Gomory Cuts

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
lt 1. On the right-hand side, I is an integer and
0 lt f lt 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), 275-278.

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 Solution
x2
4
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
1
2
3
4
5
30
LP Relaxation with Cuts 1 and 2
x2
4
3
Optimal Solution (4, 3)
2
1
x1
1
2
3
4
5