Download informs 2015 cplex 12.6.3

What’s new in
IBM ILOG CPLEX Optimization Studio 12.7
Ferenc Katai, Ph.D., Offering Manager, CPLEX optimization Studio
Xavier Nodet, Development Lead, CPLEX optimization Studio
© 2016 IBM Corporation
What’s new at a glance
 New features (and their consequences on performance)





(automatic) Benders decomposition algorithm
Variability tester for CPLEX/CPO (in the interactive)
Modeling Assistance in CPLEX
PWL in C, Python and file forma
CP Optimizer’s new functions/features
• Interactive for CP Optimizer (a la CPLEX)
• Variability tester in the interactive
• File format for starting points (aka warm-start) for CP Optimizer
 Performance
 CPLEX
 CPO
2
© 2016 IBM Corporation
BENDERS METHOD
3
© 2016 IBM Corporation
Benders decomposition – a short view
 Benders’ decomposition method is well-known in the math programming profession since the
1962, when Jacque Benders published it.
 It’s a kind of 2 stage method with sub-problems and master problem
 It works well when the model is very block-structured and in this case the speed up can be
very significant (in some cases even +20x)
 The problem types where Benders works/speeds up well include e.g.
• Especially Stochastic-related problems where there are many scenarios
• Facility location problems (aka network design, like LNP)
• In fact anything with some possibility to block-triangulation
 Use
 Novice use (switch type)
• where the user simply say that Benders should be performed but gives no indication
about the blocks themselves. In this case CPLEX will figure out how to separate the
problem and performs the Benders accordingly
 advanced use (familiarity with Benders, therefore more knowledge provided by the user)
• The user can define which variables are belonging to what stage of Benders
(sub/master) thru an API
4
© 2016 IBM Corporation
Principle of the Benders decomposition
 Iterative approach
 Complete problem decomposed into a single master (MIP) problem, and a series of LP subproblems.
 Master problem defined by a set of variables, and contains all the constraints that involve
these variables.
 Given a solution of the master problem, the sub-problems allow to generate:
• An optimality cut that refines their contribution to the objective value, or
• A feasibility cut if that solution is not feasible for a sub-problem
• Cuts derive from duality arguments, hence the need for the sub-problems to be LPs
Master
SP1
SP2
SP3
5
© 2016 IBM Corporation
Typical use cases
 The complete model is big, but the sub-problems decompose into manageable pieces.
Example: scenario-based stochastic problems
 No decomposition for the sub-problem, but the master and the sub-problem have a much
simpler structure. Example: facility location, where Benders decomposition turns tons of
variable upper bounds (99% of the constraints in the model) into simple bounds.
In the complete problem:
𝑦𝑖𝑗 ≤ 𝑥𝑖
These constraints disappear from the master model, and appear in the sub-problem as
𝑦𝑖𝑗 ≤ 0,1 , depending on the solution for 𝑥𝑖 in the master problem
 Similar to above, when master can enumerate much faster than the complete model and
yet the bounds that it provides are not much worse. Example: some network design models.
6
© 2016 IBM Corporation
Annotations
 Annotations provide information to CPLEX that doesn’t belong to the definition of the model
 Example: defining the block structure for Benders decomposition algorithm
 All elements of a model (objective, variables, constraints) can be annotated
 A given annotation is defined by a name (the key), a type (for the value), and a default
value
 For a given model element, and a given key, a single value can be attached
 For Benders:
 Key is “cpxBendersPartition”
 Type is long
 Value 0: this variable is a master variable
 Value k>0: this variable belongs to sub-problem k.
7
© 2016 IBM Corporation
benders strategy parameter
 benders strategy 1: user-specified decomposition
 CPLEX will use the decomposition specified as is
 benders strategy 2: CPLEX refines user’s decomposition
 The master variables are left
 Each sub-problem will be further decomposed if possible
 User can simply specify the master variables, and let CPLEX do the rest
 benders strategy 3: full automatic decomposition
 A user-specified decomposition is ignored
 Integer variables go into the master problem
 Continuous variables in a sub-problem
 Decompose this sub-problem if possible
 benders strategy 0 (default)
 If a decomposition was specified: refine it, and run benders algorithm
 Otherwise, run regular Brand-and-Bound
8
© 2016 IBM Corporation
Computational results
Benders B&C compared to “regular” B&C
1.06x
0.94
0.8
0.6
1.48x
0.40
5.47x
0.2
5.69x
0.18
0.18
stochastic
non stochastic
all models
3.07x
3.57x
0.28
0.4
0.68
2.53x
6.40x
0.16
0.33
1
5.41x
0.18
0
>1s
>10s
>100s
Stochastic
# models
# wins/losses
Non stochastic
# timeout
# models
# wins/losses
(regular B&C /
Benders B&C)
9
# timeout
(regular B&C /
Benders B&C)
> 1sec
168
130/37
76/17
152
61/80
6/4
> 10 secs
163
126/37
76/17
86
52/32
6/4
> 100 secs
149
117/32
76/17
44
30/13
6/4
Date:
Testset:
Machine:
Timelimit:
6 November 2016
390 models
Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads
10,000 sec
© 2016 IBM Corporation
PERFORMANCE VARIABILITY
10
© 2016 IBM Corporation
Performance variability:
seemingly neutral changes lead to a big variation in solution time
By now a well known issue in MIP:

Danna (MIP workshop, 2008)

Koch et al. (Math. Prog. Comp. 2011)

Fischetti and Monaci (Op. Res. 2014)

Achterberg and Wunderling (Facets of Comb. Optimization, 2013)

