Optimizing over the Split Closure
Anureet Saxena
ACO PhD Student,
Tepper School of Business,
Carnegie Mellon University.
(Joint Work with Egon Balas)
Talk Outiline
• Cutting Planes Commercial
• Split Closure Separation Problem
• PMILP & Deparametrization
• Computational Results
• Support Size & Sparsity
• Support Coefficients
• Cuts Statistics
• arki001 solved
Anureet Saxena, TSoB
1
MIP Model
min cx
Ax ¸ b
xj 2 Z 8 j2N1
Contains
xj ¸ 0 j2N
xj · uj j2N1
N1: set of integer variables
Incumbent
Fractional
Solution
Anureet Saxena, TSoB
2
Taxonomy of Cutting Planes
Fractional
Basic
MIR
Fractional
Gomory
Split Cuts
Fractional
Basic Feas
Chvatal
MIG
Mixed Integer
Basic Feas
Mixed Integer
Basic
Intersection
Basic +
Strengthening
Intersection
Basic
L&P
L&P +
Strengthening
Intersection
Basic Feas +
Strengthening
Intersection
Basic Feas
Anureet Saxena, TSoB
Simple
Disjunctive
3
Taxonomy of Cutting Planes
Fractional
Basic
MIR
Fractional
Gomory
Split Cuts
Fractional
Basic Feas
Chvatal
MIG
Mixed Integer
Basic Feas
Mixed Integer
Basic
Intersection
Basic +
Strengthening
Intersection
Basic
L&P
L&P +
Strengthening
Intersection
Basic Feas +
Strengthening
Intersection
Basic Feas
Anureet Saxena, TSoB
Simple
Disjunctive
4
Taxonomy of Cutting Planes
Fractional
Basic
MIR
Fractional
Gomory
Split Cuts
Fractional
Basic Feas
Chvatal
MIG
Mixed Integer
Basic Feas
Mixed Integer
Basic
Intersection
Basic +
Strengthening
Intersection
Basic
L&P
L&P +
Strengthening
Intersection
Basic Feas +
Strengthening
Intersection
Basic Feas
Anureet Saxena, TSoB
Simple
Disjunctive
5
Taxonomy of Cutting Planes
Fractional
Basic
MIR
Fractional
Gomory
Split Cuts
Fractional
Basic Feas
Chvatal
MIG
Mixed Integer
Basic Feas
Mixed Integer
Basic
Intersection
Basic +
Strengthening
Intersection
Basic
L&P
L&P +
Strengthening
Intersection
Basic Feas +
Strengthening
Intersection
Basic Feas
Anureet Saxena, TSoB
Simple
Disjunctive
6
Taxonomy of Cutting Planes
Fractional
Basic
MIR
Fractional
Gomory
Split Cuts
Elementary
Chvatal Closure
Fractional
Basic Feas
MIG
Elementary closure of P w.r.t a family
 of cutting planes is defined by
Mixed Integer
Basic Feas
intersecting P with all rank-1 cuts in
Intersection
.
Mixed Integer
Basic
Eg: CG
Intersection
Basic Feas +
Strengthening
Intersection
Basic Feas
Basic +
Strengthening
L&P +
Strengthening
Closure, Split Closure
Intersection
Basic
Anureet Saxena, TSoB
L&P
Simple
Disjunctive
7
Elementary Closures
Fractional
Basic
MIR
Fractional
Gomory
Split Cuts
Fractional
Basic Feas
CG Closure
Chvatal
Split Closure
MIG
Mixed Integer
Basic Feas
Mixed Integer
Basic
Intersection
Basic Feas +
Strengthening
Intersection
Basic Feas
Intersection
Basic +
Strengthening
L&P +
Strengthening
L&P Closure
Intersection
Basic
Anureet Saxena, TSoB
L&P
Simple
Disjunctive
8
Elementary Closures
Operations Research
Inference Dual
max v
x2PI )P cx¸v
P2 
Elementary Closures
Rank-1 cuts have
short polynomial
length proofs
Proof Family
Constraint Programming
Complexity Theory
Anureet Saxena, TSoB
9
Elementary Closures
How much duality gap can be closed by
optimizing over elementary closures?
L&P Closure
CG Closure
Split Closure
Bonami and Minoux
Fischetti and Lodi
?
