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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Determination of Optimal Vertices from Feasible
Solutions in Unimodular Linear
Shinji
Programming
Mizuno, Romesh Saigal and James B. Orlin
Working Paper No. 3231-90-MSA
December, 1990
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Determination of Optimal Vertices from Feasible
Solutions in Unimodular Linear
Shinji
Programming
Mizuno, Romesh Saigal and James B. Orlin
Working Paper No. 3231-90-MSA
W.I.T.
December, 1990
uBwras
Determination of Optimal Vertices from Feasible Solutions in
Unimodular Linear Programming
Shinji
MIZUNO
Department of Prediction and Control
The
4-6-7
Institute of Statistical
Mathematics
Minami-Azabu Minato-ku, Tokyo 106 Japan
Romesh SAIGAL
Department
of Industrial
The
Ann
and Operations Engineering,
University of Michigan,
Arbor, Michigan 48109-2117,
ORLIN
James B.
USA
*
Sloan School of Management,
Massachusetts Institute of Technology,
Cambridge,
MA
02139,
USA
August 1989, revised November 1990
Abstract
In this paper
modular,
we consider a
sind the other
linear
programming problem with the underlying matrix
uni-
data integer. Given arbitrary near optimum feasible solutions to the
primal and the dual problems, we obtain conditions under which statements can be
made about
the value of certain variables in optimal vertices. Such results have applications to the prob-
lem of determining the stopping
scaling
method and
Key words:
criterion in interior point
the path following
methods
for linear
methods
like
the primal-dual affine
programming.
Linear Programming, duality theorem, unimodular, totally unimodular, interior
point methods,
Abbreviated
title:
*This author's research
is
Feasible solutions
partially
supported by
and optimal vertices
NSF
grant
DDM-8921835 and
Airforce Grant
AFOSR-88-0088
1
We
Introduction
consider the following linear programming problem:
Minimize
c^x,
subject to
Aa;
a;
where
0),
A
and
an unimodular matrix
is
(i.e.,
>
6,
0,
A
the determinant of each Basis matrix of
c are integers. For such linear programs,
6,
=
is
it
known that
well
all
is
—1,
or
1,
extreme vertices are
integers.
In this paper
we consider the problem of determining optimal solutions of
from information derived from a given pair of primal and dual near optimum
example of such a
result
the strong duality theorem which asserts that
is
value of the given primal solution
is
this linear
program
feasible solutions.
An
the objective function
if
equal to the objective function value of the given dual solution,
then we can declare the pair to be optimal for the respective problems. Here we investigate the
problem of determining optimal vertices of the two problems given that the difference
objective function values
(
i.e.,
the duality gap
)
is
greater than zero.
unimodular systems, under the hypothesis that the duality gap
we obtain
a result
(
program
is
is
)
is
that
unique, then the
if
the duality gap
optimum
These results have applications
in
is
less
the
For the special case of
small
(
not necessarily zero
results that assert the integrality of variables in optimal solutions.
Corollary 3
in
An example
),
of such
than 1/2, and the optimum solution of the
vertex can be obtained by a simple rounding routine.
determining stopping rules
in interior
study of these methods was initiated by the seminal work of Karmarkar
point methods.
Our
[2].
The
results are
particularly applicable to the methods which work in both the primal and the dual feasible regions.
These include the methods of Kojima, Mizuno and Yoshise
and Ye
[14].
These
results can also be used in the primal
objective function value
objective function
integers)
when the
it
is
is
available; and,
to the dual
Monteiro and Adler
[9],
Saigal
[10],
methods where a lower bound on the
methods where an upper bound on the
available. In case the data of the linear
program
is
integral
(
i.e.,
A,
6,
c are
can be shown that an optimum solution of the linear program can be readily identified
duality gap
becomes smaller than 2~^'^', where L
to code all the integer
for the
[3],
is
the size of the binary string needed
data of the linear program. Compare this to
tiie
result just
quoted above
unimodular systems, which include the transportation and assignment problems.
such systems,
first
such results were obtained by Mizuno and Masuzawa
1
[8]
in the
For
context of the
transportation problem, Masuzawa, Mizuno and Mori
problem, and Saiga!
[11] in
show how
the context of the
minimum
cost flow
the context of the assignment problem.
