N 22 x˜N 2 = b2 ;
(29)
y 1 + s˜B1 = cB1 ;
(30)
T
A B12 y 1 + s˜B2 = cB2 ;
(31)
y 2 + sN 1 = cN 1 ;
(32)
T
T
A N 12 y 1 + A N 22 y 2 + sN 2 = cN 2 :
(33)
Here, vectors with indices B1; B2; N 1 and N 2 are the vectors of dimensions r; |B| − r, m − r and |N | − m + r,
respectively. Since we consider (29) for any ¿0, we have b2 = 0, and so the equation can be divided by
to give
x˜N 1 + A N 22 x˜N 2 = 0:
(29′ )
Eqs. (30) and (32) can be written in the form
y 1 = cB1 − s˜B1 ;
(30′ )
y 2 = cN 1 − sN 1
(32′ )
and thus the variables y 1 and y 2 can be eliminated from (31) and (33) to obtain
T
A B12 (cB1 − s˜B1 ) + s˜B2 = cB2 ;
(31′ )
T
T
A N 12 (cB1 − s˜B1 ) + A N 22 (cN 1 − sN 1 ) + sN 2 = cN 2 :
(33′ )
T
Eq. (31′ ) holds for any ¿0 and thus A B12 cB1 = cB2 and
T
− A B12 s˜B1 + s˜B2 = 0:
(31′′ )
M. Halicka / Operations Research Letters 28 (2001) 9–19
17
Now (28), (31′′ ), (29′ ), (33′ ) together with (24) – (27) form the full system of 2n equations for the 2n
variables xB1 ; s˜B2 ; x˜N 1 ; sN 2 ; s˜B1 ; xB2 ; sN 1 ; x˜N 2 . The Jacobian of this system with respect to the variables in the
given order is
..
I
0
0
0
.
0
A
0
A
B12
N 12
r
..
T
B12
0
I
0
0
.
−
A
0
0
0
|B|−r
..
0
.
0
0
0
A N 22
0
Im−r
0
..
..
T
T
0
I
A
.
S
0
−
0
0
0
I
.
−
A
n
|N |−m+r
N 12
N 22
=: · · · · · · · · ·
···
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
:
..
..
˜
S B1
0
0
0
.
XB1
0
0
0
D1 . D2
.
..
0
0
0
S˜B2
0
0
XB2
0
..
0
0
.
0
0
X˜ N 1
0
0
SN 1
..
˜
.
0
0
0
SN 2
0
0
0
X N2
The values of the above Jacobian along the “tilde” path are denoted by JII (). Similarly as in the rst
approach, the nonsingularity of JII () is evident for ¿ 0.
Theorem 2. For any ¿0 the “tilde” path (xB (); x˜N (); s˜B (); sN ()) is a solution to the system (28); (29′ );
(31′′ ); (33′ ); (24)–(27) of 2n equations; and the nonsingularity of the Jacobian JII () extends to = 0.
Proof. To prove the nonsingularity of JII () at = 0 we use the following formula for the determinant of
block matrices
!
Z 1 Z2
(34)
= det(Z1 ) det(Z4 − Z3 Z1−1 Z2 );
det
Z3 Z4
where Z1 and Z4 are square matrices and Z1 is nonsingular. According to (34), det (JII ()) = det(M ), where
M := M () = D2 − D1 S. Through simple matrix manipulations we obtain
..
˜
˜
XB1
−S B1 AB12
.
0
−S B1 AN 12
..
..
˜
XB2 A TB12
.
M2 ()
S B2
.
0
0
M1 ()
···
··· :
···
···
···
···
···
M () =
= ···
..
..
.
M4 ()
−SN 1 A N 22
M3 ()
.
X˜ N 1
0
0
..
T
T
X˜ N 2 A N 12
0
.
X˜ N 2 A N 22
SN 2
Since M2 (0) = 0 and M3 (0) = 0, we have det(M (0)) = det(M1 (0)) · det(M4 (0)). We now apply formula (34)
to M1 (0) and M4 (0). In this way, we obtain
T
−1 ˜
S B1 A B12 )
det(M1 (0)) = det(XB1 )det(S˜B2 + XB2 A B12 XB1
T
−1 ˜
−1 ˜
S B2 + A B12 XB1
S B1 A B12 ) 6= 0;
= det(XB1 )det(XB2 )det(XB2
18
M. Halicka / Operations Research Letters 28 (2001) 9–19
T
−1 ˜
−1 ˜
S B2 are positive de nite and A B12 XB1
S B1 A B12 is positive
where the last relation holds because XB1 ; XB2 ; XB2
semide nite. Analogously, det(M4 (0)) 6= 0 and so det(JII (0)) 6= 0.