Anureet Saxena, TSoB
10
Elementary Closures
How much duality gap can be closed by
optimizing over elementary closures?
L&P Closure
CG Closure
Split Closure
Bonami and Minoux
Fischetti and Lodi
Balas and Saxena
Anureet Saxena, TSoB
11
Split Disjunctions
•
•
•
 2 ZN, 0 2 Z
j = 0, j2 N2
0 <  < 0 + 1
 x · 0
 x ¸ 0 + 1
Split Disjunction
Anureet Saxena, TSoB
12
Split Cuts
u
u0
Ax ¸ b
 x · 0
Ax ¸ b
 x ¸ 0+1
L x ¸ L
R x ¸ R
x¸
Anureet Saxena, TSoB
v
v0
Split Cut
13
Split Closure
Elementary Split Closure of P = { x | Ax ¸ b } is the
polyhedral set defined by intersecting P with the valid
rank-1 split cuts.
C = { x2 P |  x ¸  8 rank-1 split cuts  x¸ }
Without Recursion
Anureet Saxena, TSoB
14
Algorithmic Framework
Add Cuts
min cx
Ax ¸ b
t x¸ t t2
Solve
Master LP
Integral Sol?
Unbounded?
Infeasible?
Yes
MIP Solved
No
Split Cuts
Generated
Rank-1 Split Cut
Separation
No Split Cuts
Generated
Anureet Saxena, TSoB
Optimum over
Split Closure
attained
15
Algorithmic Framework
Add Cuts
min cx
Ax ¸ b
t x¸ t t2
Solve
Master LP
Integral Sol?
Unbounded?
Infeasible?
Yes
MIP Solved
No
Split Cuts
Generated
Rank-1 Split Cut
Separation
No Split Cuts
Generated
Anureet Saxena, TSoB
Optimum over
Split Closure
attained
16
Split Closure Separation Problem
Theorem: lies in the split closure of P if and only if the optimal value
of the following program is non-negative.
Cut Violation
•
•
•
•
•
=1
Disjunctive
Cut
u.e + v.e + u0 + v0 = 1
u0 + v0 = 1
y=1
||2=1
Split Disjunction
Normalization Set
Anureet Saxena, TSoB
17
Split Closure Separation Problem
Theorem: lies in the split closure of P if and only if the optimal value
of the following program is non-negative.
Mixed Integer
Non-Convex
Quadratic Program
u0 + v0 = 1
Anureet Saxena, TSoB
18
SC Separation Theorem
Theorem: lies in the split closure of P if and only if the optimal
value of the following parametric mixed integer linear program is
non-negative.
Parameter
Parametric
Mixed Integer
Linear Program
Anureet Saxena, TSoB
19
Deparametrization
Parameteric Mixed
Integer Linear
Program
Anureet Saxena, TSoB
20
Deparametrization
Parameteric Mixed
Integer Linear
Program
If  is fixed,
then PMILP reduces
to a MILP
Anureet Saxena, TSoB
21
Deparametrization
MILP( )
Deparametrized
Mixed Integer
Linear Program
Maintain a dynamically
updated grid of parameters
Anureet Saxena, TSoB
22
Separation Algorithm
Initialize Parameter Grid (  )
For  2 ,
Diversification
•Solve MILP() using CPLEX 9.0
• Enumerate  branch and bound nodes
• Store all the separating split disjunctions which
are discovered
Strengthening
At least one
Grid Enrichment
no
split disjunction
yes
STOP
discovered?
