Tutorial 4: Outline of Solutions
Q1. Consider the standard form polyhedron
P
=
{
x
∈
R
n

Ax
=
b
,
x
≥
0
}
. Suppose the
matrix
A
has dimensions
m
×
n
and its rows are linearly independent. For each of the following
statements, state whether it is true or false. If true, provide an argument, else, provide a counter
example.
(a) If
n
=
m
+ 1, then
P
has at most two basic feasible solutions.
(b) The set of all optimal solutions is bounded.
(c) At every optimal solution, no more than
m
variables can be positive.
(d) If there is more than one optimal solution, then there are uncountably many optimal solu
tions.
(e) If there are several optimal solutions, then there exist at least two basic feasible solutions
that are optimal.
Solution
:
(a) TRUE. Since
n

m
= 1, the feasible region can be visualized in one dimensional space with
n
inequality constraints. Clearly in 1D, this can have at most two basic feasible solutions.
(b) FALSE. The following LP
min
0
x
1
s.t.
x
1
= 1
x
1
,x
2
≥
0
has an unbounded set of optimal solutions expressed as
x
*
1
= 1
,x
*
2
≥
0.
(c) FALSE. In the example in (b),
m
= 1,
n
= 2.
Every optimal solution of the form
x
*
1
= 1
,x
*
2
>
0 has two strictly positive variables. The statement is true only for basic fea
sible solutions (in this case
x
*
1
= 1
,x
*
2
= 0).
(d) TRUE. Given a set of optimal solutions
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 Spring '10
 TanBanPin
 Optimization, Zagreb, Optimal Solutions, sible solutions

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