9.7
We are given that (x1 , x2 , x3 ) = (2, 0, 4) is an optimal solution to the following LP:
max
s.t.
4x1 + 2x2 + 3x3
2x1 + 3x2 + x3 ≤ 12
(1)
x1 + 4x2 + 2x3 ≤ 10
(2)
3x1 + x2 + x3 ≤ 10
(3)
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
Therefore, the optimal value of this LP is 20. The dual of this LP is
min
12y1 + 10y2 + 10y3
s.t.
2y1 + y2 + 3y3 ≥ 4
(4)
3y1 + 4y2 + y3 ≥ 2
(5)
y1 + 2y2 + y3 ≥ 3
(6)
y1 ≥ 0, y2 ≥ 0, y3 ≥ 0
Let (y1 , y2 , y3 ) be an optimal solution to the dual. Since constraint (1) is not active at (2, 0, 4), by
complementary slackness, we must have y1 = 0. Since x1 = 2 6= 0 and x3 = 4 6= 0, again, by complementary
slackness, we must have that the dual constraints (4) and (6) are active. Putting this all together, the
optimal solution (y1 , y2 , y3 ) to the dual must satisfy:
y1 = 0
2y1 + y2 + 3y3 = 4
y1 + 2y2 + y3 = 3
Solving this system of equations, we obtain (y1 , y2 , y3 ) = (0, 1, 1).
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