1. Gomory Cuts Updated 25 March 2009

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

3. Example ILP in Standard Form 3

4. Linear Programming Relaxation 4

5. LP Relaxation: Final Tableau 5

6. Row 1 Equation for x 2 Every feasible ILP solution satisfies this constraint. Cuts off the continuous LP optimum (4.5, 3.5). 6

7. Row 2 Equation for x 1 7

8. 8

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

10. Equation for z 10

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

12. General Form of Gomory Cuts 12

13. General Form of Gomory Cuts Integer Part Fractional Part 13

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

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), 275-278. 15

16. Cutting Plane Algorithm for ILP 1.Solve LP Relaxation with the Simplex Method 2.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 16

17. Cutting Plane Algorithm Example: Cut 1 17

18. Cutting Plane Algorithm Example: Cut 1 18

19. Dual Simplex Method 1.Select a basic variable with a negative value in the RHS column to leave the basis 2.Let r be the row selected in Step 1 3.Select a non-basic variable j to enter the basis such that a)The entry in row r of column j, a rj, is negative b)The ratio -a 0j /a rj is minimized 4.Pivot on entry in row r of column j. 19

20. Cutting Plane Algorithm Example: Cut 1 20

21. Cutting Plane Algorithm Example: Cut 1 21

22. Cutting Plane Algorithm Example: Cut 2 22

23. Cutting Plane Algorithm Example: Cut 2 23

24. Cutting Plane Algorithm Example: Cut 2 24

25. Cutting Plane Algorithm Example: Cut 2 25

26. Cutting Plane Algorithm Example: Cut 2 26

27. Cutting Plane Algorithm Example: Cut 2 Optimal ILP Solution: x 1 = 4, x 2 = 3, and z =58 27

28. LP Relaxation: Graphical Solution x2x2 x1x1 1 2 3 12345 4 Optimal Solution: (4.5, 3.5) 28

29. LP Relaxation with Cut 1 x2x2 x1x1 1 2 3 12345 4 Optimal Solution: (4 4/7, 3) 29

30. LP Relaxation with Cuts 1 and 2 x2x2 x1x1 1 2 3 12345 4 Optimal Solution: (4, 3) 30