LIBRARY
OF THE
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
ALFRED
P.
SLOAN SCHOOL OF MANAGEMENT
GENERALIZED LAGRANGE MULTIPLIERS
IN INTEGER PROGRAMMING
by
Jeremy
March
F.
7,
Shapiro
1969
Working Paper 371-69
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
GENERALIZED LAGRANGE MULTIPLIERS
IN INTEGER PROGRAMMING
by
Jeremy
March
F.
7,
Shapiro
1969
Working Paper 371-69
RECEIVED
JUL 25 1969
M.
I.
T.
Li.
5
f.
„,
......a
p.
2,
line 5
p.
13
—
1
—
—
Add [2] to ths refsrencss iiatacl
"In [iOj. Nemhausar and,.,"
Add the following footnote to the second
sevij-.ence
of LEILlA 3.
K
"1.
If
E
V,
k-1
^
= 0, then division by this sum 1e of csurse not
:^
p.
14
—
19
—
n
_
I
a.v,,,.
'^
-
and
„
}1
In the second sum on the i&ft of tha first inequality, read
".
"v
p.
the inequalities are
'
liua from the bottom- "example in [9]"
-
c.v,,,.
ABSTRACT
The integer programming problem is reformulated using group
theory thereby allowing a new Lagrangian optimization problem for integer
programming to be constructed.
its relationship
The properties of this problem and
to group theoretic branch and bound and cutting plane
algorithms are discussed.
Necessary and sufficient conditions for the
existence of optimal multipliers are also given.
GENERALIZED LAGRANGE MULTIPLIERS
IN INTEGER PROGRAMMING
1
.
Introduction
Several authors ([3], [4], [8], [9], [10], [14]) have proposed
generalized Lagrangian methods for finding good or optimal solutions to
integer programming problems.
The capital budgeting problem of Lorie
Savage [9], essentially the 0-1 multi-dimensional Knapsack problem,
and
has received particular attention in this regard.
In [9], Nemhauser and
Ullman prove the somewhat negative result that the approach of Everett
[4]
applied to the capital budgeting problem by Kaplan in [8] can yield
an optimal solution only if there is an optimal linear programming solution
that is integer.
[12],
[13])
In this paper, we use group theory
([5],
[6],
to reformulate the integer programming problem,
[7],
[11],
thereby
obtaining a Lagrangian problem which appears to offer greater combinatorial
resolution than previous methods.
Conversely, the usefulness of the group
theoretic approach is enhanced by the Lagrangian problem.
In the next section we construct the Lagrangian problem and discuss
its properties.
The following section is concerned with the relationship
of cuts generated by the Lagrangian problem to the strong cuts developed in
[6].
Necessary and sufficient conditions for the existence of optimal
multipliers are also given.
concluding remarks.
There follows a numerical example and a few
There is one appendix in which some of the previous results
are specialized to the zero-one integer programming problem.
6^9G42
2.
Construction of the Lagrangian Problem and Its Properties
Consider the Integer programming problem In the form
s.t.
mln
cw
Aw =
b
(1)
w non-negative integer
where
lx(m+n) vector of integers, A is an mx(m+n) matrix of
is a
c
integers, and b is an
denoted by a.,
j
raxl
The columns of A are
vector of integers.
= 1,..., m+n.
Let B be an optimal LP basis for (1)
and rearrange the columns of A so that it can be partitioned as (R,B)
and
c
as
(c^^,
The generic n-vector of non-basic variables is denoted
c^).
by X and Is called a correction.
with addition modulo
The columns of B
abelian group consisting of D elements (see
the group and let
Identity element.
X
,
k = 0,1,..., D-I,
[
I
5
generate a finite
Let G be
]).
be its elements where
X
is the
Finally, let g be the function which maps integer
m-vectors Into group elements.
It can easily be shown that problem
(I)
is equivalent to
min
E
c.x,
(2a)
If B is an arbitrary basis, the analysis below is valid with minor
modifications.
n
s.t.
Z
r..x.
<
a.x. =
B
b
,
= l,...,m,
i
,
(2b)
n
E
(2c)
j=l
X.
where
and
c.
6
= c.
- c B~ a
= g(b).
j=l,...,n
=0,1,2,...;
R = (r
,
.
.
