gomory’s cutting plane algorithm for integer programming
DESCRIPTION
Gomory’s cutting plane algorithm for integer programming. Prepared by Shin-ichi Tanigawa. Rounding does not give any useful result. We first solve the LP-relaxation. Optimize using primal simplex method. Optimize using primal simplex method. Optimize using primal simplex method. - PowerPoint PPT PresentationTRANSCRIPT
Gomory’s cutting plane algorithm for integer
programming
Prepared by Shin-ichi Tanigawa
Rounding does not give any useful result
2
214
213
230236
xzxxxxxx
We first solve the LP-relaxation
2
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230236
xzxxxxxx
Optimize using primal simplex method
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214
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230236
xzxxxxxx
41
412
413
21
23
21
230
66
xxz
xxx
xxx
Optimize using primal simplex method
2
214
213
230236
xzxxxxxx
41
412
413
21
23
21
230
66
xxz
xxx
xxx
43
432
431
41
41
23
41
41
23
61
611
xxz
xxx
xxx
Optimize using primal simplex method
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432
431
41
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23
41
41
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61
611
xxz
xxx
xxx
The optimal solution is fractional
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432
431
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41
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611
xxz
xxx
xxx
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41 43 xxz (1)
Generating an objective row cut
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432
431
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41
23
41
41
23
61
611
xxz
xxx
xxx
23
41
41 43 xxz
23
41
41 43
xxz (weakening)
(1)
Generating an objective row cut
43
432
431
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41
23
41
41
23
61
611
xxz
xxx
xxx
23
41
41 43 xxz
23
41
41 43 xxz
23
41
41 43
xxz (weakening)
(1)
(2)(for integers)
Generating an objective row cut
43
432
431
41
41
23
41
41
23
61
611
xxz
xxx
xxx
23
41
41 43 xxz
21
41
41
43 xx
23
41
41 43 xxz
23
41
41 43
xxz (weakening)
(1)
(2)(for integers)
(2) - (1)
Generating an objective row cut
Cutting plane is violated by current optimum solution
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432
431
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41
23
41
41
23
61
611
xxz
xxx
xxx
23
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41 43 xxz
21
41
41
43 xx
12 x
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41
41 43 xxz
23
41
41 43
xxz (weakening)
(1)
(2)(for integers)
(2) - (1)
(substitute for slacks)
Generating an objective row cut
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230236xxxxxx
The first cutting plane: 12 x21
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41
43 xx
435 41
41
21 xxx
A new slack variable is added:
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43 xx
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435
432
431
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41
23
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41
21
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41
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61
611
xxz
xxx
xxx
xxx
The new cut is added to the dictionary
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43 xx
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435
432
431
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41
23
41
41
21
41
41
23
61
611
xxz
xxx
xxx
xxx
Re-optimize using dual simplex method
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432
431
41
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23
41
41
21
41
41
23
61
611
xxz
xxx
xxx
xxx
5
453
52
451
1 4 2
1 31
32
32
xzxxx
xx
xxx
Re-optimize using dual simplex method
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453
52
451
1 4 2
1 31
32
32
xzxxx
xx
xxx
A new fractional solution has been found
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453
52
451
1 4 2
1 31
32
32
xzxxx
xx
xxx
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31
32
451 xxx (1)Generating a constraint row cut
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453
52
451
1 4 2
1 31
32
32
xzxxx
xx
xxx
32
31
32
451
xxx
32
31
32
451 xxx (1)
(weaken)
Generating a constraint row cut
5
453
52
451
1 4 2
1 31
32
32
xzxxx
xx
xxx
32
31
32
451
xxx
32
31
32
451 xxx (1)
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31
32
451 xxx (2)
Generating a constraint row cut
(valid for integers)
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453
52
451
1 4 2
1 31
32
32
xzxxx
xx
xxx
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31
32
451
xxx
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31
32
451 xxx (1)
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31
32
451 xxx (2)
(2) – (1)
Generating a constraint row cut
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32
32
45 xx
5
453
52
451
1 4 2
1 31
32
32
xzxxx
xx
xxx
021 xx
32
31
32
451
xxx
32
31
32
451 xxx (1)
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31
32
451 xxx (2)
(2) – (1)
Generating a constraint row cut
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32
45 xx
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230236xxxxxx
The second cutting plane
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32
45 xx
021 xx
456 32
32
32 xxx
Add a new slack variable:32
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32
45 xx
5
456
453
52
451
1 32
32
32
4 2 1
31
32
32
xz
xxx
xxxxx
xxx
The new cut is inserted into the optimum dictionary
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32
32
45 xx
5
456
453
52
451
1 32
32
32
4 2 1
31
32
32
xz
xxx
xxxxx
xxx
5
654
653
52
651
1 32 1
235 1
1 211
xz
xxx
xxx
xx
xxx
Re-optimize using dual simplex method
5
654
653
52
651
1 32 1
235 1
1 211
xz
xxx
xxx
xx
xxx
The new optimum solution is integral