After presenting the notation and assumptions
results that
in
[6]
section
in
to identify an optimal solution
in
2,
section 3
we prove our main
from the duality gap; and,
in section 4,
we
present the concluding remarks.
2
We
Notation and Definitions
consider the following primal and dual linear programming problems.
(P)
(D)
Minimize
c^x,
subject to
a;
Maximize
b y,
subject to
(y, z)
G Gx = {x
Ax =
:
e Fy^ =
b,x > 0}.
{{y, z)
A^ y +
:
z
=
c,
2
>
0}.
Throughout the paper, we impose the following assumptions on (P) and (D).
Assumption
1
The vectors b and c are integral and the matrix
Assumption 2 The
From Assumption
tion 2
feasible regions
1, all
Gx and Fyz
are
linear
unimodular.
is
nonempty.
Gx and Fyz
the vertices of the polyhedral sets
and the duality theorem of
A
are integral.
programming, the problems (P) and (D) have optimal
solutions and their optimal values are the same, say v'
.
Let
Sx and Sy^ denote the optimal
solution sets of (P) and (D), respectively:
We
We
Jx
=
Sx
=
{xeGx-.
Syz
=
{{y,z)GFyz:b'^y=v'}.
define the orthogonal projective sets of
v'},
Fy^ and Sy^ onto the space of
Fz
=
{z:{y,z)eFyz},
Sz
=
{z:{y,z)eSyz}-
also define
Gxix°)
From Assump-
= {xGGx:
L.c°J
<
X,
<
[j;°]
for each
j}
z:
for
each x° G
G^ and
F,(z°)
E F^, where
for each z^
[xj
= {zeF,:
and
[x]
<z,<
[z^j
for each
[2°]
j}
denote the largest integer smaller than or equal to x and the
smallest integer larger than or equal to x, respectively.
The Main Results
3
Suppose
common optimum
denotes their
ex' —
between
[x']
is
some primal
x' denotes
v'
,
Theorem
objective function value.
(the measure of non-optimality of x'
In particular, one part of the
.
some dual
feasible solution, z' denotes
theorem asserts that
—
an optimal integer solution x'(j) such that Xj(j)
to establish that in the case the optimal solution
is
if
—
fx']
.
[x' J is less
These
unique, and
and
v*
2 developes a relationship
and the distance of
),
x'
feasible solution,
x'.
from
ex' — v"
than
[x'J
and
then there
results are used in Corollary 3
ex' —
v*
<
1/2,
rounding x' to
the nearest integer gives the optimal solution.
Except
the case that the optimal solution
in
guarantee that there
is
X* with the property that ij
Theorem
6 gives
=
Assume
•
•
•
,
It
0.
to the dual
1,
Theorem
a solution x* obtained by rounding each
x' can be simultaneously rounded. In particular.
region
unique.
and Corollary
2
componant of
x'
3
do not
to the nearest
However, Theorem 4 gives conditions under which some componants of a feasible solution
integer.
have Xj
is
bounds on a primal
satisfy Zj
=
0.
is
x'j
feasible solution
We now
that feasible solutions x°
Gx{x^)
m)
the closest intger to
bounds on a dual
also gives
must
is
Theorem
z'j
4 states that there
for all
j
satisfying
|
x*
is
an optimal solution
—
""'
x'j
\<
—
'
Z^
.
under which an optimal solution x* must
feasible solution x'
under which an optimal solution
prove these theorems.
G Gx and {y^,z^) G Fyz
are available.
Since the feasible
a bounded polyhedral convex set and x^ G Gj:(^x^), there exit vertices u'
of Gi:(x°) such that
[i
=
m
i=l
m
«=i
A,
where
m
vertex u'
<
is
1
+
dim(Gx(a;''));
>0
for
f=
l,2,---,m.
dim(5) denotes the dimension of the
integral (see, for example, Schrijver [12]), so
set S.
By Assumption
1,
each
for
t
not.
=
We
1, 2,
•
•
•
,
m
and j
=
1, 2,
•
•
•
,
Some
n.
u"s
of the vertices
divide the vertices into two index sets Iq and
are optimal, but the others are
(possibly /q
/^v
=
<?!>
or
/yv
=
<P)
and rewrite
the relation above as follows:
^
a;0=
^
A.«'4-
x:a.+^
A,
>
A.m',
(1)
'&Jn
>6/o
for
i
=
A.
i,
€ /o U In,
where
/o
=
{i
:
Ij.^
—
\^i
:
«'
e
VL
^ SX
Sj:,
Similarly, there exist optimal vertices iw'
i«'
€
G Jn)
i^z(2'')\'S3 (i
Kruskal
[1])
1,2,-
i
=
1, 2,
€ 5^
fl
•
-.m},
•
•
,
m}.