)
= B~ R,
(2d)
b =
(h
= B~ b, a.
)
= g(a
)
Condition (2b) is equivalent to the requirement that
the basic variables (relative to B) be non-negative, and condition
(2c)
is equivalent
to the requirement that they be Integer.
If we let
X = {x|x satisfies (2c) and (2d)},
Then problem (2) can be rewritten as
n
min
E
j
s.t.
Z
1
r..x.
<
c.x.
=l
J
J
b.,
1
= l,...,m,
(2
)
=1
xe)C.
Problem (2') has the same form as Everett's problem
of
[2
]
.
The
[
4
]
and problem (2)
approach suggested by Everett is to weight the
constraints (2b) by multipliers and place them in the objective function.
1
"
We assume the existence of at least one x such that .Z.a.x. =g.
J J
Jproblem (1) is infeasible.
Otherwise,
m
In particular,
for given u.
^
>_0,
i
= l,...,m, satisfying c. +
T.
1
=
j
^
u?r
^J
^
i=l
l,...,n, we construct the new Lagrangian optimization problem
n
rain
E
3=1
,(c
m
.
+
J
Z
—
u.r
i=l ^
,
.
)x.
^J
J
.
(4)
s.t. xeX
Problem (4) is a group optimization problem identical in form to the
group optimization problem originally derived in
[
12
(4)
]
for a further discussion of this problem.
is denoted by x
,
[
5
]
See
.
The requirement that c. +
J
11
]
and
An optimal solution to
or more generally by x(u) when u
of multipliers in (4).
[
T;
i=l
is the vector
u.r..
^
^J
>
~
0, j
=
l,...,n, is necessary for (4) to have a bounded solution.
Notice that we could have constrained explicitly some variables in
(1)
to be zero or one.
The modifications to the construction above in
this case are given in Appendix A.
0,
We begin our analysis with a restatement in the context of this
paper of Everett's theorem
1
4
[
P ={i|u°
LEMMA
i
=
An optimal solution x
1:
;
p.
401].
>
0}
.
to
First, for given u
>^
(5)
(4)
is optimal in (2) with b^,
l,...,m, replaced by
n
(i)
E
3=1
if i e P
r. .X?
'' '
n
(ii)
where
Proof:
q
.
is any
+
r. .X?
E
if i
q.
non-negative integer,
i
^
«!
P,
P.
Suppose the contrary; that is, suppose there exists a non-
negative integer vector ycX with the properties that
n
E
n
c.y.
J
j=l
<
c.x°,
E
j=l
^
J
J
and
Since u. >^0,
i
=
n
_
E
r, .y.
n
<
E
0, let
_
r^ .x?
l,...,m, there obtains
for ieP.
m
n
(c.
Z
j=l
+
n
_
u°r..)y. =
Z
J
1=1^
(c.
+
^J
n
<
J
j=l
J
=
u°r..)x°
Z
icP
^J
^
.
+
J
u°r..)y.
Z
J
icP
(E.
+
^J
^
m
n
_
Z
(c
Z
j=l
Z
J
j=l
Z
J
_
u°r..)x°
i=l
^
^J
J
The last inequality is a contradiction to the optlmality of x
When (4) is solved by
algorithm of
tlie
[11
],
K
,
k = 0,1,..., D-1,
replaced by
COROLLARY
A,
K
A,
1:
X
?^
K
Let x° (A
e.
)
(4).
the algorithm also
finds optimal solutions to (A) when the groun constant
A
in
B
in
(/c)
is replaced by
be the optimal solution when
3
is
K.
.
An optimal solution x (A.) to (4) with group constant
is optimal in (2) with b.,
i
= l,...,m,
replaced by
1
n
(i)
Z
j=l
if ieP
r..x°. (A, )
^J J
^
n
(ii)
where
q.
Z
?..x°(A,
)
+
q.
is any non-negative integer,
if
i
ii^P
^
P.
Thus, problem (4) is algorithmically useful in several ways.
=0
Suppose
5 ] ) fails to solve
(i.e., the group optimization problem of
n
_
Then (4) can be resolved
Z r..x. > b..
(2) because for at least one i,
^
j=l ^J J
(4)
with
u
[
with new u^ for which some u^
(2b) with X = X
is violated.