5. Conclusion
The results of Sections 3 and 4 imply the analyticity of the central path. In fact, the variables xB ; x˜N ; y; s˜B ; sN
and enter into both systems (described in Theorems 1 and 2) analytically and thus the application of the
analytic version of the implicit function theorem yields the analyticity of the “tilde” path as the function of
at any ¿0. Since xN () = x˜N (), sB () = s˜B (), the analyticity of xN () and sB () is evident.
For some applications it is often important to consider the central path (i.e. the solution to (1)) not only as
a function of the parameter ¿ 0 but also as a function of the weight vector ! ¿ 0. Since ! enters into the
systems described in Theorems 1 and 2 analytically, we immediately obtain the analyticity of (x(; !); s(; !))
at any ¿0 and ! ¿ 0. Obviously, the analyticity of the central path implies some weaker results on the
limiting properties of its derivatives and these limiting properties (together with the analyticity of the central
path at !) have applications to the analysis and construction of polynomial-time algorithms which are also
locally superlinearly or quadratically convergent. (For more details see e.g. [14].)
The proofs of analyticity of the central path for LP presented in this paper are simpler, not only than the
proofs of analyticity from [7,15], but even than the proofs of the weaker results on the limiting properties of
derivatives from [1,16,4]. For this reason, the proofs presented here can be useful both as a teaching and as
a research tool.
Acknowledgements
The author wishes to thank Milan Hamala for many stimulating discussions on the subject of this paper.
Thanks also to Pavol Brunovsky, Joseph Gruendler and two anonymous referees for their comments and
suggestions which resulted in the improvement of the readability of this paper. This work was supported in
part by VEGA grants 1/4302/97 and 1/7675/20.
References
[1] I. Adler, R.D.C. Monteiro, Limiting behavior of the ane scaling continuous trajectories for linear programming problems, Math.
Programming 50 (1991) 29–51.
[2] A. Goldman, A. Tucker, Theory of linear programming, in: H. Kuhn, A. Tucker (Eds.), Linear Inequalities and Related Systems,
Vol. 38, Princeton University Press, Princeton, NJ, 1956, pp. 53–97.
[3] C. Gonzaga, Path-following methods for linear programming, SIAM Rev. 34 (2) (1992) 167–224.
[4] O. Guler, Limiting behavior of weighted central paths in linear programming, Math. Programming 65 (1994) 347–363.
[5] O. Guler, D. den Hertog, C. Roos, T. Terlaky, T. Tsuchiya, Degeneracy in interior point methods for linear programming: A survey,
Ann. Oper. Res. 46 (1993) 107–138.
[6] O. Guler, C. Roos, T. Terlaky, J.-Ph. Vial, A survey of the implications of the behavior of the central path for the duality theory
of linear programming, Management Sci. 41 (1995) 1922–1934.
[7] M. Halicka, Analytical properties of the central path at boundary point in linear programming, Math. Programming 84 (1999)
335–355.
[8] B. Jansen, C. Roos, T. Terlaky, J.P. Vial, Interior-point methodology for linear programming: duality, sensitivity analysis and
computational aspects, K. Frauendorfer, H. Glavitsch, R. Bacher (Eds.), Optimization in Planning and Operation of Electric Power
Systems, Lecture Notes of the SVOR/ASRO Tutorial, Thun, October 14 –16, 1992, Physica–Verlag Heidelberg, 1993, pp. 57–123.
[9] L. McLinden, An analogue of Moreau’s proximation theorem, with applications to the nonlinear complementarity problem, Paci c
Journal of Mathematics 88 (1) (1980) 101–161.
[10] C. Roos, T. Terlaky, J.-Ph. Vial, Theory and Algorithms for Linear Optimization: An Interior Point Approach, Wiley, New York,
NY, 1997.
M. Halicka / Operations Research Letters 28 (2001) 9–19
19
[11] C. Roos, J.-Ph. Vial, Interior point methods, J.E. Beasley (Ed.), Advances in Linear and Integer Programming, Oxford Lecture Series
in Mathematics and its Applications, Vol. 4, Oxford University Press, Oxford, Great Britain, 1996, pp. 47–102.
[12] J. Stoer, M. Wechs, On the analyticity properties of infeasible-interior point paths for monotone linear complementarity problems,
Numer. Math. 81 (1999) 631–645.
[13] J. Stoer, M. Wechs, Infeasible-interior-point paths for sucient linear complementarity problems and their analyticity, Math.
Programming 83 (1998) 403–423.
[14] J. Stoer, M. Wechs, S. Mizuno, High order infeasible-interior-point methods for solving sucient linear complementarity problems,
Math. Oper. Res. 23 (1998) 832–862.
[15] M. Wechs, The analyticity of interior-point-paths at strictly complementary solutions of linear programs, Optimization, Methods and
Software 9 (1998) 209–243.
[16] C. Witzgall, P.T. Boggs, P.D. Domich, On the convergence behavior of trajectories for linear programming, Contemporary
Mathematics 114 (1990) 161–187.