Bifurcation
Anureet Saxena, TSoB
23
Implementation Details
Processor Details
• Pentium IV
• 2Ghz, 2GB RAM
COIN-OR
CPLEX 9.0
Core Implementation
• Solving Master LP
• Setting up MILP
• Disjunctions/Cuts Management
• L&P cut generation+strengthening
Anureet Saxena, TSoB
Solving MILP(  )
24
Computational Results
• MIPLIB 3.0 instances
• OR-Lib (Beasley) Capacitated Warehouse Location
Problems
Anureet Saxena, TSoB
25
MIPLIB 3.0 MIP Instances
Instance
10teams
dcmulti
egout
flugpl
gen
gesa2_o
khb05250
misc06
qnet1
qnet1_o
rgn
vpm1
fixnet6
gesa2
14 Instances
# Int Var
1800
75
55
11
150
720
24
112
1417
1417
100
168
378
408
# Cont Var
225
473
86
7
720
504
1326
1696
124
124
80
210
500
816
% Gap Closed
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
99.76%
98.66%
98-100% Gap Closed
Anureet Saxena, TSoB
26
MIPLIB 3.0 MIP Instances
Instance
10teams
dcmulti
egout
flugpl
gen
gesa2_o
khb05250
misc06
qnet1
qnet1_o
rgn
vpm1
fixnet6
gesa2
14 Instances
# Int Var
1800
75
55
11
150
720
24
112
1417
1417
100
168
378
408
# Cont Var
225
473
86
7
720
504
1326
1696
124
124
80
210
500
816
% Gap Closed
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
99.76%
98.66%
98-100% Gap Closed
Anureet Saxena, TSoB
27
MIPLIB 3.0 MIP Instances
Instance
pp08aCUTS
modglob
pp08a
gesa3
gesa3_o
set1ch
bell5
arki001
vpm2
mod011
qiu
11 Instances
# Int Var
64
98
64
384
672
240
58
538
168
96
48
Unsolved MIP Instance
In MIPLIB 3.0
# Cont Var
176
324
176
768
480
472
46
850
210
10862
792
% Gap Closed
97.01%
96.48%
95.81%
95.78%
95.31%
89.41%
87.44%
83.05%
81.22%
80.72%
77.51%
75-98% Gap Closed
Anureet Saxena, TSoB
28
MIPLIB 3.0 MIP Instances
Instance
rout
bell3a
blend2
3 Instances
# Int Var
315
71
264
# Cont Var
241
62
89
% Gap Closed
70.73%
55.19%
46.77%
25-75% Gap Closed
Anureet Saxena, TSoB
29
MIPLIB 3.0 MIP Instances
Instance
danoint
dano3mip
pk1
3 Instances
# Int Var
56
552
55
# Cont Var
465
13321
31
% Gap Closed
7.44%
0.12%
0.00%
0-25% Gap Closed
Anureet Saxena, TSoB
30
MIPLIB 3.0 MIP Instances
Summary of MIP Instances (MIPLIB 3.0)
Total Number of Instances: 34
Number of Instances included: 33
No duality gap: noswot, dsbmip
Instance not included: rentacar
Results
98-100% Gap closed in 14 instances
75-98% Gap closed in 11 instances
25-75% Gap closed in 3 instances
0-25% Gap closed in 3 instances
Average Gap Closed: 82.53%
Anureet Saxena, TSoB
31
MIPLIB 3.0 Pure IP Instances
Instance
air03
gt2
mitre
mod008
mod010
nw04
p0548
p0282
fiber
9 Instances
# Variables
10757
188
10724
319
2655
87482
548
282
1254
%Gap Closed
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
100.00%
99.90%
98.50%
98-100% Gap Closed
Anureet Saxena, TSoB
32
MIPLIB 3.0 Pure IP Instances
Instance
lseu
p2756
l152lav
p0033
4 Instances
# Variables
89
2756
1989
33
%Gap Closed
93.75%
92.32%
87.56%
87.42%
75-98% Gap Closed
Anureet Saxena, TSoB
33
MIPLIB 3.0 Pure IP Instances