Fj(z°)
(i
€ Jo) and nonoptimal
established
is
in
Theorem
1,
vertices
Hoffman and
such that
^
^
=
ix,w^
/i,+
Hi
1
-
(integrality of these vertices
z°
Theorem
,
i
Let x^
>
6 Gx and (y°,z°) G
x^ and z^ are expressed as (1) and
for
+ Y.
Yl
i
M.
(2)
M."'''
=
1-
E Jo U Jn-
i^yz, airf /e< f' be the
(2), respectively.
optimal value of (PJ. Suppose that
Then we have
ieJN
Proof:
We
easily see that
Jx° -V* =
where the
last inequality follows
Y
^.c^«'
from c^u'
=
+
Y
^.c^«'
v' for each
i
-
^*
G lo and c^u' >
v*
+
1
for
each
i
G
/yv-
same way, we
In the
from the
follows
first
also have the second inequaUty of the theorem.
The
third inequahty
two inequalities and
=
+ (u'-6^y°).
(c^x°-i;')
D
The above theorem can be used
Theorem
(a) If
is
x'j
=
X*
J
2 Let x° G Gx, and
integral
and c x
v'
lower bound on the optimal value v* of (P).
he a
<
then there exists an optimal solution x* G
1
—
v'
<
x^
(cj If c
x°
—
v'
<
[xj]
Suppose that
If Xj is integral
—
|_XjJ
—
Xj
a;° is
then
u'
then there exists an optimal solution x*{j)
u'
=
cJ^O
x
—
„i
t;
<
1
expressed as
=
x^ for each
then Iq
/
i.e.,
4>,
U
i.
Iq
=
then
4>
Theorem
Jx^ -V* >Y,^' =
^
[xj] for each
x*
£,
=
5^ such
that x*{j)
=
1
implies
1-
there exists an optimal solution u'
Thus
(b) x° cannot be integral.
rx°i
X*
that x^(j)
(1).
x°
Under the condition of
If
that
G Sx such
then there exists an optimal solution x''{j) G
-v' >
if
S^ such
J
x^
Hence,
solution.
x^.
(h) If c
Proof:
let v'
—
some information about an optimal
to obtain
=
[x'jj
+
1.
S^, we have
u'j
^
[xjl
for
each
i
G
lo,
u'j
=
[x°J
for each
i
€
Iq-
or equivalently
{i
G lo) such that
Then we
see
'&In
16/0
Hence we have
In the
As a
<
[x°J
+
(c^x° -
t)')
<
L.oj
+
(c^x° -
O-
1)
(b).
same way, we can prove
(c).
special case of the above theorem,
we
get the following useful result.
Corollary 3 Let x° e Gx and (y°,2°) G Fy^.
x*(j)
Theorem
(by
G Si such
If {x°)'^ z°
<
1/2, for each j,
there exists an
that
'
-'(J)
=
L^?J
^/
-^^
-
L-^?J
[xfl
«/
x]
-
<
1/2,
[xOj
>
1/2.
€
S-,;,
(3)
In case the problem (P) has a unique optimal solution x*
we can compute each coordinate
of the optimal solution by (3).
Proof:
If
x^
is
integral.
the case where Xj
is
Theorem
not integral.
-
rx°l
Since 6 y°
x°-
-
|_x^J
is
>
a lower bound of
1/2,
Theorem
Theorem
If x^
x°
t;*,
>
—
[x°J
1/2
>
<
1/2,
=
[xjj for
an x* ^ Sx- So we only consider
we have
(x°)^z°
= c^x° -
6^y°.
by Theorem 2(c), there exists x* G 5^ such that i^
=
[x^J
.
If
we have
Xj
Hence, by
2(a) implies that x*
—
[XjJ
>
1/2
2(b), there exists x'
2 gives information
>[x)z=cx— by.
G S^ such that
ij
=
\xj].
about an element of an optimal solution. The next theorem shows
a relation between a feasible solution and coordinates of an optimal solution.
