ORIE 3300/5300
ASSIGNMENT 10
Fall 2008
Individual work.
Due: 3 pm, Friday November 21.
1. We have spent much time on the idea of a “certiﬁcate of optimality,”
which can be used to show that a proposed solution to a linear pro
gramming problem is optimal. In this question you will see how to
produce a certiﬁcate that a linear programming problem is infeasible.
Suppose the constraints of a linear programming problem in standard
equality form are
Ax
=
b,
x
≥
0
,
where
A
is an
m
×
n
matrix.
(a) Suppose there is a vector
y
∈ <
m
satisfying
A
T
y
≥
0
,
b
T
y <
0
.
Show that the linear programming problem has no feasible solu
tion. (Hint: consider
y
T
Ax
= (
A
T
y
)
T
x
for a feasible
x
.)
Hence such a
y
can be viewed as a certiﬁcate of infeasibility. As
with optimality, we want to show that whenever a linear program
ming problem is infeasible, such a certiﬁcate of infeasibility exists.
(b) Suppose for simplicity that
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 Fall '08
 TODD
 Operations Research, Optimization, linear programming problem, Infeasibility

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