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Gomory Cuts. Updated 25 March 2009. Example ILP. Example taken from “Operations Research: An Introduction” by Hamdy A. Taha (8 th Edition). Example ILP in Standard Form. Linear Programming Relaxation. LP Relaxation: Final Tableau. Row 1 Equation for x 2. - PowerPoint PPT Presentation
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Gomory Cuts
Updated 25 March 2009
Example ILP
integerand0,,35763s.t.
107max
21
21
21
21
xxxxxxxxz
Example taken from “Operations Research: An Introduction” by Hamdy A. Taha (8th Edition) 2
Example ILP in Standard Form
integer and 0,,,
357
63s.t.
107max
4321
421
321
21
xxxx
xxx
xxx
xxz
3
Linear Programming Relaxation
0,,,
357
63s.t.
107max
4321
421
321
21
xxxx
xxx
xxx
xxz
4
LP Relaxation: Final Tableau
1
2
4321
21422
322
10102
21322
122
71001
216622
3122
630010
BVRHSRow
x
x
z
xxxxz
5
Row 1 Equation for x2
2
13
22
1
22
7432 xxx
3integer and 0 22 xx
2
1
22
1
22
743 xx
Every feasible ILP solution satisfies this constraint.Cuts off the continuous LP optimum (4.5, 3.5).
6
Row 2 Equation for x1
2
14
22
3
22
1431 xxx
2
14
22
3
22
211 431
xxx
22
211
22
1complement
7
Row 2 Equation for x1
2
14
22
3
22
211 431
xxx
2
14
22
3
22
214331 xxxx
2
14
22
3
22
214331
xxxx
8
Row 2 Equation for x1
4negativity-non andy integralit 31 xx
2
1
22
3
22
2143 xx
2
14
22
3
22
214331
xxxx
Every feasible ILP solution satisfies this constraint.Cuts off the continuous LP optimum (4.5, 3.5).
9
Equation for z
2
166
22
31
22
6343 xxz
2
166
22
91
22
192 43
xxz
2
166
22
9
22
192 4343
xxxxz
10
Equation for z
2
166
22
9
22
192 4343
xxxxz
2
1
22
9
22
1943 xx
Every feasible ILP solution satisfies this constraint.Cuts off the continuous LP optimum (4.5, 3.5).
11
General Form of Gomory Cuts
2
166
22
9
22
192 4343
xxxxz
2
14
22
3
22
214331
xxxx
2
13
22
1
22
7432
xxx
12
General Form of Gomory Cuts
2
14
22
3
22
214331
xxxx
Integer Part
Fractional Part
13
General Form of Gomory Cuts
fxffIxfxcn
iii
n
iii
n
iii
111
Integer 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.
Fractional Part
Gomory Cut
14
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
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
Cutting Plane Algorithm Example: Cut 1
2
13
22
1
22
7432
xxx
2
1
22
1
22
743 xx
2
1
22
1
22
7543 xxx
17
Cutting Plane Algorithm Example: Cut 1
5
1
2
54321
21122
122
70003
214022
322
10102
213022
122
71001
2166022
3122
630010
BVRHSRow
x
x
x
z
xxxxxz
18
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, arj, is negative
b) The ratio -a0j /arj is minimized
4. Pivot on entry in row r of column j.
19
Cutting Plane Algorithm Example: Cut 1
5
1
2
54321
21122
122
70003
214022
322
10102
213022
122
71001
2166022
3122
630010
BVRHSRow
x
x
x
z
xxxxxz
20
Cutting Plane Algorithm Example: Cut 1
3
1
2
54321
7417
227
110003
7447
17
100102
31001001
629100010
BVRHSRow
x
x
x
z
xxxxxz
21
Cutting Plane Algorithm Example: Cut 2
3
1
2
54321
7417
227
110003
7447
17
100102
31001001
629100010
BVRHSRow
x
x
x
z
xxxxxz
22
Cutting Plane Algorithm Example: Cut 2
7
44
7
61
7
1541
xxx
7
44
7
6
7
15451
xxxx
7
44
7
6
7
15541 xxxx
23
Cutting Plane Algorithm Example: Cut 2
7
44
7
6
7
15451
xxxx
7
4
7
6
7
154 xx
7
4
7
6
7
1654 xxx
24
Cutting Plane Algorithm Example: Cut 2
7
44
7
6
7
15451
xxxx
7
1 Row32 Row4
21
52
51
xx
xxxx
25
Cutting Plane Algorithm Example: Cut 2
6
3
1
2
654321
7417
67
100004
74107
227
110003
74407
17
100102
301001001
6209100010
BVRHSRow
x
x
x
x
z
xxxxxxz
26
Cutting Plane Algorithm Example: Cut 2
6
3
1
2
654321
476100004
114010003
411000102
301001001
5873000010
BVRHSRow
x
x
x
x
z
xxxxxxz
Optimal ILP Solution: x1 = 4, x2 = 3, and z =5827
LP Relaxation: Graphical Solutionx2
x1
1
2
3
1 2 3 4 5
4 Optimal Solution: (4.5, 3.5)
28
LP Relaxation with Cut 1x2
x1
1
2
3
1 2 3 4 5
4
Optimal Solution: (4 4/7, 3)
29
LP Relaxation with Cuts 1 and 2x2
x1
1
2
3
1 2 3 4 5
4
Optimal Solution: (4, 3)
30