Lodi and Tramontani (INFORMS 2013 Tutorial)
MIP solver ingredients

Heuristics

Cutting planes


Branching variable selection
...
Changes in platform, code, parameter settings …
→ may have (big) impact on any ingredient
→ on the whole solution process
→ (sometimes) hard to get robust performance
11
© 2016 IBM Corporation
Performance variability exists and has massive effects
3 instances from MIPLIB2010 test set, run on 5 different random seeds
model
map14
rococoB10011000
neos911880
12
seed 1
seed 2
seed 3
seed 4
seed 5
min
max
max/min
65.41
61.77
62.82
62.35
64.96
61.77
65.41
1.06
1582.72
811.05
844.39
2223.25
740.78
740.78
2223.25
3.00
315.22
10000.14
354.19
1.71
27.40
1.71
10000.14
5848.04
© 2016 IBM Corporation
Performance variability exists and has massive effects
3 instances from MIPLIB2010 test set, run on 5 different random seeds
model
seed 1
map14
rococoB10011000
neos911880
seed 2
seed 3
seed 4
seed 5
min
max
max/min
65.41
61.77
62.82
62.35
64.96
61.77
65.41
1.06
1582.72
811.05
844.39
2223.25
740.78
740.78
2223.25
3.00
315.22
10000.14
354.19
1.71
27.40
1.71
10000.14
5848.04
All solvable instances, run on 5 different seeds, max/min truncated at 1000, log scale
1000
100
10
1
13
© 2016 IBM Corporation
Performance variability exists and has massive effects
3 instances from MIPLIB2010 test set, run on 5 different random seeds
model
seed 1
map14
rococoB10011000
neos911880
seed 2
seed 3
seed 4
seed 5
min
max
max/min
65.41
61.77
62.82
62.35
64.96
61.77
65.41
1.06
1582.72
811.05
844.39
2223.25
740.78
740.78
2223.25
3.00
315.22
10000.14
354.19
1.71
27.40
1.71
10000.14
5848.04
All solvable instances, run on 5 different seeds, max/min truncated at 1000, log scale
1000
rough classification
100
10
≤2
ok
131
]2, 4]
shaky
53
>4
bad
28
1
14
© 2016 IBM Corporation
Performance variability exists and has massive effects
3 instances from MIPLIB2010 test set, run on 5 different random seeds
model
map14
rococoB10011000
neos911880
seed 1
seed 2
seed 3
seed 4
seed 5
min
max
max/min
65.41
61.77
62.82
62.35
64.96
61.77
65.41
1.06
1582.72
811.05
844.39
2223.25
740.78
740.78
2223.25
3.00
315.22
10000.14
354.19
1.71
27.40
1.71
10000.14
5848.04
To assess performance variability with CPLEX:
15

new command in the interactive: tools→runseeds

run one model with different random seeds

surrogate for random perturbation of the model
© 2016 IBM Corporation
CPLEX> read beasleyC3.mps
16
1. Read model
© 2016 IBM Corporation
CPLEX> read beasleyC3.mps
CPLEX> set mip tolerance mipgap 0
CPLEX> set mip tolerance absmipgap 0

17
1. Read model
2. Set parameters (if any)
parameters set here apply to every run!
© 2016 IBM Corporation
CPLEX>
CPLEX>
CPLEX>
CPLEX>
read beasleyC3.mps
set mip tolerance mipgap 0
set mip tolerance absmipgap 0
tools runseeds




18
1. Read model
2. Set parameters (if any)
3. Launch
runs model with different random seeds
default is 30 random seeds
each solve starts from scratch
CPLEX compiles statistics for each run
© 2016 IBM Corporation
CPLEX>
CPLEX>
CPLEX>
CPLEX>
. . .
19
read beasleyC3.mps
set mip tolerance mipgap 0
set mip tolerance absmipgap 0
tools runseeds
1. Read model
2. Set parameters (if any)
3. Launch
4. Wait
© 2016 IBM Corporation
CPLEX>
CPLEX>
CPLEX>
CPLEX>
read beasleyC3.mps
set mip tolerance mipgap 0
set mip tolerance absmipgap 0
tools runseeds
. . .
1. Read model
2. Set parameters (if any)
3. Launch
4. Wait
5. Inspect results
====== runseeds statistics of 30 runs
exit sol
run code stat
1
0 101
2
0 101
3
0 101
. . .
30
0 101
objective
value
754
754
754
gap
(%)
0.00
0.00
0.00
iteration
count
478311
378060
421786
node
count
4051
3991
4021
runtime
seconds
10.68
8.91
10.04
dettime
ticks
6941.20
5653.56
6489.83
754
0.00
413233
3969
9.06
5775.45
node
count
runtime
seconds
dettime
ticks
4009
43
10.84
1.07
7021.98
779.37
Exit codes:
0 : No error
Optimization status codes:
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average:
754
0.00
459225
std dev: 1.87205e-10
0.00
66658
20
© 2016 IBM Corporation
CPLEX>
CPLEX>
CPLEX>
CPLEX>
read beasleyC3.mps
set mip tolerance mipgap 0
set mip tolerance absmipgap 0
tools runseeds
. . .
1. Read model
2. Set parameters (if any)
3. Launch
4. Wait
5. Inspect results
====== runseeds statistics of 30 runs
exit sol
run code stat
1
0 101
2
0 101
3
0 101
. . .
30
0 101
objective
value
754
754
754
gap
(%)
0.00
0.00
0.00
iteration
count
478311
378060
421786
node
count
4051
3991
4021
runtime
seconds
10.68
8.91
10.04
dettime
ticks
6941.20
5653.56
6489.83
754
0.00
413233
3969
9.06
5775.45
node
count
runtime
seconds
dettime
ticks
4009
43
10.84
1.07
7021.98
779.37
Exit codes:
0 : No error
Optimization status codes:
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average:
754
0.00
459225
std dev: 1.87205e-10
0.00
66658
21
© 2016 IBM Corporation
CPLEX>
CPLEX>
CPLEX>
CPLEX>
read beasleyC3.mps
set mip tolerance mipgap 0
set mip tolerance absmipgap 0
tools runseeds
. . .
1. Read model
2. Set parameters (if any)
3. Launch
4. Wait
5. Inspect results
====== runseeds statistics of 30 runs
exit sol
run code stat
1
0 101
2
0 101
3
0 101
. . .
30
0 101
objective
value
754
754
754
gap
(%)
0.00
0.00
0.00
iteration
count
478311
378060
421786
node
count
4051
3991
4021
runtime
seconds
10.68
8.91
10.04
dettime
ticks
6941.20
5653.56
6489.83
754
0.00
413233
3969
9.06
5775.45
Exit codes:
0 : No error
solution information, should be consistent
otherwise → modeling assistance
Optimization status codes:
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average:
754
0.00
459225
std dev: 1.87205e-10
0.00
66658
22
node
count
runtime
seconds
dettime
ticks
4009
43
10.84
1.07
7021.98
779.37
© 2016 IBM Corporation
CPLEX>
CPLEX>
CPLEX>
CPLEX>
read beasleyC3.mps
set mip tolerance mipgap 0
set mip tolerance absmipgap 0
tools runseeds
. . .
1. Read model
2. Set parameters (if any)
3. Launch
4. Wait
5. Inspect results
====== runseeds statistics of 30 runs
exit sol
run code stat
1
0 101
2
0 101
3
0 101
. . .
30
0 101
objective
value
754
754
754
gap
(%)
0.00
0.00
0.00
iteration
count
478311
378060
421786
node
count
4051
3991
4021
runtime
seconds
10.68
8.91
10.04
dettime
ticks
6941.20
5653.56
6489.83
754
0.00
413233
3969
9.06
5775.45
node
count
runtime
seconds
dettime
ticks
4009
43
10.84
1.07
7021.98
779.37
Exit codes:
0 : No error
solution effort,
indicates variability
Optimization status codes:
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average:
754
0.00
459225
std dev: 1.87205e-10
0.00
66658
23
© 2016 IBM Corporation
CPLEX>
CPLEX>
CPLEX>
CPLEX>
read beasleyC3.mps
set mip tolerance mipgap 0
set mip tolerance absmipgap 0
tools runseeds
. . .
1. Read model
2. Set parameters (if any)
3. Launch
4. Wait
5. Inspect results
====== runseeds statistics of 30 runs
exit sol
run code stat
1
0 101
2
0 101
3
0 101
. . .
30
0 101
objective
value
754
754
754
gap
(%)
0.00
0.00
0.00
iteration
count
478311
378060
421786
node
count
4051
3991
4021
runtime
seconds
10.68
8.91
10.04
dettime
ticks
6941.20
5653.56
6489.83
754
0.00
413233
3969
9.06
5775.45
node
count
runtime
seconds
dettime
ticks
4009
43
10.84
1.07
7021.98
779.37
Exit codes:
0 : No error
Optimization status codes:
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average:
754
0.00
459225
std dev: 1.87205e-10
0.00
66658
24
© 2016 IBM Corporation
CPLEX>
CPLEX>
CPLEX>
CPLEX>
read beasleyC3.mps
set mip tolerance mipgap 0
set mip tolerance absmipgap 0
tools runseeds
. . .
1. Read model
2. Set parameters (if any)
3. Launch
4. Wait
5. Inspect results
====== runseeds statistics of 30 runs
exit sol
run code stat
1
0 101
2
0 101
3
0 101
. . .
30
0 101
objective
value
754
754
754
gap
(%)
0.00
0.00
0.00
iteration
count
478311
378060
421786
node
count
4051
3991
4021
runtime
seconds
10.68
8.91
10.04
dettime
ticks
6941.20
5653.56
6489.83
754
0.00
413233
3969
9.06
5775.45
node
count
runtime
seconds
dettime
ticks
4009
43
10.84
1.07
7021.98
779.37
Exit codes:
0 : No error
Optimization status codes:
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average:
754
0.00
459225
std dev: 1.87205e-10
0.00
66658
25
© 2016 IBM Corporation
CPLEX>
CPLEX>
CPLEX>
CPLEX>
read beasleyC3.mps
set mip tolerance mipgap 0
set mip tolerance absmipgap 0
tools runseeds
1. Read model
2. Set parameters (if any)
3. Launch
. . .
4. Wait
Statistics per run
====== runseeds statistics of 30 runs
5. Inspect results
exit sol
run code stat
1
0 101
2
0 101
3
0 101
. . .
30
0 101
objective
value
754
754
754
gap
(%)
0.00
0.00
0.00
iteration
count
478311
378060
421786
node
count
4051
3991
4021
runtime
seconds
10.68
8.91
10.04
dettime
ticks
6941.20
5653.56
6489.83
754
0.00
413233
3969
9.06
5775.45
node
count
runtime
seconds
dettime
ticks
4009
43
10.84
1.07
7021.98
779.37
Failed runs
Exit codes:
0 : No error
Aggregated statistics
Optimization status codes:
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average:
754
0.00
459225
std dev: 1.87205e-10
0.00
66658
26
© 2016 IBM Corporation
Some example results