[17] G. Zhao, J. Zhu, Analytical properties of the central trajectory in interior points methods, D.-Z. Du, J. Sun (Eds.), Advances in
Optimization and Approximation, Kluwer Academic Publishers, Dordrecht, 1994, pp. 362–375.
www.elsevier.com/locate/dsw
Two simple proofs for analyticity of the central path
in linear programming
Margareta Halicka
Department of Applied Mathematics, Faculty of Mathematics and Physics, Comenius University, Mlynska dolina,
842 15 Bratislava, Slovakia
Received 2 December 1998; received in revised form 1 January 2000
Abstract
Several papers have appeared recently establishing the analyticity of the central path at the boundary point for both
linear programming (LP) and linear complementarity problems (LCP). While the proofs for LP are long, proceeding from
limiting properties of the corresponding derivatives, the proofs for LCP are very simple, consisting of an application of
the implicit function theorem to a certain system of equations. Inspired by the approach for LCP, this paper gives two
simple ways of proving the analyticity of the central path for LP. One follows the idea for LCP, the other is based on a
c 2001 Elsevier Science B.V. All rights reserved.
proper partition of the system de ning the central path.
MSC: primary: 90C05; secondary: 90C33
Keywords: Linear programming; Interior point methods; (weighted) Central path; Limiting behavior; Analyticity
1. Introduction
Consider the following pair of dual linear programming (LP) problems:
(P)
min{cT x | Ax = b; x¿0};
(D)
max{bT y | AT y + s = c; s¿0};
where A is an m × n matrix, rank(A) = m ¡ n; c; x; s ∈ Rn ; b; y ∈ Rm . We restrict our attention to problems
satisfying the assumption
(AS)
there exist
x ¿ 0; s ¿ 0
such that
Ax = b;
AT y + s = c:
E-mail address: halicka@fmph.uniba.sk (M. Halicka).
c 2001 Elsevier Science B.V. All rights reserved.
0167-6377/01/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 0 0 ) 0 0 0 6 5 - 1
10
M. Halicka / Operations Research Letters 28 (2001) 9–19
In the context of interior point methods for solving (P) and (D), it is important to study the properties of certain interior point paths. We recall here the de nition and some properties of these paths, called
!-weighted central paths.
For any ! ¿ 0; ! ∈ Rn , de ne the following parameterized system
Ax = b;
x¿0;
AT y + s = c;
s¿0;
Xs = !;
(1)
where X := diag(x) and ∈ R; ¿0. Note that for = 0 system (1) gives necessary and sucient conditions
for optimality of both (P) and (D). Under assumption (AS), for any ¿ 0 there exists the unique solution
(x(); y(); s()) of (1) such that x() ¿ 0; s() ¿ 0. The set of points {x(); y(); s()}¿0 is called an
(!-weighted) central path. Due to the rank condition for A, we have a one-to-one correspondence between y
and s in (1). This enables us to omit y() from the de nition of an (!-weighted) central path.
The usual central path is the !-weighted central path corresponding to the weight vector ! = (1; : : : ; 1)T .
In this paper we consider the (!-weighted) central path corresponding to the arbitrarily chosen weight vector
! ¿ 0. We study the properties of this central path under the natural parameterization given in (1). That is,
we are interested in the properties of (x(); s()) as a function of ¿ 0.
It is easy to see that the central path has nice analytical properties. Actually, the function
Ax − b
G(x; y; s; ) = AT y + s − c
Xs − !
is real analytic and its Jacobian with respect to (x; y; s) is nonsingular at those points where all components
of both x and s are non-zero. Moreover, the central path satis es
G(x(); y(); s(); ) = 0
(2)
for each ¿ 0. Thus, by the implicit function theorem, the central path is analytic in for ¿ 0. That is,
it is in nitely di erentiable and the Taylor series of (x(); s()) for any 0 ¿ 0 converges to (x(); s()) at
a neighborhood of 0 . (More analytical properties of the central path for ¿ 0 were studied in [17,15].)
The limiting property of the !-weighted central path is that there exists a nite limit of (x(); s()) as
↓ 0, and this limit value forms a strictly complementary solution of (P) and (D). For the proof of this
assertion see [9,6,8]. Therefore, we can extend the domain of the central path (as a function of ) to the
closed interval [0; ∞) by
(x(0); s(0)) := lim (x(); s()):
↓0
(3)
Now, Eq. (2) holds even for = 0, but the study of the analytical properties of the central path becomes
more complicated since some information contained in system (1) vanishes at = 0. In fact, system (1) can
have many solutions at = 0, and the Jacobian, G ′ (x; y; s; 0), at these solutions is singular. Consequently, the
implicit function theorem does not apply to Eq. (2) at = 0. Thus e ort was concentrated on the analysis
of limiting properties of the kth derivatives, (x(k) (); s(k) ()), as ↓ 0. The existence of such limits was
established in [1,16] for k = 1 and, in [4] for any k¿1.
