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 <= 1

[1 1.5]x2

x1

X1 >= 2

[2 1.5] , z=3.5, z=2.5

[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

[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

Example: Maximize x1+x2

x2

x1

SSbar

Sums edges out of S >= 2

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.

Objective Function to be minimized

Unbounded

Infeasible solution

LP feasible, but integer infeasible