Instance
p0201
air04
air05
seymour
misc03
cap6000
6 Instances
Ceria, Pataki et al
closed around 50%
of the gap using
10 rounds of L&P
cuts
# Variables
201
8904
7195
1372
159
6000
%Gap Closed
74.93%
62.42%
62.05%
61.94%
51.47%
37.63%
25-75% Gap Closed
Anureet Saxena, TSoB
34
MIPLIB 3.0 Pure IP Instances
Instance
misc07
fast0507
stein27
stein45
4 Instances
# Variables
259
63009
27
45
%Gap Closed
19.48%
18.08%
0.00%
0.00%
0-25% Gap Closed
Anureet Saxena, TSoB
35
MIPLIB 3.0 Pure IP Instances
Summary of Pure IP Instances (MIPLIB 3.0)
Total Number of Instances: 25
Number of Instances included: 24
No duality gap: enigma
Instance not included: harp2
Results
98-100% Gap closed in 9 instances
75-98% Gap closed in 4 instances
25-75% Gap closed in 6 instances
0-25% Gap closed in 4 instances
Average Gap Closed: 71.63%
Anureet Saxena, TSoB
36
MIPLIB 3.0 Pure IP Instances
Instance
air03
air04
air05
cap6000
enigma
fast0507
fiber
gt2
l152lav
lseu
misc03
misc07
mitre
mod008
mod010
nw04
p0033
p0201
p0282
p0548
p2756
seymour
stein27
stein45
24 Instances
# Variables
10757
8904
7195
6000
100
63009
1254
188
1989
89
159
259
10724
319
2655
87482
33
201
282
548
2756
1372
27
45
%Gap Closed (SC)
100.00%
62.42%
62.05%
37.63%
18.08%
98.50%
100.00%
87.56%
93.75%
51.47%
19.48%
100.00%
100.00%
100.00%
100.00%
87.42%
74.93%
99.90%
100.00%
92.32%
61.94%
0.00%
0.00%
71.63%
%Gap Closed (CG)
100.00%
27.60%
15.50%
26.90%
4.70%
98.50%
100.00%
69.20%
91.30%
51.20%
16.10%
100.00%
100.00%
100.00%
100.00%
85.40%
60.50%
99.90%
100.00%
69.20%
23.50%
0.00%
0.00%
62.59%
% Gap Closed by
First Chvatal Closure
(Fischetti-Lodi Bound)
Anureet Saxena, TSoB
37
MIPLIB 3.0 Pure IP Instances
Instance
air03
air04
air05
cap6000
enigma
fast0507
fiber
gt2
l152lav
lseu
misc03
misc07
mitre
mod008
mod010
nw04
p0033
p0201
p0282
p0548
p2756
seymour
stein27
stein45
24 Instances
# Variables
10757
8904
7195
6000
100
63009
1254
188
1989
89
159
259
10724
319
2655
87482
33
201
282
548
2756
1372
27
45
%Gap Closed (SC)
100.00%
62.42%
62.05%
37.63%
18.08%
98.50%
100.00%
87.56%
93.75%
51.47%
19.48%
100.00%
100.00%
100.00%
100.00%
87.42%
74.93%
99.90%
100.00%
92.32%
61.94%
0.00%
0.00%
71.63%
Anureet Saxena, TSoB
%Gap Closed (CG)
100.00%
27.60%
15.50%
26.90%
4.70%
98.50%
100.00%
69.20%
91.30%
51.20%
16.10%
100.00%
100.00%
100.00%
100.00%
85.40%
60.50%
99.90%
100.00%
69.20%
23.50%
0.00%
0.00%
62.59%
38
MIPLIB 3.0 Pure IP Instances
Instance
air03
air04
air05
cap6000
enigma
fast0507
fiber
gt2
l152lav
lseu
misc03
misc07
mitre
mod008
mod010
nw04
p0033
p0201
p0282
p0548
p2756
seymour
stein27
stein45
24 Instances
# Variables
10757
8904
7195
6000
100
63009
1254
188
1989
89
159
259
10724
319
2655
87482
33
201
282
548
2756
1372
27
45
%Gap Closed (SC)
100.00%
62.42%
62.05%
37.63%
18.08%
98.50%
100.00%
87.56%
93.75%
51.47%
19.48%
100.00%
100.00%
100.00%
100.00%
87.42%
74.93%
99.90%
100.00%