Recently, almost simultaneously, several papers have appeared proving the analyticity of the central path
at the boundary [7,12–15]. While two of them deal with LP problems [7,15], the others treat linear complementarity problems (LCP) where the linear programming problem is regarded as a special case. The proofs
for LP by Halicka [7] and Wechs [15] proceed from the investigation of the limits of explicit formulas for
high-order derivatives for ¿ 0 and also establish the geometric growth of the derivatives at the boundary.
M. Halicka / Operations Research Letters 28 (2001) 9–19
11
Although these proofs for LP are constructive and insightful, they are rather technical and quite long. On
the contrary, the proof by Stoer and Wechs [12] (see also [13]) for the monotone LCP is, surprisingly, very
simple. It consists of an application of the implicit function theorem to a system of equations, the existence of
which was deduced from the system of equations de ning the central path. Let us note that the construction
of that system is described in the follow-up paper by Stoer et al. [14].
Inspired by the approach for LCP, this paper gives two simple ways of proving the analyticity of the
central path for LP. The main idea of both proof techniques is common: we rewrite the system of equations
describing the central path, so that the corresponding Jacobian remains nonsingular as ↓ 0. One proof
technique (Section 3) follows the ideas for LCP from [12–14] and allows geometric interpretation. The other
(Section 4) uses a partition of A, rst described in [7]. Here, the system is rewritten in a di erent form and
the resulting Jacobian exhibits some (skew-)symmetric properties. A common conclusion of both these proof
techniques is given in Section 5.
2. Preliminaries
The main tool in developing limiting properties of the central path is the concept of an optimal partition
(B; N ) of (P) and (D). Recall that the optimal partition is a partition of the index set {1; : : : ; n} such that
every strictly complementary optimal solution (x; s) of (P) and (D) has the following property: xi ¿ 0; si = 0
for i ∈ B and xi = 0; si ¿ 0 for i ∈ N . For further details of the optimal partition concept see [2], or [11].
We use the notation xB and xN to refer to the restriction of any vector x ∈ Rn to the coordinate sets B and N ,
respectively. Correspondingly, the submatrices AB and AN are formed by taking the B and N columns of A,
respectively. Thus, we can write A = (AB ; AN ) by reordering the columns in A, if necessary. Finally, by |B|
and |N | we denote the cardinality of the index sets B and N , respectively. In this notation, we can rewrite
Eqs. (1) as
AB xB + AN xN = b;
ATB y + sB = cB ;
ATN y + sN = cN ;
XB sB = !B ;
XN sN = !N :
(4)
Since (x(0); s(0)) is strictly a complementary optimal solution of (P), (D) and (B; N ) is the optimal partition,
we have
xB (0) ¿ 0;
xN (0) = 0;
sB (0) = 0;
sN (0) ¿ 0:
(5)
We now use a technique, as used by Stoer and Wechs [12,13] in the context of LCP’s, and de ne the new
“tilde” variables
s˜B := XB−1 !B ⇒ sB = s˜B ;
x˜N := SN−1 !N ⇒ xN = x˜N :
(6)
Then, instead of (xB (); xN (); sB (); sN ()), we study the corresponding “tilde” path (xB (); x˜N (); s˜B ();
sN ()) for ¿0. It is easy to see that for each ¿0 the “tilde” path is the solution to the system
AB xB + AN x˜N = b;
(7a)
12
M. Halicka / Operations Research Letters 28 (2001) 9–19
ATB y + s˜B = cB ;
(7b)
ATN y + sN = cN ;
(7c)
XB s˜B = !B ;
(7d)
X˜ N sN = !N :
(7e)
More precisely, (xB (); x˜N (); s˜B (); sN ()) is even the unique solution to (7) such that xB () ¿ 0; sN () ¿ 0.
Moreover, by (5) and (6), we have s˜B (0) ¿ 0 and x˜N (0) ¿ 0.
The Jacobian of (7) is
AB 0
0
AN
0
0
0
0
IN
ATB
IN
0
0 ;
ATN
J (xB ; y; sN ; x˜N ; s˜B ; ) = 0
S˜
0
0
XB
B 0
0
0
X˜ N
SN
0
and its values along the “tilde” path are denoted by J (), i.e. J () := J (xB (); y(); sN (); x˜N (); s˜B (); )
for ¿0. It is easy to see that J () is nonsingular for ¿ 0, but,
at =
0, its nonsingularity depends on the
AB 0
is nonsingular, which is equivalent
m × |B| matrix AB . In fact, J (0) is nonsingular if and only if
0 ATB
to the two conditions |B| = m; rank(AB ) = m. Note that these conditions are also equivalent to the uniqueness
of the optimal solutions of (P) and (D) (see [5,10]). The case of |B| =
6 m or rank(AB ) 6= m is treated in the
next two sections.
