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UIUC MATH 482 Practice Test 2 (Spring 2011)
Instructions:
This test has 7 pages including this cover sheet.
Answer 4 of the 5 problems given below. (I will grade all 5
problems and score you on the best 4.)
Each problem is scored out of 5 points.
This test is out of 20 points.
Justify all of your steps to ensure full credit.
No calculators or other aids are permitted.
Write your name and ID below and on every page.
Make your student ID available.
NAME:
STUDENT ID:
1
2
Q1. Use the dual simplex algorithm to solve the following linear program.
Minimize z = 4x1 + 6x2 + 8x3 + 10x4
subject to
x1 + 2x2 + 4x3 + 8x4 1
x1 + x2 x3 + x4 1
2x1 x2 x3 + x4 1
and x1 , x2 , x3 , x4 0.
3
Q2. Given the usual primal problem (P)
max c x s.t. Ax b, x 0
one has the usual dual problem (D)
min y b s.t. y A c , y 0.
Prove that the dual of (D) is (P).
4
Q3. The following linear program P is infeasible. Provide a certicate
that proves that P is infeasible using Farkas lemma. (Be sure to clearly
state how you are using Farkas Lemma, e.g., by giving a statement of
the Lemma.)
min 6x1 +2x2
subject to x1 x2
x1 x2
x1 +2x2
x1 ,
x2
=
z
3
2
1
0
5
Q4. For the LP
Maximize a + 3b + 2c + d subject to
a 2 b + 3c + 3d 7
2a + b + c + 4d 10
4 a + b + 3c d 8
and a, b, c, d 0.
State the dual minimization problem. Suppose you guess (a, b, c, d) =
(0, 42/5, 0, 2/5) is optimal. Use complementary slackness to prove your
guess and the solution to the dual problem is correct.
6
Q5. Prove Farkas Lemma from class.
7
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