aflow30a
A stable
model
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average:
1158
0.00
110298
std dev:
0
0.00
24585
27
node
count
runtime
seconds
dettime
ticks
3980
779
3.19
0.30
1883.01
181.44
© 2016 IBM Corporation
Some example results

aflow30a
A stable
model
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average:
1158
0.00
110298
std dev:
0
0.00
24585

runtime
seconds
dettime
ticks
3980
779
3.19
0.30
1883.01
181.44
node
count
runtime
seconds
dettime
ticks
gmu-35-40
A model
with high variability
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average: -2.40673e+06
0.00
36824708
std dev:
0
0.00
94077727
28
node
count
4624403
12739496
566.02 339839.59
1607.25 935535.50
© 2016 IBM Corporation
Some example results

aflow30a
A stable
model
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average:
1158
0.00
110298
std dev:
0
0.00
24585

runtime
seconds
dettime
ticks
3980
779
3.19
0.30
1883.01
181.44
node
count
runtime
seconds
dettime
ticks
A model with high variability
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average: -2.40673e+06
0.00
36824708
std dev:
0
0.00
94077727

node
count
4624403
12739496
566.02 339839.59
1607.25 935535.50
Hurink_edata_mt06_lazy
A
suspicious model
objective
gap iteration
node
value
(%)
count
count
101 : integer optimal solution (30 times)
average:
49.2667
0.00 499151265 72138619
std dev:
5.52695
0.00 532709392 107808642
runtime
seconds
dettime
ticks
5946.96 3269407.19
7681.00 4400455.20
standard deviation in objective → indicator for numerical problems
→ modeling assistance
29
© 2016 IBM Corporation
MODELING ASSISTANCE
30
© 2016 IBM Corporation
Modeling assistance


detect problems in model data (small numbers, …)
detect bad modeling constructs (big-M, ...)
→ improve numerical robustness




