Industrial Engineering 634
Fall 2011
Integer Programming
Lecturer: Nelson Uhan
Scribe: Mahesh Ramaraj
August 24, 2011
Lecture 2. Linear Programs and Polyhedra
Following our discussion on integer programs from last class, we will focus this lecture on linear
programs and polyhedra. What is a linear program? Given a matrix
A
∈
R
m
×
n
and vectors
b
∈
R
m
and
c
∈
R
n
, a
linear program
is an optimization problem of following form
max
cx
(1)
s.t.
Ax
≤
b
(2)
Equality constraints can be rewritten as two inequality constraints, and by multiplying by

1, we
can change the direction of the constraints and the objective function.
A
feasible solution
to a linear program of the form (1)(2) is a vector
x
such that
Ax
≤
b
. An
optimal solution
of a linear program of the form (1)(2) is a feasible solution with maximum value of
cx
. A linear program of the form (1)(2) is
infeasible
if
{
x
∈
R
n
:
Ax
≤
b
}
=
∅
and it is
unbounded
if for all
α
∈
R
, there exists a
y
∈ {
x
∈
R
n
:
Ax
≤
b
}
with
cy > α
.
In rest of the lecture we will focus on the feasible sets of linear programs. A
polyhedron
in
R
n
is a set of type
P
=
{
x
∈
R
n
:
Ax
≤
b
}
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.
 Spring '97
 RAVINDRAN
 Industrial Engineering, Optimization, linear program, Polytope, c ∈ Rn, xk − x1

Click to edit the document details