^
0;
e.g., yl
>
only if the
i
constraint in
Let x^ be the new optimal solution to (4).
.
"First, suppose
There are three cases to consider.
j
this case, ex
1
r..x.
E
=l
^^
-
^b.,
1
,
i
e
In
P.
the cost of an optimal correction to (2)>
n _
because of the constraints Z r,.Xj <_
j
X
e
is a lower bound on z*,
This lower bound is greater than ex
" E r
i
^
^
P,
= l ^^
-•
implied by (4) which were not in effect when (4) was pre-
viously solved with u
The improvements of bounds could be very impor-
= 0.
tant to the algorithm of
[
12
]
in which group theoretic bounds are used to
limit the search of the non-basics for an optimal correction.
" 1
Z r..x.
^J J
3=1
The second possibility is that
- o
feasible correction.
-
b.,
<
-
i
=
l,...,m, or x
-
.
Since ex
the non-negative quantity ex
z*,
<
1
is a
^
an upper bound on the loss from terminating with x
- o
- ex
is
1
.
.
Finally, if neither of the two cases above obtain, the cut
(c
is a valid cut
+ u^R)x
>
+ u^R)x^
(c
which could be added to (1).
(6)
The relationship of the
Lagrangian methods to the cutting plane method are discussed in the next
section.
If the requirements vector b can be relaxed,
to
q
(1)
may be derived by replacing
b
with Rx
can be any integer vector satisfying
q
>_
(A,
K
+ Bq for some
and q. =
solution to the new (approximate) problem is (x
We are now assuming u
)
then a suitable
(>.,),
if
i
e
A
K
P.
solution
where
The optimal
q)
is an arbitrary non-negative vector in (4).
In order to put this relaxation scheme in better perspective, con-
sider the isomorphic representation of G as M(I)/m(B) where
m(I) is the
group consisting of all integer points in m-space with ordinary addition,
and WV(B) is the subgroup consisting of all integer points which can be
spanned by integer combinations of the basic activities
(2c)
is then an
equivalence class of m-vectors.
.
The element
Because of the combin-
atorial structure of the given problem, it may be that vectors from
awkward as
hand side
B
are
vectors in (1); e.g., low cost corrections for group right
b
B
in
3
may tend to be long and infeasible in (2).
It is tempting to try to construct an algorithm which, starting with
u
o
= 0,
solves (4) with a monotonic increasing sequence of u
- k
so that Rx
converges to b
k
k
-
from above.
k = 0,1,2,...,
,
Unfortunately, the Rx
k
vectors are
not sufficiently well behaved to make such an algorithm feasible.
There
- k
is, however, a certain degree of regular behavior exhibited by the Rx
LEMMA
2:
Any optimal solutions x
multipliers u
and u
and x
to
(4)
respectively must satisfy
(u^ - u°) R (x° -
xS
>
with non-negative
.
10
The proof is immediate from the inequalities
Proof.
(c
+ u°R)x° 1
(c
+ u°R)x\
(c
+ u^R)x^
(c
+ u^R)x°.
and
Thus, if
n
L
—
o
r..x..
>
u,
k
u,
k
<
and u. = u.,
= l,...,m,
i
^ k,
i
1
1
Unfortunately, with u
1
J-1
control the magnitude of the sums
then
Z r..x. <
..ill—
j=l
J
-^
so defined there is no way to explicitly
n _
^
Y.
r..x.,
j=l ^J
i
?*
k.
Although future research
J
may lead to algorithms with well behaved properties for using (4) to solve
(2),
it appears at this time that
is as a heuristic.
the primary usefulness of (4) in solving (2)
In particular, we are currently implementing an adaptive
integer programming system based on the ideas in [7], and we plan to experiment with the setting of multipliers in (4) in the near future.
Finally, we remark that the multipliers in (4) have an unusual
interpretation for the integer programming problem with the formulation
(2).
In particular,
c.
+ ua
J
where
ij;
=
uB
.
.
= c.
J
J
-
Ila
.
J
+ ua
.
J
is
adjusted by
-
-i|;
.
- i>)a.
(IT
J
J
The usual LP interpretation of
cost savings to be gained by reducing b
Jl.
= c.