31
issues collected from years of CPLEX support and development

issues with problem data

issues in the model formulation

issues with parameter settings

issues during the solution process
CPLEX scans your model and looks for these issues
for each issue found a warning is printed
enabled by parameter setting (disabled by default)
© 2016 IBM Corporation
Modeling assistance


detect problems in model data (small numbers, …)
detect bad modeling constructs (big-M, ...)
→ improve numerical robustness





issues collected from years of CPLEX support and development

issues with problem data

issues in the model formulation

issues with parameter settings

issues during the solution process
CPLEX scans your model and looks for these issues
for each issue found a warning is printed
enabled by parameter setting (disabled by default)
CPX_PARAM_DATACHECK 0 – off
input
32
solve
© 2016 IBM Corporation
Modeling assistance


detect problems in model data (small numbers, …)
detect bad modeling constructs (big-M, ...)
→ improve numerical robustness





issues collected from years of CPLEX support and development

issues with problem data

issues in the model formulation

issues with parameter settings

issues during the solution process
CPLEX scans your model and looks for these issues
for each issue found a warning is printed
enabled by parameter setting (disabled by default)
CPX_PARAM_DATACHECK 0 – off
1 – check for invalid data (dup indices, NaNs, ...)
input &
check
33
solve
© 2016 IBM Corporation
Modeling assistance


detect problems in model data (small numbers, …)
detect bad modeling constructs (big-M, ...)
→ improve numerical robustness
issues collected from years of CPLEX support and development

issues with problem data

issues in the model formulation

issues with parameter settings

issues during the solution process

CPLEX scans your model and looks for these issues

for each issue found a warning is printed

enabled by parameter setting (disabled by default)

CPX_PARAM_DATACHECK 0 – off
modeling assistance 1 – check for invalid data (dup indices, NaNs, ...)
2 – check for common modeling issues
input &

check
solve
modeling assistance
34
© 2016 IBM Corporation
Modeling assistance
Quick reminder: Assessing numerics in CPLEX (for MIP)
35
© 2016 IBM Corporation
Modeling assistance
Quick reminder: Assessing numerics in CPLEX (for MIP)

Solution quality:
CPLEX> display solution quality
MILP objective
MILP solution norm |x| (Total, Max)
MILP solution error (Ax=b) (Total, Max)
MILP x bound error (Total, Max)
MILP x integrality error (Total, Max)
MILP slack bound error (Total, Max)
36
2.1961300000e+05
4.22897e+06 5.34130e+05
1.02709e-09 4.36557e-11
0.00000e+00 0.00000e+00
0.00000e+00 0.00000e+00
3.55271e-15 2.22045e-16
© 2016 IBM Corporation
Modeling assistance
Quick reminder: Assessing numerics in CPLEX (for MIP)

Solution quality:
CPLEX> display solution quality
MILP objective
MILP solution norm |x| (Total, Max)
MILP solution error (Ax=b) (Total, Max)
MILP x bound error (Total, Max)
MILP x integrality error (Total, Max)
MILP slack bound error (Total, Max)

2.1961300000e+05
4.22897e+06 5.34130e+05
1.02709e-09 4.36557e-11
0.00000e+00 0.00000e+00
0.00000e+00 0.00000e+00
3.55271e-15 2.22045e-16
Kappa statistics: track condition number of basis matrices
CPLEX> set mip strategy kappa 2
CPLEX> optimize
…
CPLEX> display solution kappa
Max condition number:
2.7435e+13
Percentage (number) of stable bases:
77.51%
(11908)
Percentage (number) of suspicious bases: 19.31%
(2966)
Percentage (number) of unstable bases:
3.18%
(489)
Percentage (number) of ill-posed bases:
0.00%
(0)
Attention level:
0.011480
37
© 2016 IBM Corporation
Modeling assistance
Quick reminder: Assessing numerics in CPLEX (for MIP)

Solution quality:
CPLEX> display solution quality
MILP objective
MILP solution norm |x| (Total, Max)
MILP solution error (Ax=b) (Total, Max)
MILP x bound error (Total, Max)
MILP x integrality error (Total, Max)
MILP slack bound error (Total, Max)

2.1961300000e+05
4.22897e+06 5.34130e+05
1.02709e-09 4.36557e-11
0.00000e+00 0.00000e+00
0.00000e+00 0.00000e+00
3.55271e-15 2.22045e-16
Kappa statistics: track condition number of basis matrices
Max condition number:
Percentage (number) of stable bases:
Percentage (number) of suspicious bases:
usually no problem
Percentage (number) of unstable bases:
double check
Percentage (number) of ill-posed bases:
be concerned
Attention level:
38
© 2016 IBM Corporation
Issues with problem formulation:
aTx ≤ My
y used to model availability
of a resource
39
big-M constraints
y binary, M huge
y = 0 → aTx ≤ 0
y = 1 → aTx ≤ M
resource not available
resource available
© 2016 IBM Corporation
Issues with problem formulation:
aTx ≤ My
y used to model availability
of a resource
big-M constraints
y binary, M huge
y = 0 → aTx ≤ 0
y = 1 → aTx ≤ M
resource not available
resource available
Typically M is much larger than ai
→ wide range of coefficients in the coefficient matrix
causes numerical trouble
40
© 2016 IBM Corporation
Issues with problem formulation:
aTx ≤ My
y used to model availability
of a resource
big-M constraints
y binary, M huge
y = 0 → aTx ≤ 0
y = 1 → aTx ≤ M
resource not available
resource available
Typically M is much larger than ai
→ wide range of coefficients in the coefficient matrix
causes numerical trouble
→ use indicator constraints
y = 0 → aTx ≤ 0
y = 1 → aTx ≤ M
often one of the two can be expressed as bound on a variable
41
© 2016 IBM Corporation
Issues with problem formulation:
big-M constraints
AFALL_12_3
model with big-M constraints
Max condition number:
Percentage (number) of
Percentage (number) of
Percentage (number) of
Percentage (number) of
Attention level:
42
1.6814e+13
stable bases:
8.06%
suspicious bases: 89.66%
unstable bases:
2.28%
ill-posed bases:
0.00%
0.015804
© 2016 IBM Corporation
Issues with problem formulation:
big-M contraints
AFALL_12_3
model with big-M constraints
Max condition number:
1.6814e+13
CPLEX
Warning
1042:
In
constraint
'c1109',
binary variable 'x257' is
Percentage (number) of stable bases:
8.06%
linked with
variableof'x2079'
having
ratio 89.66%
1e+06.
Percentage
(number)
suspicious
bases:
Percentage (number) of unstable bases:
2.28%
Percentage (number) of ill-posed bases:
0.00%
c1109:
1000000
x257
+
x2079
≤
0
Attention level:
0.015804
x257 binary, x2079 ≥ 0
43
© 2016 IBM Corporation
Issues with problem formulation:
big-M constraints
AFALL_12_3
model with big-M constraints
Max condition number:
1.6814e+13
CPLEX
Warning
1042:
In
constraint
'c1109',
binary variable 'x257' is
Percentage (number) of stable bases:
8.06%
linked with
variableof'x2079'
having
ratio 89.66%
1e+06.
Percentage
(number)
suspicious
bases:
Percentage (number) of unstable bases:
2.28%
Percentage (number) of ill-posed bases:
0.00%
c1109:
1000000
x257
+
x2079
≤
0
x257 = 0 → x2079 = 0
Attention level:
0.015804
x257 = 1 → x2079 ≤ 1000000
x257 binary, x2079 ≥ 0
44
© 2016 IBM Corporation
Issues with problem formulation:
big-M constraints
AFALL_12_3
model with big-M constraints
Max condition number:
1.6814e+13
CPLEX
Warning
1042:
In
constraint
'c1109',
binary variable 'x257' is
Percentage (number) of stable bases:
8.06%
linked with
variableof'x2079'
having
ratio 89.66%
1e+06.
Percentage
(number)
suspicious
bases:
Percentage (number) of unstable bases:
2.28%
Percentage (number) of ill-posed bases:
0.00%
c1109:
1000000
x257
+
x2079
≤
0
x257 = 0 → x2079 = 0
Attention level:
0.015804
x2079 ≤ 1000000
x257 binary, x2079 ≥ 0
45
© 2016 IBM Corporation
Issues with problem formulation:
big-M constraints
AFALL_12_3
model with big-M constraints
Max condition number:
1.6814e+13
CPLEX
Warning
1042:
In
constraint
'c1109',
binary variable 'x257' is
Percentage (number) of stable bases:
8.06%
linked with
variableof'x2079'
having
ratio 89.66%
1e+06.
Percentage
(number)
suspicious
bases:
Percentage (number) of unstable bases:
2.28%
Percentage (number) of ill-posed bases:
0.00%
c1109:
1000000
x257
+
x2079
≤
0
x257 = 0 → x2079 = 0
Attention level:
0.015804
x2079 ≤ 1000000
x257 binary, x2079 ≥ 0
Max condition number:
Percentage (number) of
Percentage (number) of
Percentage (number) of
Percentage (number) of
Attention level:
1.6814e+13
stable bases:
8.06%
suspicious bases: 89.66%
unstable bases:
2.28%
ill-posed bases:
0.00%
0.015804
2.2289e+13
80.15%
19.83%
0.02%
0.00%
0.002048
→ numerics improve significantly
46
© 2016 IBM Corporation
Issues with problem data:
Single precision input data
CPLEX uses IEEE double precision arithmetic (quad precision in some places)

