ORIE 3300/5300
ASSIGNMENT 10
Fall 2010
Individual work.
Due: 3 pm, Friday November 19.
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
b
≥
0.
Show that if the linear pro
gramming problem is infeasible, then the auxiliary problem
max
(

e
)
T
v
subject to
Ax
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 '08
 TODD
 Operations Research, Optimization, linear programming problem, Infeasibility

Click to edit the document details