92.32%
61.94%
0.00%
0.00%
71.63%
Anureet Saxena, TSoB
%Gap Closed (CG)
100.00%
27.60%
15.50%
26.90%
4.70%
98.50%
100.00%
69.20%
91.30%
51.20%
16.10%
100.00%
100.00%
100.00%
100.00%
85.40%
60.50%
99.90%
100.00%
69.20%
23.50%
0.00%
0.00%
62.59%
39
MIPLIB 3.0 Pure IP Instances
Instance
air04
air05
cap6000
fast0507
l152lav
lseu
misc03
misc07
p0033
p0201
p2756
seymour
12 Instances
# Variables
8904
7195
6000
63009
1989
89
159
259
33
201
2756
1372
%Gap Closed (SC)
62.42%
62.05%
37.63%
18.08%
87.56%
93.75%
51.47%
19.48%
87.42%
74.93%
92.32%
61.94%
62.42%
%Gap Closed (CG)
27.60%
15.50%
26.90%
4.70%
69.20%
91.30%
51.20%
16.10%
85.40%
60.50%
69.20%
23.50%
45.09%
Comparison of Split Closure vs CG Closure
Total Number of Instances: 24
CG closure closes >98% Gap: 9
Ratio
2.261
4.003
1.347
3.847
1.265
1.027
1.005
1.210
1.024
1.239
1.334
2.636
1.850
Results (Remaining 15 Instances)
Split Closure closes significantly more gap in 9 instances
Both Closures close almost same gap in 6 instances
Anureet Saxena, TSoB
40
OrLib CWLP
• Set 1
– 37 Real-World Instances
– 50 Customers, 16-25-50 Warehouses
• Set 2
– 12 Real-World Instances
– 1000 Customers, 100 Warehouses
Anureet Saxena, TSoB
41
OrLib CWLP Set 1
Summary
CWLP
Instances
(Set 1)
Instance of OrLib
% Gap Closed
Instance
% Gap Closed
100.000%
100.000%
cap41
cap93
Number of Instances:
37
100.000%
100.000%
cap42
cap94
100.000%
100.000%
cap43
cap101
Number of Instances
included:
37
100.000%
100.000%
cap44
cap102
cap51
cap61
cap62
cap63
cap64
cap71
cap72
cap73
cap74
cap81
cap82
cap83
cap84
cap91
cap92
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
cap103
cap104
cap111
cap112
cap113
cap114
cap121
cap122
cap123
cap124
cap131
cap132
cap133
cap134
Results
100% Gap closed in 37 instances
Anureet Saxena, TSoB
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
100.000%
42
OrLib CWLP Set 2
Instance
% Gap Closed
Summary of capa_8000
OrLib CWFL Instances
(Set 2)
87.00%
87.57%
Number of Instances:capa_10000
12
91.53%
Number of Instancescapa_12000
included: 12
capa_14000
capb_5000
Results
capb_6000
capb_7000
>90% Gap closed in 10
instances
85-90% Gap closed incapb_8000
2 instances
capc_5000
Average Gap Closed:capc_5750
92.82%
capc_6500
capc_7250
97.53%
94.18%
92.80%
92.42%
93.74%
93.42%
93.90%
94.69%
95.06%
Anureet Saxena, TSoB
43
Algorithmic Framework
Add Cuts
min cx
Ax ¸ b
t x¸ t t2
Solve
Master LP
Integral Sol?
Unbounded?
Infeasible?
Yes
MIP Solved
No
Split Cuts
Generated
Rank-1 Split Cut
Separation
No Split Cuts
Generated
Anureet Saxena, TSoB
Optimum over
Split Closure
attained
44
Algorithmic Framework
Add Cuts
min cx
Ax ¸ b
t x¸ t t2
Solve
Master LP
What are the
characteristics of
the cuts which are
binding at the final
optimal solution?
Integral Sol?
Yes
MIP Solved
Unbounded?
Infeasible? What can one say about
the split disjunctions
which were used to
generate cuts?