II
is that n.
by one unit.
reflects the
In the above expression,
which can be interpreted as an added cost because the
distance with respect to the
i
basis activity a
from b to the boundary
11
of the cone
{ H
B
:
£
} is
>_
insufficient.
This last statement can be
made more precise by the following lemma from [13].
Given any set
that if b.
(1)
^
I
^.
t*,
i
{ 1
e
,
.
.
I,
.
,m}
,
The lemma states:
there exist non-negative integers t* such
then (2) with rows
i
e
I
omitted in (2b) solves
in the sense that there is an optimal solution to the reduced
problem which is an optimal correction in (1).
the condition of this theorem for some set I.
Suppose (2) satisfies
Then problem (4) formed
from the reduced problem (2) is the same as problem (4) formed from the
original problem (2) with
u.
= 0,
i
e
I.
12
Cutting Plane and Lagrangian Methods
3.
In
5
[
Gomory uses the group theoretic approach to develop
],
theorems for characterizing and constructing strong cuts for the cutting
If we let
plane method.
[X]
of interest are the faces of
denote the convex hull of X
,
then the cuts
Our purpose in this section is to
[X].
relate problem (4) and the Lagrangian cuts to these faces.
Suppose Ax
A
a
and
—>
—>
A
o
0.
Lagrangian cut
kAx
>_
>_
(6)
A
is a face of
It
is natural
this problem is:
[
6
;
16]
p.
that
to ask if it is possible to obtain
equivalent to this face.
is an equivalent cut,
kA
Gomory shows
[X].
Since for any k
0,
>
the correct mathematical statement of
Does there exist a u
>_
0,
k
>
0,
such that
uR - kA = -c?
(7)
The system (7) consists of n equations in m+1 variables which does not
always possess a solution of the form we seek.
Although this inability to produce selected strong cuts appears
to be a
[X]
drawback of the Lagrangian method, we remark that the faces of
are generated with respect only to the group identities of the non-
basic activities a.,
l,...,n, and
=
j
the requirements vector b.
The
Lagrangian approach, on the other hand, incorporates Information about
the real space identities of the a. and b.
would be to (i) select a boundary
for a specified u
use the methods of
contain
x
.
of
[X]
by solving problem (4)
chosen according to the magnitude of b
?*
[
point x
A synthesis of the two approaches
6
]
,
and (11)
to generate cuts beginning with those which
13
Does there
Of course, a more fundamental existential question is:
exist a u
>^
such that (4) solves (2)?
A solution
needed.
implies
s
t
e 2^
To answer this, a definition is
is said to be irreducible if s
<^
t
and
Let T be the set of all irreducible solutions.
= t.
say T = {t
easy to demonstrate that T is finite;
Suppose X* is an optimal correction.
+ uR
>
X
e
It is
•
Then it is clear that x* is
if and only if there exists a u
an optimal solution in (A)
c
)i^_i
s
such that
>^
(8)
0,
and
(c
Note that each x
where
(c
e
t
+
uR)t'*^
(c
>
+ uR)x*,
k = 1,...,K.
(9)
X can be decomposed in at least one way as x =
e
T and s is non-negative integer.
Thus,
(c
+ uR)x
>^
(c
+ uR)x* and condition (9) is sufficient as well as necessary.
t
+
s
+ uR)t
>_
The
following lemma is a direct consequence of Farkas' lemma applied to (8)
for u
and (9)
LEMMA
3:
^
0.
Suppose x* is an optimal correction.
Then there exists a u
such that X* is optimal in (4) if and only if for every
V ^
such that
"
^ - k
Z Rt
k=l
+
v,
K
K
Z
k=l
a.v„.
J^,
v
Z
^
k=l
.
<
Rx*,
K-^n
vector v
>_
>
and
14
there obtains
ct^
I
+
V,
k=l
^
ex*.
-
K
K
k=l
.5:,c.v
\"
"
^k
"
k=l
Suppose, however, that
This lemma is difficult to interpret as it stands.
vector v
there exists a
and v ^
—>
with v
= 0,
j
= l,...,n, and
r>.+j
also such that
K
_
L
Rt
_
v
.
<_
Rx*,
(10)
<
ex*.