3. The rst approach
As mentioned above, Stoer and Wechs proved the analyticity of the central path for the monotone LCP
in [12] and for the sucient LCP in [13]. The main idea of their proof resides in adding some equations to
the original system of equations describing the central path. Then, the enlarged (overdetermined) system of
equations has a Jacobian with full column rank, and the existence of a subsystem with a nonsingular Jacobian
is ensured. Thus, the analyticity follows by an application of the implicit function theorem. The study of this
problem was completed in [14], where the authors describe explicitly how to select a subsystem from the
enlarged system with a nonsingular Jacobian at = 0.
It is well-known that LP problems can be formulated in the LCP format. So, the results (found in [12–
14]) concerning the analyticity of the central path for LCP are also valid for LP. As mentioned above, the
analyticity of the central path was proven directly for LP in [7,15], but the proofs are long. Thus, the value of
[12–14] to LP is that the analyticity of the central path can be proved in a very simple manner, speci cally the
system of equations for the central path may be rewritten, so that the corresponding Jacobian is non-vanishing
at = 0. Obviously, the proof technique and the resulting system described in [12–14] depend on the LCP
format, so it is not clear what the proof and the resulting system would look like in terms of LP. In this
section the idea of the proof by Stoer and Wechs for LCP is applied directly to the LP problem (without the
reformulation of LP to LCP). Moreover, while the procedure of Stoer and Wechs is described in two di erent
steps ( rst adding some equations to the original system, then the selection of the subsystem), here the result
will be obtained in one step.
13
M. Halicka / Operations Research Letters 28 (2001) 9–19
Denote
P :=
AB
0
0
ATB
!
;
Q :=
AN
0
0
IB
!
and
q :=
b
cB
!
:
Then (7a) and (7b) can be rewritten as
!
!
x˜N
xB
+ Q
= q:
P
y
s˜B
(8)
Let rank(P) = r ¡ m + |B|. Then there exists a nonsingular (m + |B|) × (m + |B|) matrix W such that
!
W1
and W2 P = 0:
W=
W2
Here W2 is an (m + |B| − r) × (m + |B|) matrix, and we have
W2 w = 0 ⇔ ∃u:
w = Pu:
(9)
To see this, W2 P = 0 implies Col(P) ⊆ Null(W2 ). However, since W is nonsingular, rank(W2 ) = m + |B| − r,
which implies dim(Null(W2 )) = r. Now, since rank(P) = r, we get Col(P) = Null(W2 ), from which (9) follows.
Multiplying (8) by W on the left we obtain
!
!
xB
x˜N
(10a)
+ W1 Q
= W1 q;
W1 P
y
s˜B
W2 Q
x˜N
s˜B
!
= W2 q:
From (10b) at = 0 we get W2 q = 0. Thus instead of (10b) we have
!
x˜N
= 0:
W2 Q
s˜B
(10b)
(11)
We now have that for any ¿0, the “tilde” path is a solution to the system of Eqs. (10a), (11) and (7cde).
Denote JI () the Jacobian of system (10a), (11) and (7cde) with respect to (xB ; y; sN ; x˜N ; s˜B ), evaluated at
(xB (); y(); sN (); x˜N (); s˜B ()) and ¿0. It is easy to see that JI () is nonsingular for ¿ 0, since this
is a property of the Jacobian J () of original system (7). The next theorem summarizes these results and
establishes the nonsingularity of JI (0).
Theorem 1. For any ¿0 the “tilde” path (xB (); x˜N (); y(); s˜B (); sN ()) is a solution to system (10a); (11)
and (7cde) of 2n + m equations; and the nonsingularity of the Jacobian JI () extends to = 0.
Proof. It suces to prove that if JI (0) dh = 0 for some vector dh, then dh = 0.
Let dh = (d xB ; dy; dsN ; d x˜N ; d s˜B ), then the equations of JI (0) dh = 0 are
!
d xB
= 0;
W1 P
dy
W2 Q
d x˜N
d s˜B
!
= 0;
(12a)
(12b)
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M. Halicka / Operations Research Letters 28 (2001) 9–19
ATN dy + dsN = 0;
(12c)
S˜B (0) d xB + XB (0) d s˜B = 0;
(12d)
X˜ N (0) dsN + SN (0) d x˜N = 0:
(12e)
First, we consider (12a). Since W2 P = 0, we have WP
d xB
dy
!
= 0 and due to the nonsingularity of W we
obtain
P
d xB
dy
!
= 0:
(13a)
Further, we consider (12b). According to (9) applied to (12b), there exist u1 and u2 such that
!
!
d x˜N
u1
= 0:
+Q
P
d s˜B
u2
(13b)
Moreover, setting u3 = −ATN u2 , we have
ATN u2 + u3 = 0:
(14)
From (13a) and (13b) we obtain
!
