1. Gomory Cuts
Updated 25 March 2009

2. Example ILP
Example taken from “Operations Research: An Introduction” by Hamdy A. Taha (8th Edition)

3. Example ILP in Standard Form

4. Linear Programming Relaxation

5. LP Relaxation: Final Tableau

6. Row 1 Equation for x2
Every feasible ILP solution satisfies this constraint.
Cuts off the continuous LP optimum (4.5, 3.5).

7. Row 2 Equation for x1

8. Row 2 Equation for x1

9. Row 2 Equation for x1
Every feasible ILP solution satisfies this constraint.
Cuts off the continuous LP optimum (4.5, 3.5).

10. Equation for z

11. Equation for z Every feasible ILP solution satisfies this constraint.
Cuts off the continuous LP optimum (4.5, 3.5).

12. General Form of Gomory Cuts

13. General Form of Gomory Cuts
Integer Part
Fractional Part

14. General Form of Gomory Cuts
Integer Part
Gomory Cut
Fractional Part
For each variable xi, ci is an integer and 0  fi < 1.
On the right-hand side, I is an integer and 0 < f < 1.

15. Comments on Gomory Cuts
Also called fractional cuts
Assume all variables are integer and non-negative
Apply to pure integer linear programs with integer coefficients
Strengthen linear programming relaxation of ILP by restricting the feasible region
“Outline of an algorithm for integer solutions to linear programs” by Ralph E. Gomory. Bull. Amer. Math. Soc. Volume 64, Number 5 (1958),

16. Cutting Plane Algorithm for ILP
Solve LP Relaxation with the Simplex Method
Until Optimal Solution is Integral Do
Derive a Gomory cut from the Simplex tableau
Add cut to tableau
Use a Dual Simplex pivot to move to a feasible solution

17. Cutting Plane Algorithm Example: Cut 1

18. Cutting Plane Algorithm Example: Cut 1

19. Dual Simplex Method
Select a basic variable with a negative value in the RHS column to leave the basis
Let r be the row selected in Step 1
Select a non-basic variable j to enter the basis such that
The entry in row r of column j, arj, is negative
The ratio -a0j /arj is minimized
Pivot on entry in row r of column j.

20. Cutting Plane Algorithm Example: Cut 1

21. Cutting Plane Algorithm Example: Cut 1

22. Cutting Plane Algorithm Example: Cut 2

23. Cutting Plane Algorithm Example: Cut 2

24. Cutting Plane Algorithm Example: Cut 2

25. Cutting Plane Algorithm Example: Cut 2

26. Cutting Plane Algorithm Example: Cut 2

27. Cutting Plane Algorithm Example: Cut 2
Optimal ILP Solution: x1 = 4, x2 = 3, and z =58

28. LP Relaxation: Graphical Solution4
Optimal Solution: (4.5, 3.5) 3 2 1 x1 1 2 3 4 5

29. LP Relaxation with Cut 1 x2 4 3 Optimal Solution: (4 4/7, 3) 2 1 x1 15

30. LP Relaxation with Cuts 1 and 24 3
Optimal Solution: (4, 3) 2 1 x1 1 2 3 4 5