This includes but is not limited to copying or

Regulations and will not be tolerated. This includes, but is not limited to: copying or
sharing answers on tests or assignments, plagiarism, having someone else do your academic
work or working with someone on homework when not permitted to do so by the instructor.
Depending on the act, a student could receive an F grade on the test/assignment, F grade for
the course, and could be suspended or expelled from the University. See the Conduct Code at
for more details and a full explanation of the Academic Misconduct
policies.
1

1. (5 points) Prove that if (i) x is feasible to max { c
>
x : Ax ≤ b, x ≥ 0 } , (ii) y is feasible to
min { b
>
y : A
>
y ≥ c, y ≥ 0 } , and (iii) c
>
x = b
>
y , then x and y are both optimal.
2. (5 points) Prove that exactly one of the following two alternatives holds for the LP max { c
>
x :
Ax ≤ 0 , x ≥ 0 } : (a) x
*
= 0 is an optimal solution, (b) LP is unbounded.
3. (10 points) Find the dual of the following LP using the primal-dual table without reformu-
lation. Solve both primal and dual LPs using a computer solver, and check the optimality
condition to verify the optimality of the solutions. Please submit (1) the dual LP, (2) primal
optimal solution and optimal objective value, (3) dual optimal solution and optimal objective
value, (4) optimality condition, (5) whether or not the optimal solutions satisfy the optimality
condition.
max x
1
+ 2 x
2
s . t . x
1
- 2 x
2
- x
3
+ x
4
≥ 0
4 x
1
- 3 x
2
+ 4 x
3
- 2 x
4
≤ 3
- x
1
+ x
2
+ 2 x
3
+ x
4
= 1
x
2
≤ 0 , x
3
≥ - 1 .