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PARAMETRIC LINEAR PROGRAMMING AND
ANTI-CYCLING PIVOTING RULES
by
Thomas L. Magnanti*
James B. Orlin**
OR 144-85
*
**
October 1985
Support in part by grant ECS-83/6224 from the Systems Theory and
Operations Research Division of the National Science Foundation.
Support in part by Presidential Young Investigator grant
8451517-ECS of the National Science Foundation.
ABSTRACT
The traditional
perturbution
resolving degeneracy in
that eliminate ties in
restrict
(or lexicographic)
linear programming impose decision rules
the simplex ratio rule and, therefore,
the choice of exiting basic variables.
combinatorial
variables.
methods for
Bland's
pivoting rule also restricts the choice of exiting
Using ideas from parametric linear programming, we
develop anti-cycling pivoting rules that do not
exiting variables beyond the simplex ratio rule.
limit the choice of
That is, any
variable that ties for the ratio rule can leave the basis.
A
similar approach gives pivoting rules for the dual simplex method
that
do not restrict
the choice of entering variables.
The primal simplex method for minimization problems permits an
entering variable at each iteration to
the exiting variable to be any
negative reduced cost and permits
variable that satisfies
any
implementation of
the minimum ratio rule.
is
methods
for resolving degeneracy.
nondegenerate.
As
is well-known,
is guaranteed to converge if
the procedure
problem
perturbation
be any variable with a
the
In addition, there are two well-known
(or equivalently,
The first of these,
the
method,
the lexicographic)
avoids
cycling by refining the selection rule for the exiting variable
(Charnes
1952],
Dantzig
rule,
the combinatorial
[1951],
Wolfe [1963]).
developed by Bland
The second method,
[1979],
avoids cycling by
refining the selection rule for both the exiting and entering
variables.
The situation raises the following natural question:
there a simplex pivoting procedure
not
restrict the minimum ratio
this note,
for avoiding cycling that does
rule choice of exiting variables?
cycling by refining the selection
that
In
we answer this question affirmatively by describing an
anti-cycling rule based on a "homotopy principle"
variable.
Is
that avoids
rule for only the entering
We also describe an analogous
avoids cycling by refining only
dual pivoting procedure
the choice of
exiting
variables.
Our procedures are based upon a few elementary observations
concerning parametric simplex methods.
some
importance in
some theoretical
These observations may be of
their own right, since they may
shed light on
issues encountered in several recent analyses
2
of average case performance of parametric simplex methods
(e.g., Adler
[1983]).
[1983], Borgward
[1982], Haimovich [1983],
Smale
In particular, whenever the probability distribution of a
parametric linear program is chosen, as is frequently the case, so
that the problem satisfies a property that we call dual
nondegeneracy, then the parametric algorithm converges finitely even
if it is degenerate.
Parametric Linear Programming
Consider the following parametric linear programming problem:
(c. + ed)x
Minimize
Ax = b
subject to
x
where A is
>
P(e)
0
an mxn constraint matrix with (for notational
convenience) full row rank.
For a given value e, we say that P(e)
is nearly dual nondegenerate if for each primal feasible basis B
there
is at most one nonbasic variable x i whose reduced cost
ci + edi
is 0.
We say that the parametric problem P is dual
is nearly dual nondegenerate for all eR.
nondegenerate if P(e)
Consider the usual parametric simplex algorithm for solving
P(e)
for all values of
, starting with a basis that is optimal for
all sufficiently large values of e.
In the case that P is dual
nondegenerate, we show that the procedure will not cycle
any perturbations).
(without
We then apply this result to give new primal
simplex pivot rules that (1) are guaranteed to avoid cycling, and
3
(2)
rely only on the selection of the entering variable;
i.e.,
any
basic variable satisfying the minimum ratio rule may leave the
basis.
The following procedure is a version of the usual parametric
simplex method as applied to a minimization problem.