!
d x˜N
u1 + d xB
+Q
=0
P
u2 + dy
d s˜B
for all ∈ R:
(15)
Thus, from (15), (12c) and (14) it follows that
AB (u1 + d xB ) + AN d x˜N = 0;
ATB (u2 + dy) + d s˜B = 0;
ATN (u2 + dy) + (u3 + dsN ) = 0
Since the vectors
!
u1 + d xB
d x˜N
and
for all ∈ R:
d s˜B
u3 + dsN
(16)
!
are orthogonal we have
u1T d s˜B + d x˜TN u3 + (d xBT d s˜B + d x˜TN dsN ) = 0
for all ∈ R
which implies
d xBT d s˜B + d x˜TN dsN = 0:
From (12d) and (12e) we now obtain
d s˜B = −XB (0)−1 S˜B (0) d xB
(17)
(18a)
and
dsN = −X˜ N (0)−1 SN (0) d x˜N :
(18b)
Substituting (18a) and (18b) into (17) yields
(19)
− d xBT XB (0)−1 S˜B (0) d xB − d x˜TN X˜ N (0)−1 SN (0) d x˜N = 0
−1 ˜
−1
˜
and since XB (0) S B (0) and X N (0) SN (0) are positive de nite, we obtain d xB =0 and d x˜N =0. Consequently,
(18a) and (18b) yield d s˜B = 0 and dsN = 0.
M. Halicka / Operations Research Letters 28 (2001) 9–19
15
Finally, by (12c) and dsN = 0, we get ATN dy = 0 and from (13a) we obtain ATB dy = 0. Now, the full row
rank of A yields dy = 0. Thus dh = 0 and the theorem is proved.
The results of this section are closely related to the geometric interpretation of the central path as described
by Adler and Monteiro [1] (see also [3]). By this interpretation the (!-weighted) central path is, at any ¿0,
the (!-weighted) analytic center of a certain set P(), where P(0) is the set of optimal solutions for (P) and
(D). In our case, we de ne P(); ¿0, as the set of variables xB ¿0; y; sN ¿0 satisfying (10a) and (7c)
where x˜N = x˜N () and s˜B = s˜B () are xed. Then it is straightforward to verify that Eqs. (10a), (11) and
(7cde) represent necessary and sucient conditions describing the (!-weighted) analytic center of P() for
all ¿0.
Since the analytic center of P(); ¿0, is de ned uniquely, the above interpretation implies the uniqueness
of the (positive) solutions of (10a), (11) and (7cde) for all ¿0.
4. The second approach
We now present an alternative procedure that yields a di erent algebraic description for the central path
with non-vanishing Jacobian. First we note that, prior to this section, the symbol := was used to de ne the
object on its left-hand side. In this section we also use the symbol =: to de ne the object on the right.
Let rank(AB ) =: r. From the rank conditions on A and AB one has that A can be partitioned in the form
(by reordering the columns in AB or in AN , if necessary)
"
..
AB11
A = [AB . AN ] =:
AB21
|{z}
r
where
R :=
"
AB11
AN 11
AB21
AN 21
#
AB12
AB22
|{z}
..
.
..
.
|B|−r
AN 11
AN 12
AN 21
|{z}
AN 22
|{z}
m−r
#
}r
}m − r;
|N |−m+r
is a nonsingular m × m matrix. Multiplying A by R−1 on the left, we obtain
.
.
A := R−1 A = R−1 [AB .. AN ] =: [A B .. A N ]
and also
.
A = [A B .. A N ] =:
Ir
A B12
0(m−r)×r
A B22
..
.
..
.
0r×(m−r)
Im−r
Note that A B22 = 0, since rank(AB ) = rank(A B ) = r and thus
#
"
"
#
0
A N 12
Ir A B12
;
A N =
:
A B =
0
0
Im−r A N 22
AN 12
:
A N 22
(20)
De ne b := R−1 b and y := RT y. Using the corresponding partition of all vector variables Eqs. (7) can be
written in the form
# " #
"
#
"
b1
x˜N 1
xB1
;
(21)
=
+ AN
AB
x˜N 2
xB2
b2
16
M. Halicka / Operations Research Letters 28 (2001) 9–19
T
A B
"
T
A N
"
y 1
y 2
y 1
y 2
#
+
#
"
+
"
s˜B1
s˜B2
sN 1
sN 2
#
#
=
=
"
"
cB1
cB2
cN 1
cN 2
#
#
;
;
(22)
(23)
XB1 s˜B1 = !B1 ;
(24)
XB2 s˜B2 = !B2 ;
(25)
X˜ N 1 sN 1 = !N 1 ;
(26)
X˜ N 2 sN 2 = !N 2 :
(27)
Substituting (20) into (21) – (23) yields
xB1 + A B12 xB2 + A N 12 x˜N 2 = b1 ;
(28)
