IP problem and LP relaxation problem
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Consider the IP problem given below.
Minimize Summation (i=1 to 4) x_i,
subject to
x1 + x2 + x3 >= 1,
x1 + x2 + x4 >= 1,
x1 + x3 + x4 >= 1,
x2 + x3 + x4 >= 1,
x1, x2, x3, x4 belong to {0,1},
and the LP relaxation, which allows 0 <= x_i <= 1, for i = 1,2,3,4. Show that,
(x1*,x2*,x3*,x4*) is an optimal solution to the LP relaxation, {please refer to the attachment for complete question}.
Solve the IP problem and LP relaxation problem.
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Solution Summary
This solution contextualizes an IP and LP relaxation problem. The minimize summation is provided. The expert determines the optimal solution to the LP relaxation.
Solution Preview
We first prove that the optimal solution to LP relaxation is (1/3, 1/3, 1/3, 1/3).
In fact, by looking at the objective function and the constraints, you will see that this LP is symmetric, meaning that xi are equally important.
To minimize the objective function while satisfying the constraints, we can let xi=1/3, i=1,2,3,4, because of the symmetry of the LP. Then all the four constraints become equalities. We can not reduce ...
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