not every rational number has an exact representation
→ round-off error will occur

Consider fraction 1/k and a simple constraint
∑i in {1, ..., k} 1/k xi = 1
47
© 2016 IBM Corporation
Issues with problem data:
Single precision input data
CPLEX uses IEEE double precision arithmetic (quad precision in some places)

not every rational number has an exact representation
→ round-off error will occur

Consider fraction 1/k and a simple constraint
∑i in {1, ..., k} 1/k xi = 1
∆ to exact value
only 1/k in single
rest in double!
single
double
1/k
Δ
Δ(∑)
Δ
Δ(∑)
1/3
9.9e-9
3.0e-8
1.9e-17
5.6e-17
1/5
3.0e-9
1.5e-8
1.1e-17
5.6e-17
1/6
5.0e-9
3.0e-8
9.3e-18
5.6e-17
xi exact integer
→infeasible by at least 1e-8 with single precision data
→in practice often a lot worse
48
© 2016 IBM Corporation
Issues with problem data:
Single precision input data
xi exact integer
→infeasible by at least 1e-8 with single precision data
→in practice often a lot worse

Obvious warning
CPLEX Warning 1036: Coefficient for %s in %s looks like %d/%d in
single precision
→ Fix data generation
49
© 2016 IBM Corporation
Issues with problem data:
Single precision input data
xi exact integer
→infeasible by at least 1e-8 with single precision data
→in practice often a lot worse

Obvious warning
CPLEX Warning 1036: Coefficient for %s in %s looks like %d/%d in
single precision
→ Fix data generation

But also
CPLEX Warning 1047: Coefficients in %s are fractions and can
be scaled with %d/%d
→ scale row to get exact values
∑i in {1, ..., k} 1/k xi = 1
50
→
∑i in {1, ..., k} xi = k
© 2016 IBM Corporation
Issues with problem data:
Single precision input data
ns2118727.mps.gz
A model
with single precision data
Max condition number:
Percentage (number) of
Percentage (number) of
Percentage (number) of
Percentage (number) of
Attention level:
51
1.0978e+14
stable bases:
67.59%
suspicious bases: 31.28%
unstable bases:
0.97%
ill-posed bases:
0.16%
0.007666
© 2016 IBM Corporation
Issues with problem data:
Single precision input data
A model with single precision data
Max condition number:
1.0978e+14
CPLEX
Warning
1036:
Coefficient
for
variable
'C7932' in constraint 'R1024'
Percentage (number) of stable bases:
67.59%
looks
like 1/24
in single
precision.bases: 31.28%
Percentage
(number)
of suspicious
Percentage (number) of unstable bases:
0.97%
Percentage
(number)
ill-posed
bases:
0.16%
R1024:
C0017 C0700 - of
C0926
+ C2957
+ C5189 C5190 - 0.0416667 C7932 = 0
Attention level:
0.007666
definitely looks
truncated/rounded
52
© 2016 IBM Corporation
Issues with problem data:
Single precision input data
A model with single precision data
Max condition number:
1.0978e+14
CPLEX
Warning
1036:
Coefficient
for
variable
'C7932' in constraint 'R1024'
Percentage (number) of stable bases:
67.59%
looks
like 1/24
in single
precision.bases: 31.28%
Percentage
(number)
of suspicious
Percentage (number) of unstable bases:
0.97%
Percentage
(number)
ill-posed
bases:
0.16%
R1024:
C0017 C0700 - of
C0926
+ C2957
+ C5189 C5190 - 0.0416667 C7932 = 0
Attention level:
0.007666
definitely looks
truncated/rounded
Max condition number:
Percentage (number) of
Percentage (number) of
Percentage (number) of
Percentage (number) of
Attention level:
53
1.0978e+14
stable bases:
67.59%
suspicious bases: 31.28%
unstable bases:
0.97%
ill-posed bases:
0.16%
0.007666
8.7299e+06
100.00%
0.00%
0.00%
0.00%
© 2016 IBM Corporation
Issues with problem data:
Single precision input data
A model with single precision data
Max condition number:
1.0978e+14
CPLEX
Warning
1036:
Coefficient
for
variable
'C7932' in constraint 'R1024'
Percentage (number) of stable bases:
67.59%
looks
like 1/24
in single
precision.bases: 31.28%
Percentage
(number)
of suspicious
Percentage (number) of unstable bases:
0.97%
Percentage
(number)
ill-posed
bases:
0.16%
R1024:
C0017 C0700 - of
C0926
+ C2957
+ C5189 C5190 - 0.0416667 C7932 = 0
Attention level:
0.007666
definitely looks
truncated/rounded
Max condition number:
Percentage (number) of
Percentage (number) of
Percentage (number) of
Percentage (number) of
Attention level:
1.0978e+14
stable bases:
67.59%
suspicious bases: 31.28%
unstable bases:
0.97%
ill-posed bases:
0.16%
0.007666
solution time from 120 seconds
54
8.7299e+06
100.00%
0.00%
0.00%
0.00%
90 seconds
© 2016 IBM Corporation
Issues with problem data:
Single precision input data
A model with single precision data
Max condition number:
1.0978e+14
CPLEX
Warning
1036:
Coefficient
for
variable
'C7932' in constraint 'R1024'
Percentage (number) of stable bases:
67.59%
looks
like 1/24
in single
precision.bases: 31.28%
Percentage
(number)
of suspicious
Percentage (number) of unstable bases:
0.97%
Percentage
(number)
ill-posed
bases:
0.16%
R1024:
C0017 C0700 - of
C0926
+ C2957
+ C5189 C5190 - 0.0416667 C7932 = 0
Attention level:
0.007666
Attention
blindly changing coefficients may change feasible region
→ may end up solving a different model
Max condition
number:model generation, not
1.0978e+14
→ change
model
Percentage (number) of stable bases:
67.59%
Percentage (number) of suspicious bases: 31.28%
Percentage (number) of unstable bases:
0.97%
Percentage (number) of ill-posed bases:
0.16%
Attention level:
0.007666
solution time from 120 seconds
55
definitely looks
truncated/rounded
8.7299e+06
100.00%
0.00%
0.00%
0.00%
90 seconds
© 2016 IBM Corporation
Issues with problem data:

