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Branch and Bound

Branch and Bound. See Beale paper. Example: Maximize z=x1+x2. x2. x1. Solve First LP problem:. Solution is [1.5 2.5]. x2. x1. [1.5 2.5]. X1 >= 2. X1 <= 1. [2 1.5] , z=3.5. [1 1.5]. , z=2.5. x2. x1. [1.5 2.5]. X1 >= 2. X1 <= 1. [2 1.5] , z=3.5. [1 1.5]. , z=2.5.

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Branch and Bound

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  1. Branch and Bound See Beale paper

  2. Example: Maximize z=x1+x2 x2 x1

  3. Solve First LP problem: • Solution is [1.5 2.5] x2 x1

  4. [1.5 2.5] X1 >= 2 X1 <= 1 [2 1.5] , z=3.5 [1 1.5] , z=2.5 x2 x1

  5. [1.5 2.5] X1 >= 2 X1 <= 1 [2 1.5] , z=3.5 [1 1.5] , z=2.5 X2<= 1 x2>= 2 [2.25, 1], z=3.25 No solution x2 x1

  6. [2 1.5] , z=3.5 X2<= 1 x2>= 2 [1 1.5] , z=2.5 [2.25, 1], z=3.25 No solution X1 <= 2 x1 >= 3 [2,1], z=3 No solution x2 x1

  7. Example: Maximize x1+x2 x2 x1

  8. S Sbar Sums edges out of S >= 2

  9. 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.

  10. Unbounded Objective Function to be minimized

  11. Infeasible solution

  12. LP feasible, but integer infeasible

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