Begin
let
B0
let
i=1;
while
be an optimal
d
K
basis for P(e)
for all
e
> 00;
0 do
begin
let B = Bi-1;
c,
let
d denote the reduced costs for the vectors c and d
with
select
respect
index s so
to basis B;
that -cs/ds = max
(-cj/dj:dj
> 0);
let ei = -/ds;
if B-lA s
< O then quit
for all
else
let Bi
(since P(e)
is unbounded from below
e < e');
be obtained from B by pivoting in x
pivoting out any variable chosen by the usual
pivot
rule
(i.e.,
ties
in the
arbitrarily);
let
i =
i+1;
end.
end.
4
and
simplex
ratio rule can be broken
We note
if d < 0 at any point
that
c + ed > c + ei-ld for all e < e i
< ei- 1.
for all
-1
in
then
this algorithm,
and hence B is an optimal basis
e
The iterative modification of
in the parametric
programming procedure can be conceptualized differently as expressed
in the following observation.
REMARK
for
that B=Bi -
Suppose
1.
*
some
and let e i
be
is an optimal basis
to
parametric algorithm when applied
and d are defined by B).
PROOF.
If e < e i ,
by our choice
of
Then e i
then cs + e
"while loop" of
this basis B
= min
(:
c
(consequently,
for P(e)).
B is optimal
< 0 and B is non-optimal.
d
the
Also,
ei,
cj
But then each cj +
+ ei dj
Cs
+ eids
C
+
if dj > 0
>
idj > O,
since c
+
for P(e*).
because B is an optimal basis
every variable
in the
selected as
for problem P(e*)
<
.
ids = 0 and cj
+
*dJ
if
dj
Since cj
j corresponding to a column from B,
+ eidj
0
*dj
= 0 for
B is optimal
for
p(ei)).
Let B
be an optimal
basis for
be defined recursively as in
Ci =
s
into B i- I
and let
i and Bi
the parametric procedure.
reduced cost with respect
pivoted
P(e ° )
to
to the basis
obtain Bi.
5
Bi-
1
for i > 1
Moreover,
let
of the variable
(i)
Bi
(ii)
If P(e)
(iii)
c1
Part
PROOF:
that
Let
is optimal
is
(i) is
in
[ei+l,
all
i such that e i
- 1
i
are optimal at
so that
i be selected
ei+ 1
cp +
>
0 and dp < 0
dp > 0 for
implies that ct
+
contradicting the
To see
<
i).
i+l
dt
+ ei dt
i < ei
or else
and
Also
obtain Bi.
to
xp
Then xt
i dp = 0 and
the assumption ei+l
=
is
p + e
i
dp =
= ei
0,
degeneracy of P(ei).
ei
note that
This proposition
i = 0
1
to obtain Bi+
8
Bi
(:
via a contradiction.
respec't to Bi.
Moreover,
= ct
(ii)
basis Bi -1
(dp < 0 since cp +
near dual
(iii),
ei;
and the fact that
into basis Bi
the reduced cost for d with
dt
,
i and either
let xp be the variable pivoted out of
because
<
> 0.
We prove
=
Let xt be the variable pivoted
let d be
i+ 1
then
a consequence of our previous remark, the fact
is an interval.
for P(e)}
ei];
dual nondegenerate,
< 0 for
both Bi and B i
optimal
i > O,
For all
1.
PROPOSITION
=
-ci/di
s
implies that
S
and di
S
> O.
the sequence of
pivots generated
by the parametric algorithm defines a simplex algorithm for the
nonparametric problem minimize
{cx: Ax = b, x > 0).
We record this
result as follows:
COROLLARY.
the
Let BO, ...
,
parametric algorithm, and
less than
O.
Bt be a sequence of bases obtained by
let
t be
Then
6
first
index for which
t +l
is
- 1
optimal for min
(i)
Bt is
(ii)
the bases BO,
(iii)
the pivot
rule,
...
,
sequence satisfies
> O)
iteration of
the usual
O};
pivot selection
3
if we replace
remains valid
Note that Proposition
(-cjldj:dj
>
the entering variable has a negative reduced
viz.,
nondegeneracy with
b, x
Bt are distinct;
cost for cost vector c.
REMARK 2.
Ax =
{cx:
the slightly weaker assumption
is unique and hence
the index s is
dual
that the argmax
unique at each
the parametric algorithm.
