Branch and Bound
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Transcript of Branch and Bound
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Branch and Bound
See Beale paper
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Example: Maximize z=x1+x2
x2
x1
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Solve First LP problem:
• Solution is [1.5 2.5]
x2
x1
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[1.5 2.5]
X1 <= 1
[1 1.5]x2
x1
X1 >= 2
[2 1.5] , z=3.5, z=2.5
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[1.5 2.5]
X1 <= 1
[1 1.5]
x2
x1
X1 >= 2
[2 1.5] , z=3.5, z=2.5
X2<= 1 x2>= 2
No solution
[2.25, 1], z=3.25
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[1 1.5]
x2
x1
[2 1.5] , z=3.5
, z=2.5X2<= 1 x2>= 2
No solution
[2.25, 1], z=3.25
X1 <= 2 x1 >= 3
[2,1], z=3
No solution
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Example: Maximize x1+x2
x2
x1
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SSbar
Sums edges out of S >= 2
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In TSP, we solve LP problem with constraint {each vertex has 2 edges
incident to it} and we add just relevant ‘subtour inequalities’ to cut off any subtour solutions. So each time we solve LP and if we get a
subtour solution, we add the specific subtour inequality to cut off that solution and resolve LP. This continues until we get a final tour
solution.
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Objective Function to be minimized
Unbounded
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Infeasible solution
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LP feasible, but integer infeasible