No
Split Cuts
Generated
Rank-1 Split Cut
Separation
No Split Cuts
Generated
Anureet Saxena, TSoB
Optimum over
Split Closure
attained
45
Support Size & Sparsity
The support of a split disjunction D(, 0) is the set of
non-zero components of 
 x · 0
 x ¸ 0 + 1
(2x1 + 3x3 – x5 · 1)
Ç
(2x1 + 3x3 – x5 ¸ 2)
Support Size = 3
Anureet Saxena, TSoB
46
Support Size & Sparsity
The support of a split disjunction D(, 0) is the set of
non-zero components of 
• Computationally Faster
• Avoid fill-in
Sparse
Split Disjunctions
Disjunctive argument
Non-negative row
combinations
Sparse
Split Cuts
Anureet Saxena, TSoB
Basis Factorization
Sparse Matrix Op
47
Support Size & Sparsity
air05 [ MIPLIB 3.0 Pure IP instance, 7195 Int Var, 62.05% Gap closed by Split Closure ]
Mean Support Size
Standard Deviation
30
25
15
10
5
157
153
149
145
141
137
133
129
125
121
117
113
109
105
101
97
93
89
85
81
77
73
69
65
61
57
53
49
45
41
37
33
29
25
21
17
9
13
5
0
1
Support Size
20
Iteration
Anureet Saxena, TSoB
48
Support Size & Sparsity
arki001** [ MIPLIB 3.0 Mixed IP instance, 538 Int Var, 83.35% Gap closed by Split Closure ]
Mean Support Size
Standard Deviation
10
9
8
6
5
4
3
2
1
91
88
85
82
79
76
73
70
67
64
61
58
55
52
49
46
43
40
37
34
31
28
25
22
19
16
13
10
7
4
0
1
Support Size
7
Iteration
Anureet Saxena, TSoB
49
Support Size & Sparsity
Pure IP Instance
nw04
air05
seymour
misc03
p0033
# Int Variables
87482
7195
1372
159
33
Mean Support Size
2.084
8.210
5.263
3.771
4.847
Empirical Observation
Substantial Duality gap can be closed
Mixed IP Instance
# Int Variables
Mean Support Size
by
using
split
cuts
generated
from 6.690
qnet1_o
1417
gesa2_o
720
4.937
sparse split disjunctions
arki001
538
3.146
vpm1
168
4.503
pp08aCUTS
64
3.850
Anureet Saxena, TSoB
50
Support Coefficients
Practice
Theory
• Determinants of sub-matrices
• Andersen, Cornuejols & Li (’05)
• Cook, Kannan & Scrhijver (’90)
• Elementary 0/1 disjunctions
• Mixed Integer Gomory Cuts
• Lift-and-project cuts
Huge Gap
det (B)
1
Anureet Saxena, TSoB
51
Support Coefficients
air05 [ MIPLIB 3.0 Pure IP instance, 7195 Int Var, 62.05% Gap closed by Split Closure ]
Mean Coef Size
Standard Deviation
8
7
5
4
3
2
1
157
153
149
145
141
137
133
129
125
121
117
113
109
105
101
97
93
89
85
81
77
73
69
65
61
57
53
49
45
41
37
33
29
25
21
17
9
13
5
0
1
Coefficient Size
6
Iteration
Anureet Saxena, TSoB
52
Support Coefficients
arki001** [ MIPLIB 3.0 Mixed IP instance, 538 Int Var, 83.35% Gap closed by Split Closure ]
Mean Coef Size
Standard Deviation
40
35
Coefficient Size
30
25
20
15
10
5
0
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
85
88
91
Iteration
Anureet Saxena, TSoB
53
Support Coefficients
Pure IP Instance
nw04
air05
seymour
misc03
p0033
#Int Variables
87482
7195
1372
159
33
Mean Coef Size
1.228
1.156
1.099
1.227
2.099
Empirical Observation
Substantial Duality gap can be closed
Mixed by
IP Instance
Variables
Mean
Coef Size
using split#Int
cuts
generated
from
qnet1_o
1417
2.381
split
disjunctions
containing
gesa2_o
720
1.767
small support coefficients.
