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

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

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

8. Example: Maximize x1+x2 x2 x1

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

10. 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.

11. Objective Function to be minimized Unbounded

12. Infeasible solution

13. LP feasible, but integer infeasible