extreme numbers
Numbers below CPLEX default tolerances
→ essentially 0
→ Fix numbers or tighten tolerance parameters





feasibility tolerance
optimality tolerance
numerical emphasis
...
Numbers too small for the solver (smaller than 1e-13)
→ essentially 0
→ Fix numbers

Numbers larger than CPX_INFBOUND
→ CPLEX will consider them as infinity
→ Fix numbers or, e.g., remove constraints with right-hand side > CPX_INFBOUND
56
© 2016 IBM Corporation
Issues with parameter settings:

relative MIP gap (in CPLEX):
relative MIP gap
best integer – best objective
1e-10 + |best integer|

57
CPX_PARAM_EPGAP: stop if relative MIP gap drops below a value
© 2016 IBM Corporation
Issues with parameter settings:

relative MIP gap (in CPLEX):
relative MIP gap
best integer – best objective
1e-10 + |best integer|


58
CPX_PARAM_EPGAP: stop if relative MIP gap drops below a value (default 1e-4)
Objective: min aTx + K
gap:
aTx* + K – (aTx' + K)
1e-10 + |aTx* + K|
→
aTx* – aTx'
1e-10 + |aTx* + K|
© 2016 IBM Corporation
Issues with parameter settings:

relative MIP gap (in CPLEX):
relative MIP gap
best integer – best objective
1e-10 + |best integer|



59
CPX_PARAM_EPGAP: stop if relative MIP gap drops below a value
Objective: min aTx + K
gap:
aTx* + K – (aTx' + K)
1e-10 + |aTx* + K|
→
aTx* – aTx'
1e-10 + |aTx* + K|
K huge → gap small, independent of aTx* and
aTx'
© 2016 IBM Corporation
Issues with parameter settings:

relative MIP gap (in CPLEX):
relative MIP gap
best integer – best objective
1e-10 + |best integer|




CPX_PARAM_EPGAP: stop if relative MIP gap drops below a value
Objective: min aTx + K
gap:
aTx* + K – (aTx' + K)
1e-10 + |aTx* + K|
→
aTx* – aTx'
1e-10 + |aTx* + K|
K huge → gap small, independent of aTx* and
aTx'
modeling assistance warning
CPLEX Warning 1037: Solution optimal within relative optimality tolerance
of %g, but not optimal if objective offset of value %g is removed
→ fix K or adapt CPX_PARAM_EPGAP

related
CPLEX Warning 1038: Solution optimal within relative optimality tolerance
of %g, but not optimal within absolute optimality tolerance of value %g
60
© 2016 IBM Corporation
Issues during solve:
big condition numbers
Numerical trouble while solving the problem
→ Can you trust the results? How much can you trust them?




CPLEX tracks condition number of basis matrices in LP solve.
Large number of bases with large condition number
→ potential numerical problems
→ inexact/incorrect answers
Solving ns2034125
CPLEX Warning
CPLEX Warning
1033: Detected 1277 suspicious condition number(s) >= 1e+07.
1034: Detected 30 unstable condition number(s) >= 1e+10.
→ Check/fix data, check solution quality, use numerical
emphasis, tighten tolerances
61
© 2016 IBM Corporation
Overview of modeling assistance warnings
Potential issues in problem data
Potential issues in problem formulation
Potential issues in parameter settings
Potential issues in solve
BIGM_COEF
one very big coefficient in row
██
BIGM_TO_IND
constraint can be replaced by indicator
██
BIGM_VARBOUND
variable bound constraint with very large bound
██
EPGAP_LARGE
solution optimal only due to large CPX_PARAM_EPGAP
███
EPGAP_OBJOFFSET
solution optimal only due to large objective offset
██
FRACTION_SCALING
rows can be scaled to get exact value
██
*_LARGE_NUM
suspiciously large number
██
*_SMALL_NUM
suspiciously small number
█
CANCEL_TOL
non-zero less than cancel tolerance
███
FEAS_TOL
rhs smaller than feasibility tolerance
███
OPT_TOL
objective coefficient smaller than optimality tolerance
███
QOPT_TOL
quadratic objective coefficient smaller than optimality tolerance
███
SINGLE_PRECISION
single precision data
█
SYMMETRY_BREAKING_INEQ
chain of symmetry breaking inequalities
██
WIDE_COEFF_RANGE
wide range of coefficients
██
KAPPA_*
bad kappa values detected during solve
62
████
© 2016 IBM Corporation
Combining modeling assistance and runseeds

momentum1.mps.gz (absolute and relative MIP gap 0):

runseeds on original model:
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average:
109143
0.00
40986027
std dev: 0.000312912
0.00
27028241
63
node
count
1128777
726250
runtime
seconds
dettime
ticks
2881.88 1725072.05
1807.30 1071736.66
© 2016 IBM Corporation
Combining modeling assistance and runseeds

momentum1.mps.gz (absolute and relative MIP gap 0):

runseeds on original model:
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average:
109143
0.00
40986027
std dev: 0.000312912
0.00
27028241

fix modeling assistance warnings:




64
node
count
1128777
726250
runtime
seconds
dettime
ticks
2881.88 1725072.05
1807.30 1071736.66
big-M constraints
Single precision data
scaled rows
wide coefficient ranges
© 2016 IBM Corporation
Combining modeling assistance and runseeds

momentum1.mps.gz (absolute and relative MIP gap 0):

runseeds on original model:
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average:
109143
0.00
40986027
std dev: 0.000312912
0.00
27028241

fix modeling assistance warnings

runseeds on model with all warnings fixed:
objective
gap iteration
value
(%)
count
101 : integer optimal solution (30 times)
average:
109143
0.00
34108977
std dev: 0.000580749
0.00
20783473
node
count
1128777
726250
node
count
921646
564391
runtime
seconds
dettime
ticks
2881.88 1725072.05
1807.30 1071736.66
runtime
seconds
dettime
ticks
2310.04 1400198.34
1237.40 750744.53
→ iteration count, solution time, number of nodes reduced
→ variability in iteration count, solution time, number of nodes reduced
65
© 2016 IBM Corporation
Be careful when auto-fixing warnings


Model ns2122603 from “unstable” MIPLIB 2010 testset
gives these warnings:
Detected constraint with wide range of coefficients. In constraint
'R12374' the ratio of largest and smallest (in absolute value)
coefficients is 2.4e+09
Coefficient for variable 'C10462' in constraint 'R12374' looks
like 1/24 in single precision.
66
© 2016 IBM Corporation
Be careful when auto-fixing warnings


Model ns2122603 from “unstable” MIPLIB 2010 testset
gives these warnings:
Detected constraint with wide range of coefficients. In constraint
'R12374' the ratio of largest and smallest (in absolute value)
coefficients is 2.4e+09
Coefficient for variable 'C10462' in constraint 'R12374' looks
like 1/24 in single precision.

Solve original model:
Max condition number:
Percentage (number) of
Percentage (number) of
Percentage (number) of
Percentage (number) of
Attention level:
1.7025e+20
stable bases:
0.00%
(0)
suspicious bases:
0.00%
(0)
unstable bases:
0.00%
(0)
ill-posed bases: 100.00%
(1)
1.000000
CPLEX stops at root with sub-optimal solution
67
© 2016 IBM Corporation
Be careful when auto-fixing warnings


Model ns2122603 from “unstable” MIPLIB 2010 testset
gives these warnings:
Detected constraint with wide range of coefficients. In constraint
'R12374' the ratio of largest and smallest (in absolute value)
coefficients is 2.4e+09
→ replace by indicator
Coefficient for variable 'C10462' in constraint 'R12374' looks
like 1/24 in single precision.
→ recompute coefficients

Solve original model:
Max condition number:
Percentage (number) of
Percentage (number) of
Percentage (number) of
Percentage (number) of
Attention level:

1.7025e+20
stable bases:
0.00%
(0)
suspicious bases:
0.00%
(0)
unstable bases:
0.00%
(0)
ill-posed bases: 100.00%
(1)
1.000000
CPLEX stops at root with sub-optimal solution
68
Solve fixed
3.4594e+15
0.55%
(17)
93.47%
(2906)
5.92%
(184)
0.06%
(2)
0.027745
not too bad
© 2016 IBM Corporation
Be careful when auto-fixing warnings


Model ns2122603 from “unstable” MIPLIB 2010 testset
gives these warnings:
Detected constraint with wide range of coefficients. In constraint
'R12374' the ratio of largest and smallest (in absolute value)
coefficients is 2.4e+09
→ replace by indicator
Coefficient for variable 'C10462' in constraint 'R12374' looks
like 1/24 in single precision.
→ recompute coefficients

Solve original model:
Max condition number:
Percentage (number) of
Percentage (number) of
Percentage (number) of
Percentage (number) of
Attention level:

1.7025e+20
stable bases:
0.00%
(0)
suspicious bases:
0.00%
(0)
unstable bases:
0.00%
(0)
ill-posed bases: 100.00%
(1)
1.000000
CPLEX stops at root with sub-optimal solution

but:


69
Solve fixed
3.4594e+15
0.55%
(17)
93.47%
(2906)
5.92%
(184)
0.06%
(2)
0.027745
not too bad
fixed model gives super-optimal solutions!
solutions are infeasible in original model!
© 2016 IBM Corporation
Be careful when auto-fixing warnings


Model ns2122603 from “unstable” MIPLIB 2010 testset
gives these warnings:
Detected constraint with wide range of coefficients. In constraint
'R12374' the ratio of largest and smallest (in absolute value)
coefficients is 2.4e+09
→ replace by indicator
Coefficient for variable 'C10462' in constraint 'R12374' looks
like 1/24 in single precision.
→ recompute coefficients

Solve original model:
Max condition number:
Percentage (number) of
Percentage (number) of
Percentage (number) of
Percentage (number) of
Attention level:

1.7025e+20
stable bases:
0.00%
(0)
suspicious bases:
0.00%
(0)
unstable bases:
0.00%
(0)
ill-posed bases: 100.00%
(1)
1.000000
CPLEX stops at root with sub-optimal solution

but:


70
fixed model gives super-optimal solutions!
solutions are infeasible in original model!
Solve fixed
3.4594e+15
0.55%
(17)
93.47%
(2906)
5.92%
(184)
0.06%
(2)
0.027745
not too bad
careful!
© 2016 IBM Corporation
PIECEWISE LINEAR
71
© 2016 IBM Corporation
PWL functions in the engine
 Before 12.7, piecewise linear functions were available through Concert
 In C++, Java, .NET, but not C, Python and file formats
400
 Version 12.7 fixes that
2
 Example of LP file format:
240
Minimize
0.5
[...]
Subject To
[...]
Pwl
f: y = x 0.5 (20,240) (40,400) 2.0
20
End
40
 This specifies that y = f(x), where f is defined by the two points (20, 240) and (40, 400), has
a slope before the first point equal to 0.5, and a slope after the last point of 2.0.
72
© 2016 IBM Corporation
‘NO CHECK’ MIP STARTS
73
© 2016 IBM Corporation
‘No check’ MIP Start level
 In can take time to check the validity of a MIP start, even when it indeed defines a complete
feasible solution: CPLEX has to solve a fixed LP.
 When the model has lazy constraints, checking the MIP start has to be delayed up to the
end of the root node: may hurt performance
 Version 12.7 introduces a new effort level: CPX_MIPSTART_NOCHECK
 When used, CPLEX just assumes that he solution given is complete and feasible.
Behavior is undefined otherwise.
74
© 2016 IBM Corporation
Other new CPO features
 Interactive for CP Optimizer (a la CPLEX)
 File format for starting points (aka warm-start) for CP Optimizer
 Variability tester in the interactive
75
© 2016 IBM Corporation
CPLEX MILP performance evolution
700
50
45
600
 1000 sec
35
30
400
 100 sec
25
300
20
200
 10 sec
total speedup
number of timeouts
500
40
15
10
100
5
0
0
10.0
(2006)
76
Date:
Testset:
Machine:
Timelimit:
11.0
(2007)
12.1
(2009)
12.2
(2010)
12.4
(2011)
6 November 2016
MILP: 3893 models
Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads, deterministic
10,000 sec
12.5.0
(2012)
12.6.0
(2013)
12.6.1
(2014)
12.6.3
(2015)
12.7.0
(2016)
© 2016 IBM Corporation
CPLEX 12.6.3 vs.12.7.0: LP performance improvement
1.01x
1.03x
1.07x
0.6
CPLEX 12.6.3
0.93
1.00
0.97
1.00
0.99
0.8
1.00
1
0.4
LP barrier
(12 threads)
CPLEX 12.7.0
Time limits:
1/0
0.2
0
>1s
>10s
626 models 299 models
1.05x
1.09x
>100s
123 models
1.21x
CPLEX 12.6.3
0.83
0.6
1.00
0.92
1.00
0.95
0.8
1.00
1
0.4
LP barrier with Xover
(12 threads)
CPLEX 12.7.0
Time limits:
3/3
0.2
0
>1s
>10s
696 models 355 models
77
Date:
Testset:
Machine:
Timelimit:
>100s
165 models
6 November 2016
LP: 2271 models
Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads, deterministic
10,000 sec
© 2016 IBM Corporation
CPLEX 12.6.3 vs.12.7.0: MILP performance improvement
1.08x
1.11x
1.18x
CPLEX 12.6.3
0.85
1.00
0.90
0.6
1.00
0.92
0.8
1.00
1
0.4
Deterministic parallel
MILP (12 threads)
CPLEX 12.7.0
Time limits:
55 / 24
0.2
0
>1s
2048
models
>10s
1254
models
>100s
667
models
 MILP improvements summary
 Root presolve: 2%
 Propagation of implications in node presolve: 3%
 Root and node probing: 3%
 Node cuts (separation and filtering): 3%
 Heuristics: 2%
 Branching: 3%
78
Date:
Testset:
Machine:
Timelimit:
6 November 2016
MILP: 4256 models
Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads, deterministic
10,000 sec
© 2016 IBM Corporation
CPLEX 12.6.3 vs.12.7.0: MIQP performance improvement
1.07x
1.13x
1.18x
CPLEX 12.6.3
0.85
1.00
1.00
0.89
0.6
0.94
0.8
1.00
1
0.4
Convex MIQP
(12 threads)
CPLEX 12.7.0
Time limits:
3/0
0.2
0
>0s
>1s
263 models 123 models
>10s
84 models
 MIQP improvements summary
 Node cuts (separation and filtering): 7%
 Tuning of node probing effort: 6%
79
Date:
Testset:
Machine:
Timelimit:
6 November 2016
MIQP: 365 models
Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads, deterministic
10,000 sec
© 2016 IBM Corporation
CPLEX 12.6.3 vs.12.7.0: MIQCP performance improvement
1.09x
1.15x
1.29x
CPLEX 12.6.3
1.00
Convex MIQCP
(12 threads)
CPLEX 12.7.0
0.78
1.00
0.87
0.6
0.92
0.8
1.00
1
0.4
Time limits:
5/1
0.2
0
>0s
>1s
259 models 154 models
>10s
88 models
 MIQCP improvements summary (all for Outer Approximation, OA)
 More aggressive reliability branching: 4%
 Better synchronization of cone cuts in the tree: 2%
 Better integration between cone cuts and QCP relaxation:
• Repeatedly solve QCP relaxation at the root, to refine OA with cone cuts: 4%
• Solve QCP relaxation in the tree, to refine the OA with cone cuts: 10%
80
Date:
Testset:
Machine:
Timelimit:
6 November 2016
MIQCP: 326 models
Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads, deterministic
10,000 sec
© 2016 IBM Corporation
CPLEX 12.6.3 vs.12.7.0: Global (MI)QP performance improvement
1.27x
1.65x
2.83x
1.00
1.00
0.2
Non-convex (MI)QP
(12 threads)
CPLEX 12.7.0
0.35
0.4
CPLEX 12.6.3
0.61
0.6
0.79
0.8
1.00
1
Time limits:
12 / 1
0
>0s
>1s
447 models 157 models
>10s
79 models
 All the improvement comes from newly added cutting planes: Cuts from ReformulationLinearization Techniques (RLT-cuts)
 H. D. Sherali and A. Alameddine, “A new reformulation-linearization technique for bilinear
programming problems”, Journal of Global Optimization 2, 379-410, 1992.
 Implemented in CPLEX within the FP7-PEOPLE-2012-ITN Marie Curie EU Project MINO, see also:
• P. Bonami, A. Lodi, J. Schweiger, A. Tramontani, "Solving standard quadratic programming by
cutting planes", Techical Report DS4DM-2016-001, 2016.
81
Date:
Testset:
Machine:
Timelimit:
6 November 2016
Non-convex (MI)QP: 742 models
Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads, deterministic
10,000 sec
© 2016 IBM Corporation
CPO performance improvements
 On integer constraint programming models:
+10%
 On Scheduling constraint programming models: +20%
82
© 2016 IBM Corporation