An Application to a Primal
Pivot Rule
We next show how to apply the previous proposition to define
primal
pivoting rules
that,
without any special
choosing the exiting variable,
Our
provision for
avoid cycling.
previous results demonstrate that the parametric
method defines
an anti-cycling pivot
minimize
subject
simplex
sequence for
cx
to
Ax = b
(P)
x >O
whenever we can choose the objective
function coefficients c and d so
that
(i)
some basis B of A is optimal
values
(ii)
of
P is dual
To establish these
for all sufficiently large
, and
nondegenerate.
two criteria,
let B be any feasible basis
7
for
(P)
corresponding
B and with positive components
columns of
to the
of N will
corresponding to the columns
ensure that P is dual nondegenerate, however,
To
(i).
satisfy property
that we avoid
requires
ties in computing the entering variables from the parametric ratio
test -Cs/d
s
= max
the nonbasic
>
small
for some
One natural
the following parametric objective
c +
(2)
c + 8e(N + IN),
(3)
c +
N with
=
these objective
and
N denotes a vector with zero components
in B and with components
in N in,
say,
1, and
(1/£)N
choice M =
order
from A.
= 6 N with
1/c.
The
n - m
Also
first two of
The third objective
1/c.
for sufficiently
for all 0
large values of
each one defines an anti-cycling pivot rule
nondegenerate.
=
...
big M method alluded to prior to the last
For each of these objective functions,
unique optimal basis
2
,
their natural
functions perturb c and d.
function is the polynomial
display with the
functions:
(l1/)N-
corresponding to columns
IN =
procedures
These
+ 8 IN,
(1)
all columns
of
cost coefficients of d as
choose the nonbasic
In these expressions,
for
from the
columns of c or d by distinct powers
distinct powers of M for some large constant M.
N
lead
following the
is to use
Alternatively, we might use a variant of the
O.
familiar big M method:
lead to
approach
perturbation theory of linear programming, we
(nonparametric)
might perturb
0)}.
>
For example,
perturbations of c or d.
usual
dj
(-cj/dj:
< 1, B
<
.
is a
Therefore,
if P is dual
To demonstrate this property, we first establish a
preliminary result by a modification of the usual
argument.
8
perturbation
PROPOSITION 2
Let A 1,
...
,
(D 1 , D 2 ,
D
A
be the columns
...
,
D)
some basis B.
of
Suppose that
is any other basis, possibly containing
Suppose that A =
that h =
[B,N],
N and
some columns
of B.
that h = h -
hDD-1A (hD is the subvector of h corresponding to
the columns
in D.)
polynomials
in
indices
D1,
D2 ,
are distinct nonzero
Then hi and h
with zero constant terms whenever i and j are
corresponding to distinct columns of A other
.. ,
than
D.
PROOF:
the columns
of D and consider
containing an (n-m) x
[0,I]
First,
independence argument.
We use a linear
the following partitioned matrix:
0
I
CD
B
N
D
(n-m) identity matrix I, and a submatrix CD of
corresponding to the columns D from
[B,N].
copy of a column of
the identity matrix I.
matrix M has
rank of n.
£n-m).
Let Q =
Let
Then h =
[O,I].
We observe that the
the vector
denote
Q.
CB
CN
0
D-lB
D-1N
I
=
9
(1,
2,
Next consider the following
matrix obtained by pivoting on the basis D in M:
M'
Therefore, each
the submatrix 0 or a
column of CD is either a copy of a column of
linear
let us duplicate
.
....
= [CB,
Let Q'
is obtained from M by pivoting.
row rank n, and M'
m columns of M'
row rank n-m because M has full
Then Q' has full
CN].
are copies of
Moreover,
the first n columns.
columns of
column of Q'
corresponding to each of these "original
a 0 vector.
Deleting these n zero vectors leaves an
nonsingular
The
must be
columns"
(n-m) x
last
(n-m)
submatrix Q''.
Finally,
observe
Consequently, the
two
that h = h -
hDD-1A
=
polynomials hi and hj
-
Q
CDD-1A =
and are thus distinct
Q''
elements of the vector
zero constant terms.