(11)
^
k=l
K
^
^k
"
k=l
and
•^
I
k=l
- k
ct
V,
^
K
^
k=l
""k
^
In this case, there does not exist a u such that x* is an optimal solution
in
(4).
in
(A)
Conditions (10) and (11) state, of course, that x* is not optimal
if there is a convex combination of the
and uses no more resource than x*.
t
which is less costly
The interested reader should compare
this with Everett's discussion on the source of gaps which can be found on
page 408 of reference 4.
15
Lemma
3
was derived without regard to whether or not the given
If x* is reducible, then necessary
optimal correction x* is irreducible.
and sufficient conditions that x* is optimal in (4) is that there exist
a u
21
such that for at least one teT for which x* =
c
( c^
+ uR
+ uR) t
t
+
s
(s
>_
0,
s
?<
0)
>_
>_
G
+ uR)
k =
t
1
, .
.
.
,
k
(7 + uR)s =
Since g(t) = g(x*) =
(c
+ uR)s
=
B,
we have g(s) =
0.
Thus, the condition that
can be interpreted as the requirement that there be at least
one costless circuit in the shortest route network representation of (4).
Any such circuit can be built up from costless elementary circuits which
can be found by the algorithm of [11].
16
A.
Numerical Example
Consider the integer programming problem
min
+ 6x, + Ax^
21x„
2
12
+
-Ix, + 13x„
s.t.
A
5
5x,
A
+ 2xc
=
(12)
10x„ - Ix. + Ix, + 3xc
3
2
X.
Activities
a,
non-negative integer,
A
j
16
5
=
8
5
= 1,2, 3, A, 5.
Problem (A) derived
and a^ constitute an optimal LP basis.
with respect to this basis is (all fractions were eliminated by multiplying
by 13)
min (lA
s.t.
-
3u
+ lu )x
+ 19x
-3x
^
+ (11 + 19u, + 37u )x
+ 2x
^
<^
+ 2u^
- 5u )x
32
(13)
- 5x
<_
2A
Ix^ + llx
+ 8x
E
11
X.
(8
-*
+ 37x
Ix
+
(mod 13)
non-negative integer,
j
=
1,2,3
The optimal solution to (13) with u° = 0, u" = 0, is x° = 0, x°
x° = 0.
= 0,
1,
This correction is not feasible in (12) as the second inequality
constraint in (13) is violated.
X.
=
Setting u
= 0,
u„ = 1, yields the solution
x„ = 0, x„ = 3, which is the optimal correction we seek.
this correction is optimal
for any u
>^
with
u.
= 0,
9/13
In fact,
^
u
<
8/5.
17
5.
Conclusion
In this paper we have tried to demonstrate how the Lagrangian
problem (4) is useful when trying to solve integer programming problems.
We saw that an optimal solution to (4) could be used in a variety of
ways to find good or optimal solutions to (1).
Unfortunately, the theor-
etical results obtained thus far are not sufficiently strong to enable
us to algorithmically control the properties of optimal solutions to
(4).
Future research may lead to an improvement in these procedures.
We emphasize once again, however, that we expect to incorporate
(4)
problem
into the adaptive group theoretic algorithm of [7] and experiment
with heuristics for setting the multipliers u..
It seems clear that the
power of the group problem from [5] can be enhanced by the approach here.
Certain results from the literature on generalized Lagrange
multipliers were not specialized for our problem although it is possible
In particular, Everett's
to do so.
reference
4
lambda and epsilon theorems in
were ignored because they appear to be not directly relevant
to our study.
Similarly, the linear programming problem in [2] and
Brooks theorems in [3] have been ignored.
In addition,
there may be some
connection between the results here and those of Balas in [1] on duality
theory in integer programming.
18
Appendix A
The Zero-one Integer Programming Problem
SC
Suppose there is a subset
1,
JF.S,
= 0,
and X
1
,2,
.
.
.
,
jtS
n
_
Z
c.x.
£
s.t.
Then (2) becomes
-J
J
<
-
b
,
i =
,
iJ
j=l
(14b)
_
(-r..)x.
Z
m
1,
i
_
n
<
-
J
1-b.,
i
= l,...,m,
1
'
(14c)
1
'
a.x. = B
Z
X.
(lAa)
r..x.
j=l
=
x^ =
or
J
J
j=l
=
Suppose further that the first m.-
.
basic variables are 0-1 variables.
min
{l,...,m+ti} such that x.
