1. Branch and Bound
See Beale paper

2. Example: Maximize z=x1+x2

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

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

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

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

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.