IEOR 162
Linear Programming
Spring 2007
Sample Midterm Solutions
1.
a.
False
. A linear program may also be infeasible.
b.
False
. There may exist optimal solutions that are not extreme points. However, we can guarantee that
if there is an optimal solution, then there will exist at least one extreme point optimal solution.
c.
True
. If the feasible region is bounded, then a linear objective function is also bounded.
d.
True
. At any solution there are
m
active constraints because all constraints are equality constraints. If
this question were to ask if there must be exactly
m
linearly independent active constraints, that would
be false because in addition to the equality constraints,
n

m
nonnegativity constraints must be at
equality.
2.
Standard Form:
max

6
x
+
1
+ 6
x

1
+
x
2

4
x
3

3
x
+
4
+ 3
x

4
s.t.

2
x
2
+
x
+
4

x

4
+
s
1
=

2
5
x
+
1

5
x

1

3
x
2
+ 2
x
+
4

2
x

4

e
2
=
1

2
x
2

x
3
+ 2
x
+
4

2
x

4
+
s
3
=

3

x
+
1
+
x

1

4
x
2

x
+
4
+
x

4
=
4
x
+
1
, x

1
, x
2
, x
3
, x
+
4
, x

4
, s
1
, e
2
, s
3
≥
0
Note that
x
1
and
x
4
were unrestricted in sign in the original formulation. They are replaced in standard
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 Spring '07
 Zhang
 Optimization, Mathematical optimization, optimal solution

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