(14d)
0,1,2,...,
or
1
,
i
e
(14e)
S^
e
j
(14f)
S
J
As before, define
Z =
For a given u
{x|x satisfies (14d),
where we now require u
the Lagrangian problem (4)
>^
(14e)
,
only if
(14f)}.
i
>_m
+
1,
we define
.
19
n
_
™
_
Z(c.+ Eur..)x.
min
i=l " ^J
J
j=l
J
(15)
xeZ
Problem (15) is a zero-one group optimization problem which can be solved
by the zero-one group algorithm of
to lemma
1
2:
u?
= m,
—>
0,
i
be an optimal solution to (15) with u
Let x
COROLLARY
1
We have the following corollary
[7].
+ l,...,m.
is optimal in
Then x
such that
(14) with (a) b
1
'
1
,
i =
l,...,m, replaced by
n
(i)
r,,x°
I
j=l ^J
if u°
>
J
n
(ii)
r. .x°
Z
j=l ^J
and (b)
1
- b
,
i =
1
,
.
.
.
,m
,
+
if u°
q,
i
J
-<
1
0;
replaced by
n
(iii)
Z
(-r. .)x°
j=l
^J
if u?
<
J
n
(iv)
where the
q.
Z
(-r. .)x° + q,
if u°
>
0,
can be any non-negative integers.
Problems (14) and (15) are the ones applicable to the LorieSavage capital budgeting model.
We mention in passing that the zero-
one group algorithm solved the example in [8] with u
= 0;
that is,
without resort to explicit use of the multipliers of this paper.
20
REFERENCES
(1)
Balas, E., "Duality in Discrete Programming," Technical Report No. 67-5,
July, 1967, Department of Operations Research, Stanford University.
Revised December, 1967.
(2)
Brooks, R. and A Geoff rion, "Finding Everett's Lagrange Multipliers by
Linear Programming," Opns. Res., l±, pp 11A9-1153, (1966).
(3)
Brooks, R. , "Some Linear Programming Applications to Stockage Problems,"
The RAND Corporation RM-5422-PR, September, 1967.
(4)
Everett, H.
"Generalized Lagrange Multiplier Method for Solving Problems
of Optimum Allocation of Resources," Opns. Res., J[l^, pp 399-417.
(5)
Gomory, R. , "On the Relation Between Integer and Non-Integer Solutions
to Linear Programs," Proceedings of the National Academy of
Sciences, 53, pp 250-256, (1965).
(6)
Gomory, R. , "Some Polyhedra Related to Combinatorial Problems," IBM
Research Report RC2145, July, 1968.
(7)
Gorry, G.A. and J. Shapiro, "An Adaptive Group Theoretic Algorithm
for Integer Programming Problems," Technical Report No. 38,
Operations Research Center, M.I.T., May, 1968.
{^)
Kaplan, S., "Solution of the Lorie-Savage and Similan Integer Programming
Problems by the Generalized Lagrange Multiplier Method," Opns. Res.,
pp 1130-1136, (1966).
,
U
(9)
Lorie, J. and L.J. Savage, "Three Problems in Capital Rationing," J.
of Business, pp 229-239, (1955).
(10)
Nemhauser, G. and Z. Ullman, "A Note on the Generalized Lagrange
Multiplier Solution to an Integer Programming Problem" Opns. Res., 16
pp 450-452, (1968).
(11)
Shapiro, J., "Dynamic Programming Algorithms for the Integer Programming
Problem I: The Integer Programming Problem Viewed as a Knapsack
Type Problem," Opns. Res. _16, pp 103-121, (1968).
(12)
Shapiro, J., "Group Theoretic Algorithms for the Integer Programming
Problem II: Extension to a General Algorithm," Opns. Res. _16
pp 928-947.
(13)
Shapiro, J., "Turnpike Theorems for Integer Programming Problems," Sloan
School of Management Working Paper 350-68, M.I.T., (1968).
—
21
(14)
Weingartner, H.M., "Capital Budgeting of Interrelated Projects: Survey
and Synthesis," Mgement Sci., U^, pp A85-516, (1966).
(15)
Weingartner, H.M. and D. Ness, "Methods for the Solution of the MultiDimensional 0/1 Knapsack Problem," Opns. Res., J[_5, pp 83-103,
(1967).
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