Theorem 4
Let x°
G Gx, and
let
v* be the optimal value of (P). If
cF x^
—
w*
<
1,
there exists
an optimal solution x' G 5^ such that
r
Ol
[xJ for each
I
-
J.
•
j
^ Ij
f
e
•
:
x^^
-
I
Ol
L^°J
<^
l-(C^X<'-t>')
,l^^S.)'
(4)
<
[xO]
for each j e \j
:
\x^]
-
x"
<
^
^l^,^^^ ^^'
Proof:
the
Suppose that
number
is
a;
expressed as
of optimal vertices
where we may assume without
than or equal to
less
is
(1),
#/o <
By Theorem
is
an index
i'
=
A,
1
- Yl
(1),
we
dim(Sj):
dim(S^).
>
A.
1
- {Jx" -
v').
G Iq such that
l
From
generaUty that
we have
1,
^
Hence there
+
l
+
1
loss of
+
dim(S^)
see
Hence we obtain
'
Since the vertex u'
CoroUary
^'
is
G^ and
.0.0 .
^Vi <
system has
-
-
^
(1
'
<
L^;J
—
w'
(y°, a°)
€ Fy^.
+
dim(S^))(l
In the
same way
and each solution
for each j £
z,
=
as
UJ
Theorem
for
each
^''-
f^^y"!
K=
j
:
x'-^
4,
J
(6)
<
{c^x°
^
-
V*)
...
,„
+ dim(Sx)
(7)
.
l
there exists a (y*, r*)
G
(5)
,,
0,
1-
J
<
for each
optimal value) and
an optimal solution of (P):
is
u >
b,
("= w*; <Ae
dim(S,))
,
I
Proof:
=
'
+
a solution
=
// [c^x^J
-{c'x^-v*)){\-{v*-h'rf ))
~:\ lj::;.: w^r:..
(l
^'
0|.
satisfies (4).
Au =
Uj
+ dim(5.) .o_,
l
- l-(c^xO-t;-)^'
A.-,
integral, x"
5 Lei x° £
ihe following
^°
<
„.'_U0,
G Syj such that
- b^rj
i+d.m(5.)
+ dim(5,)
jv'
:
.,
-
[^^J
<
"
^^^
From
and the
(5)
definition of
K
we
,
see that
'
From
and
(4)
Since z* €
l
diiTi(Sj)
x* G S^ and (y*, z*) £ Sy^ such that
(8), there exist
G^ and
+
x*
(9) holds,
is
I*
=
for each
z'
=
for
each j ^
= by*,
from which
Now we show
some coordinates of
and (D) are
vertices of (P)
Theorem
that
follows that u*
it
K
(10)
is
(7),
an optimal solution of (P).
all
=
then (7) and (10) imply u*^ z*
0,
or
n
the optimal vertices can be fixed
when
feasible
available.
€ G^ and (y
6 Let x
(9)
the solution of the system (6) and (7).
Let u* be any solution of the system (6) and
c^u'
E K,
j
,
z
)
6 Fyz,
owrf let
v'
and v"
be a
lower bound and upper bound
of the optimal value v' of (P), respectively.
—
(a) If
z'j
>
(b) If
x'j
> c^x^ —
v"
b y°, then
v',
then
x'j
=
zj=0
for any optimal vertex x'
£ S^-
for any optimal vertex (y', z*) G Sy^.
Proof: Let x* G Sx be any optimal vertex of (P), then we see
x'jz]
Uz]>v" -b^y°,
< x*^^° = x'^ic -
we have
^i
Since x^
is
In the
4
integral,
we obtain
same way, we
<
A^y'')
= v'-
^
<
b^y""
<
v"
-
b'^y°.
1-
(a).
also have (b).
Concluding Remarks
In this paper
we obtained
results
under the assumption that the linear program (P)
standard from. In the case the problem
is
given in inequality from,
we can derive
of section 3 with the added assumption that the matrix be totally unimodular.
8
all
is
in the
the results
Similar results
can also be derived for other forms of the problem,
i.e.,
problems with upper and lower bounds on
variables, etc.
These results have implications
methods. This
programming problems
via interior point
be a topic of a subsequent paper.
will
Acknowledgement:
co-editor,
for solving integer
The authors would
and an anonymous
thank Professor Masakazu Kojima, the
like to
referee for their helpful
comments.
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A.
J.
Hoffman and
Kuhn and
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