x˜N 1 + A N 22 x˜N 2 = b2 ;
(29)
y 1 + s˜B1 = cB1 ;
(30)
T
A B12 y 1 + s˜B2 = cB2 ;
(31)
y 2 + sN 1 = cN 1 ;
(32)
T
T
A N 12 y 1 + A N 22 y 2 + sN 2 = cN 2 :
(33)
Here, vectors with indices B1; B2; N 1 and N 2 are the vectors of dimensions r; |B| − r, m − r and |N | − m + r,
respectively. Since we consider (29) for any ¿0, we have b2 = 0, and so the equation can be divided by
to give
x˜N 1 + A N 22 x˜N 2 = 0:
(29′ )
Eqs. (30) and (32) can be written in the form
y 1 = cB1 − s˜B1 ;
(30′ )
y 2 = cN 1 − sN 1
(32′ )
and thus the variables y 1 and y 2 can be eliminated from (31) and (33) to obtain
T
A B12 (cB1 − s˜B1 ) + s˜B2 = cB2 ;
(31′ )
T
T
A N 12 (cB1 − s˜B1 ) + A N 22 (cN 1 − sN 1 ) + sN 2 = cN 2 :
(33′ )
T
Eq. (31′ ) holds for any ¿0 and thus A B12 cB1 = cB2 and
T
− A B12 s˜B1 + s˜B2 = 0:
(31′′ )
M. Halicka / Operations Research Letters 28 (2001) 9–19
17
Now (28), (31′′ ), (29′ ), (33′ ) together with (24) – (27) form the full system of 2n equations for the 2n
variables xB1 ; s˜B2 ; x˜N 1 ; sN 2 ; s˜B1 ; xB2 ; sN 1 ; x˜N 2 . The Jacobian of this system with respect to the variables in the
given order is
..
I
0
0
0
.
0
A
0
A
B12
N 12
r
..
T
B12
0
I
0
0
.
−
A
0
0
0
|B|−r
..
0
.
0
0
0
A N 22
0
Im−r
0
..
..
T
T
0
I
A
.
S
0
−
0
0
0
I
.
−
A
n
|N |−m+r
N 12
N 22
=: · · · · · · · · ·
···
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
:
..
..
˜
S B1
0
0
0
.
XB1
0
0
0
D1 . D2
.
..
0
0
0
S˜B2
0
0
XB2
0
..
0
0
.
0
0
X˜ N 1
0
0
SN 1
..
˜
.
0
0
0
SN 2
0
0
0
X N2
The values of the above Jacobian along the “tilde” path are denoted by JII (). Similarly as in the rst
approach, the nonsingularity of JII () is evident for ¿ 0.
Theorem 2. For any ¿0 the “tilde” path (xB (); x˜N (); s˜B (); sN ()) is a solution to the system (28); (29′ );
(31′′ ); (33′ ); (24)–(27) of 2n equations; and the nonsingularity of the Jacobian JII () extends to = 0.
Proof. To prove the nonsingularity of JII () at = 0 we use the following formula for the determinant of
block matrices
!
Z 1 Z2
(34)
= det(Z1 ) det(Z4 − Z3 Z1−1 Z2 );
det
Z3 Z4
where Z1 and Z4 are square matrices and Z1 is nonsingular. According to (34), det (JII ()) = det(M ), where
M := M () = D2 − D1 S. Through simple matrix manipulations we obtain
..
˜
˜
XB1
−S B1 AB12
.
0
−S B1 AN 12
..
..
˜
XB2 A TB12
.
M2 ()
S B2
.
0
0
M1 ()
···
··· :
···
···
···
···
···
M () =
= ···
..
..
.
M4 ()
−SN 1 A N 22
M3 ()
.
X˜ N 1
0
0
..
T
T
X˜ N 2 A N 12
0
.
X˜ N 2 A N 22
SN 2
Since M2 (0) = 0 and M3 (0) = 0, we have det(M (0)) = det(M1 (0)) · det(M4 (0)). We now apply formula (34)
to M1 (0) and M4 (0). In this way, we obtain
T
−1 ˜
S B1 A B12 )
det(M1 (0)) = det(XB1 )det(S˜B2 + XB2 A B12 XB1
T
−1 ˜
−1 ˜
S B2 + A B12 XB1
S B1 A B12 ) 6= 0;
= det(XB1 )det(XB2 )det(XB2
18
M. Halicka / Operations Research Letters 28 (2001) 9–19
T
−1 ˜
−1 ˜
S B2 are positive de nite and A B12 XB1
S B1 A B12 is positive
where the last relation holds because XB1 ; XB2 ; XB2
semide nite. Analogously, det(M4 (0)) 6= 0 and so det(JII (0)) 6= 0.
5. Conclusion
The results of Sections 3 and 4 imply the analyticity of the central path. In fact, the variables xB ; x˜N ; y; s˜B ; sN
and enter into both systems (described in Theorems 1 and 2) analytically and thus the application of the
analytic version of the implicit function theorem yields the analyticity of the “tilde” path as the function of
at any ¿0. Since xN () = x˜N (), sB () = s˜B (), the analyticity of xN () and sB () is evident.