arki001
538
3.044
vpm1
pp08aCUTS
168
64
Anureet Saxena, TSoB
1.833
1.418
54
Cuts Statistics
Instance Name
10teams
air03
air04
air05
arki001
bell3a
bell5
blend2
cap6000
dano3mip
dsbmip
egout
enigma
fiber
fixnet6
flugpl
gen
gesa2
gesa2_o
gesa3
khb05250
l152lav
lseu
misc03
n
2025
10757
7564
7195
959
100
104
334
5911
13873
2320
141
100
1298
878
35
870
1224
1224
1152
1350
1989
89
160
m
230
124
614
426
782
86
91
184
2095
3202
1182
98
21
363
478
24
780
1392
1248
1368
101
97
28
96
ncuts
37
3
185
192
227
19
26
3
22
123
32
57
4
113
321
6
32
116
140
129
72
80
19
71
Avg Cut Density
37.75%
26.74%
24.81%
31.48%
15.96%
5.79%
10.80%
26.35%
99.90%
50.92%
4.16%
2.75%
55.50%
6.92%
2.75%
20.95%
1.09%
1.10%
1.73%
1.62%
5.98%
67.62%
31.87%
31.63%
Instance Name
misc06
mitre
mod008
mod010
mod011
modglob
noswot
nw04
p0033
p0201
p0282
p2756
pk1
pp08a
pp08aCUTS
qiu
qnet1
qnet1_o
rgn
rout
set1ch
seymour
stein27
stein45
vpm1
vpm2
Anureet Saxena, TSoB
n
1808
10724
319
2655
6872
384
128
87482
33
201
282
2756
86
240
240
840
1541
1541
180
556
712
1372
27
45
378
378
m
820
2054
6
146
1535
286
182
36
16
133
241
755
45
136
246
1192
503
456
24
291
492
4944
118
331
234
234
ncuts
31
523
16
9
674
105
33
136
16
36
69
298
20
127
113
220
113
111
39
124
223
413
20
34
47
111
Avg Cut Density
1.53%
0.50%
76.25%
35.76%
5.29%
4.93%
15.63%
63.46%
26.52%
30.21%
5.15%
1.72%
61.05%
4.21%
4.64%
13.00%
5.23%
2.41%
20.11%
19.22%
1.06%
1.88%
27.59%
43.14%
2.02%
2.16%
55
isc
u
Anureet Saxena, TSoB
vp
m
1
rg
n
se
t1
ch
st
ei
n2
7
qn
et
1
pk
pp
1
08
aC
UT
S
m
06
od
00
8
m
od
01
1
no
sw
ot
p0
03
3
p0
28
2
m
lse
ge
n
ge
sa
2_
o
kh
b0
52
50
ncuts
600
be
ll5
ca
p6
00
0
ds
bm
ip
en
ig
m
a
f ix
ne
t6
4
ar
ki0
01
ai
r0
10
te
am
s
Number of Cuts
#Cuts
800
700
Average: 113.80
500
400
300
200
100
0
instances
56
#Cuts/m vs log(n)
#Cuts/m
400.00%
350.00%
300.00%
ncuts/m
250.00%
Average: 45.76%
200.00%
150.00%
100.00%
50.00%
0.00%
0.000
1.000
2.000
3.000
4.000
5.000
6.000
log(n)
Anureet Saxena, TSoB
57
Average Cut Density vs log(n)
Avg Cut Density
120.00%
Avg Cut density
100.00%
80.00%
Average: 20.82%
60.00%
40.00%
20.00%
0.00%
0.000
1.000
2.000
3.000
4.000
5.000
6.000
log(n)
Anureet Saxena, TSoB
58
Cuts Statistics
Internet Checkable Proofs
Strengthened formulations for MIPLIB 3.0 instances
available at
www.andrew.cmu.edu/user/anureets/osc/osc.htm
Google Query: anureet saxena (I’m feeling lucky)
Anureet Saxena, TSoB
59
arki001
• MIPLIB 3.0 & 2003 instance
• Metallurgical Industry
Problem Stats
• Unsolved for the past 10 years [1996-2000-2005]
1048 Rows
1388 Columns
123 Gen Integer Vars
415 Binary Vars
850 Continuous Vars
Anureet Saxena, TSoB
60
Solution Strategy
Original
Problem
CPLEX 9.0
Presolver
Preprocessed
Problem
CPLEX 9.0
Emphasis on optimality
Strong Branching
Rank-1 Split Cut
Generation
Anureet Saxena, TSoB
Strengthened
Formulation
61
Strengthening + CPLEX 9.0
Solved to optimality
Crossover Point
(227 rank-1 cuts)
Anureet Saxena, TSoB
62
Strengthening + CPLEX 9.0
Solved to optimality
arki001 Solution Statistics
% Gap closed by rank-1 split cuts: 83.05%
Time spent in generating rank-1 split cuts: 53.76 hrs
Time taken by CPLEX 9.0 after strengthening: 10.94 hrs
No. of branch-and-bound nodes enumerated by CPLEX: 643425
Total time taken to solve the instance to optimality: 64.70 hrs
Anureet Saxena, TSoB
63
CPLEX 9.0
After 100 hours:
43 million B&B nodes
22 million active nodes
12GB B&B Tree
Anureet Saxena, TSoB
64
Comparison
Crossover
Point
Anureet Saxena, TSoB
65
Thank You
Anureet Saxena, TSoB
66