Q'.
refered to in the
proposition may be obtained from
statement of the
with
the
two distinct
polynomials in c
e
Now consider the selection rule for the incoming variable at any
point in
the parametric simplex
basis is D and
respect
ratio
to this
and d are the
(3'),
E by 1/c
(1),
(2) and
(3)
+ hj)/dj:
dj
>
+ hj):
dj
> O}),
max
{-(cj
(2')
max
{-cj/(dj
(3')
max
{-cj/hj(1/E):
the
ratios.
denotes
in h.
By the usual
Consequently,
let h
of c and d =
=
N.
1N with
Then the
j
0)
h(1/c) >
polynomial
hj(l/c)
index
reduced costs
becomes:
(1')
then a single
that the current
the incoming variable for the three objective
sufficiently small constant
1),
Assume
in Proposition 2,
As
basis.
rule for choosing
functions
In
that c
method.
and
0).
in 1/c
obtained by replacing
purturbation argument,
(i.e.,
gives
for all c
<
c(D)
10
c is a
for some constant
the unique minimum
< min
if
{c(D):
in each of
<
(D)
these
D is a basis of A),
the
is
entering variable
and
1 (see Remark 2 as well)
unique and Proposition
,
and
for D as a function of
if we express
(Similarly,
its Corollary apply.
then by Proposition 2,
and is
cost is a different nonconstant polynomial in
Therefore,
the
That
is
if
is,
sufficiently small,
is 0
reduced of xj
reduced cost
each reduced
linear in
then the value of e
.
for which
is different for all nonbasic variables
j.
P is dual nondegenerate.)
Each of the perturbations
pivot rules
procedures.
Since our purpose in
and
(3)
as certain
lead to different
lexicographic selection
this note has been merely to
simplex point
rules that do not
restrict
we will not specify the details of these
leaving variable,
procedures,
(2),
(1),
that can be interpreted
establish the possibility of
the
the
nor do we discuss their computation requirements
(or claim
that they are efficient).
In concluding this section, we might note that the parametrics
seem essential
(or homotopies)
for the results given
Simply perturbing the objective function to avoid
ties in
selection of an entering variable will not suffice.
standard examples of
in this paper.
the
For example,
cycling in the simplex method, there
in
is no dual
degeneracy and the choice of an entering variable is unique.
Dual Pivot Rules
Arguments
similar
side parametrics
to those used previously apply to right-hand
in a dual simplex algorithm.
parametric problem
cx
minimize
subject
to
Ax = b + eg
x
11
>
0.
That is,
consider the
We say that this
feasible
is primal nondegenerate if for any dual
problem
basis B (i.e.,
c -
cgB B-1A > 0)
is at most one basic variable x
that B
and parameter value
whose value
is zero.
', there
Then, assuming
is an optimal basis for sufficiently large values of the
parameter e and assuming primal nondegeneracy, we can mimic our
earlier arguments
to show that the usual parametric dual
simplex
method will
(i)
compute
(it)
for
values of
(i.e.,
0° ,
optimal solutions to P'(e) for all e <
8 > 0, defines
at any step
satisfies br =
Consequently,
a dual simplex method
the leaving variable xr
(i),
[(Bi)-lb]r
< 0).
the primal nondegeneracy assumption results
in a
finitely convergent dual simplex algorithm that permits any variable
satisfying
the dual minimum ratio rule to
leave the basis.
In order
to ensure primal nondegeneracy, we can introduce right hand side
perturbations;
[here £
is
that
is consider right hand sides such as
a column vector defined by
(i)
b + eg +
(ii)
b +
Choosing the
(g
,
+
2,
....
m)]:
or
).
m-vector g so that
unique optimal basis
= (1,
the following
(Bo)-1
for sufficiently
g > 0 will ensure
that B
large values for e and that
is a
this
perturbation will ensure primal nondegeneracy.
Acknowledgment
We are grateful
improvement
in
to Robert Freund for a suggestion that led to an
the proof of
Proposition 2.
12
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2, 103-107.
Math. of Oper. Research
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26, 157-177.
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20, 160-170.
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and
of
Production
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New York.
"The simplex method is very good! - On the
Haimovich, M. (1983).
of
pivot steps and related properties of random
expected number
Report, Columbia University, New
Technical
linear programs,"
York.
"On the average number of steps of the simplex
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27, 241method of linear programming," Mathematical Programming
262.
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A technique for resolving degeneracy in linear
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13