For some applications it is often important to consider the central path (i.e. the solution to (1)) not only as
a function of the parameter ¿ 0 but also as a function of the weight vector ! ¿ 0. Since ! enters into the
systems described in Theorems 1 and 2 analytically, we immediately obtain the analyticity of (x(; !); s(; !))
at any ¿0 and ! ¿ 0. Obviously, the analyticity of the central path implies some weaker results on the
limiting properties of its derivatives and these limiting properties (together with the analyticity of the central
path at !) have applications to the analysis and construction of polynomial-time algorithms which are also
locally superlinearly or quadratically convergent. (For more details see e.g. [14].)
The proofs of analyticity of the central path for LP presented in this paper are simpler, not only than the
proofs of analyticity from [7,15], but even than the proofs of the weaker results on the limiting properties of
derivatives from [1,16,4]. For this reason, the proofs presented here can be useful both as a teaching and as
a research tool.
Acknowledgements
The author wishes to thank Milan Hamala for many stimulating discussions on the subject of this paper.
Thanks also to Pavol Brunovsky, Joseph Gruendler and two anonymous referees for their comments and
suggestions which resulted in the improvement of the readability of this paper. This work was supported in
part by VEGA grants 1/4302/97 and 1/7675/20.
References
[1] I. Adler, R.D.C. Monteiro, Limiting behavior of the ane scaling continuous trajectories for linear programming problems, Math.
Programming 50 (1991) 29–51.
[2] A. Goldman, A. Tucker, Theory of linear programming, in: H. Kuhn, A. Tucker (Eds.), Linear Inequalities and Related Systems,
Vol. 38, Princeton University Press, Princeton, NJ, 1956, pp. 53–97.
[3] C. Gonzaga, Path-following methods for linear programming, SIAM Rev. 34 (2) (1992) 167–224.
[4] O. Guler, Limiting behavior of weighted central paths in linear programming, Math. Programming 65 (1994) 347–363.
[5] O. Guler, D. den Hertog, C. Roos, T. Terlaky, T. Tsuchiya, Degeneracy in interior point methods for linear programming: A survey,
Ann. Oper. Res. 46 (1993) 107–138.
[6] O. Guler, C. Roos, T. Terlaky, J.-Ph. Vial, A survey of the implications of the behavior of the central path for the duality theory
of linear programming, Management Sci. 41 (1995) 1922–1934.
[7] M. Halicka, Analytical properties of the central path at boundary point in linear programming, Math. Programming 84 (1999)
335–355.
[8] B. Jansen, C. Roos, T. Terlaky, J.P. Vial, Interior-point methodology for linear programming: duality, sensitivity analysis and
computational aspects, K. Frauendorfer, H. Glavitsch, R. Bacher (Eds.), Optimization in Planning and Operation of Electric Power
Systems, Lecture Notes of the SVOR/ASRO Tutorial, Thun, October 14 –16, 1992, Physica–Verlag Heidelberg, 1993, pp. 57–123.
[9] L. McLinden, An analogue of Moreau’s proximation theorem, with applications to the nonlinear complementarity problem, Paci c
Journal of Mathematics 88 (1) (1980) 101–161.
[10] C. Roos, T. Terlaky, J.-Ph. Vial, Theory and Algorithms for Linear Optimization: An Interior Point Approach, Wiley, New York,
NY, 1997.
M. Halicka / Operations Research Letters 28 (2001) 9–19
19
[11] C. Roos, J.-Ph. Vial, Interior point methods, J.E. Beasley (Ed.), Advances in Linear and Integer Programming, Oxford Lecture Series
in Mathematics and its Applications, Vol. 4, Oxford University Press, Oxford, Great Britain, 1996, pp. 47–102.
[12] J. Stoer, M. Wechs, On the analyticity properties of infeasible-interior point paths for monotone linear complementarity problems,
Numer. Math. 81 (1999) 631–645.
[13] J. Stoer, M. Wechs, Infeasible-interior-point paths for sucient linear complementarity problems and their analyticity, Math.
Programming 83 (1998) 403–423.
[14] J. Stoer, M. Wechs, S. Mizuno, High order infeasible-interior-point methods for solving sucient linear complementarity problems,
Math. Oper. Res. 23 (1998) 832–862.
[15] M. Wechs, The analyticity of interior-point-paths at strictly complementary solutions of linear programs, Optimization, Methods and
Software 9 (1998) 209–243.
[16] C. Witzgall, P.T. Boggs, P.D. Domich, On the convergence behavior of trajectories for linear programming, Contemporary
Mathematics 114 (1990) 161–187.
[17] G. Zhao, J. Zhu, Analytical properties of the central trajectory in interior points methods, D.-Z. Du, J. Sun (Eds.), Advances in
Optimization and Approximation, Kluwer Academic Publishers, Dordrecht, 1994, pp. 362–375.