Identifying the Set of Always-Active
Constraints in a System of Linear
Inequalities by a Single Linear Program
Robert M. Freund
Robin Roundy
Michael J. Todd
Sloan W.P. No.
1674-85 (Rev)
October,
1985
Abstract:
Given a system of m linear
the form Ax < b, a constraint
inequalities in n unknowns of
index i is called always-active if Ax <
b has a solution and every solution satisfies Aix = b i.
lies
Our interest
in identifying the set of always-active constraints of Ax
solving one linear program generated from the data
Key Words:
Linear
Abbreviated
Title:
b by
(A,b).
Inequalities, Active Constraints, Linear Program.
Always-Active Constraints.
Given a system of m linear
inequalities in n unknowns, of the
form
Ax < b,
(*)
a constraint index i is called always-active if Ax < b has a solution
and every solution satisfies Aix = bi,
active in every solution.
i.e.
the ith constraint is
In large optimization problems where the
constraint set is given by the system (*),
identifying the always-
a-ctive constraints before processing the problem enables the user to
explicitly reduce both the dimension of the feasible region and the
number of constraints, and has the potential therefore to simplify the
original problem.
Our interest lies in identifying the set of always-
active constraints by solving just one linear program generated from
the data (A,b).
A first attempt (also noted by several of our colleagues) is to
solve
maximize eTy
x,y
Ax+y < b
(P1)
0 < y < ce
where e is the vector of ones of appropriate dimension and
positive and "sufficiently small."
however hard to determine.
A suitable value of
is
seems
If the data (A,b) is integer (or
rational), then Orlin has pointed out [3] that a sufficiently small
can be predetermined (using arguments similar to those used in the
ellipsoid method).
2
Our first solution is closely related to
homogenization as
similar to
(P1),
and employs
those reducing questions about polyhedra
to questions about convex cones.
Consider the
linear program:
maximize eTy
x,y,a
Ax+y-ba < 0
a >
(P2)
1
We have the following straightforward result:
Proposition 1.
and finite,
If the system
(*) is
feasible, then (P2)
and for any optimal solution
always-active constraint indices
(x*,y*,a*) to
is the set
ily*
is feasible
(P2), the set of
= 0).
Furthermore,
x*
is an element of the relative interior of (xeRnlAx < b}.
the system
(*) is not feasible, then
Another linear program that
(P2)
is
infeasible.
identifies all of
If
[X]
the always-active
constraints is obtained through consideration of the following
problem.
maximize t
x,t
subject
Obviously,
to:
(P3).
Ax+et < b
(P3) is always feasible,
solution to
constraints.
(*),
When t*=O,
it can easily be shown that if x* and X* are
and its dual
minimize
X
subject
if and only if there is no
> 0 if and only if there are no always-active
and t
optimal values of (P3)
t* < 0
to:
XTb
ATX = 0
eT
=
(D3), namely
(D3),
1
>O
3
then if X i
> 0,
the ith constraint
is always-active, and if Aix*
than the ith-constraint is not always-active.
we have a strictly complementary
(D3)
(i.e. X i
> 0 or bi -
pair of
for each constraint whether or not it
If we could ensure that
optimal
Aix*> 0 for each i),
< bi,
solutions to
(P3) and
then we could identify
is always-active.
This is
accomplished by solving the following linear program:
maximize e
x,X,t,e
subject to:
(1)
Ax + et
< b
(2)
ATX
= 0
(3)
eTX
=
+ t
= 0
(4)
-bTX
(5)
X-Ax-et-ee
(6)
X
(6) represent dual
> -b
> 0
feasibility, contraint
duality, and a positive value of
strict complementarity of X and
(*) has a solution.
Proposition 2.
constraints.
If
If
set
(4) represents strong
in constraints
(b-Ax) when t=0.
(5) represents
Tucker's strict
We have:
infeasible, then
(*) has no always-active
(x*,X*,t*,e*),
we have:
= 0, the set of always-active constraint indices
(i*kXi
{xERnAx
(3) and
(P4) is feasible, then it is finite, and for any
optimal solution
If t
(P4) is
(2),
ensures that a positive value of e will
complementarity theorem [4]
(i)
(P4)
(1) represent primal feasibility, constraints
Constraints
exist if
1
> 0) and x*
lies
in the relative interior of
< b).
4
is the
III
t* > O,
(ii)
If
(iii)
If t* < 0,
Instead of
there are no always-active constraints.
the system
(*) has no solution.
the system (*),
Ax=b,
[X]
we could have worked with the system
x> O,
(**)
whereby the always-active constraints would correspond to null
variables of
to
(**),
interest
(**),
i.e.
see Luenberger
in the recent
variables xj
[2].
such that xj=O
Determining null
in every solution
variables is also of
linear programming algorithm of Karmarkar
5
[1].
References
[1]
N. Karmarkar, "A new polynomial-time algorithm for linear
programming," Combinatorica 4 (1984) pp. 373-395.
[2]
D.G. Luenberger, Introduction to linear and nonlinear programming
(Addison-Wesley, Reading, Massachusetts, 1973).
[3]
J.B.
[4]
A.W. Tucker, "Dual systems of homogeneous linear relations," in:
H.W. Kuhn and A.W. Tucker, eds., Linear Inequalities and Related
Systems, Annals of Mathematics Study No. 38 (Princeton University
Press, Princeton, New Jersey, 1956) pp. 3-18.
Orlin,
private communication.
6