1. 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
IBM Big Data & Analytics © 2014 IBM Corporation

2. 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

3. Benders method

4. 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

5. Principle of the Benders decomposition
Iterative approach
Complete problem decomposed into a single master (MIP) problem, and a series of LP sub-problems.
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
Do we generate cuts at every node, at integer nodes?
Do we generate cuts from all sub-problems each time?
Master
SP1
SP2
SP3
IBM Analytics © 2015 IBM Corporation

6. 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: 𝑦 𝑖𝑗 ≤ 𝑥 𝑖 These constraints disappear from the master model, and appear in the sub-problem as 𝑦 𝑖𝑗 ≤ 0,1 , depending on the solution for 𝑥 𝑖 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.

7. 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.

8. 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

9. Computational results
Benders B&C compared to “regular” B&C
1.06x
1.48x
2.53x
3.07x
3.57x
5.47x
5.69x
6.40x
5.41x
Stochastic
Non stochastic
# models
# wins/losses
# timeout (regular B&C / Benders B&C)
> 1sec
168
130/37
76/17
152
61/80
6/4
> 10 secs
163
126/37 86
52/32
> 100 secs
149
117/32 44
30/13
Used cmpseeds.sh
Date: 6 November 2016 Testset: 390 models
Machine: Intel 2.67GHz, 24 GB RAM, 12 threads
Timelimit: 10,000 sec

10. Performance variability

11. 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

12. Performance variability exists and has massive effects
3 instances from MIPLIB2010 test set, run on 5 different random seeds
model
seed 1
seed 2
seed 3
seed 4
seed 5
min
max
max/min
map14
65.41
61.77
62.82
62.35
64.96
1.06
rococoB
811.05
844.39
740.78
3.00
neos
315.22
354.19
1.71
27.40

13. Performance variability exists and has massive effects
3 instances from MIPLIB2010 test set, run on 5 different random seeds
model
seed 1
seed 2
seed 3
seed 4
seed 5
min
max
max/min
map14
65.41
61.77
62.82
62.35
64.96
1.06
rococoB
811.05
844.39
740.78
3.00
neos
315.22
354.19
1.71
27.40
All solvable instances, run on 5 different seeds, max/min truncated at 1000, log scale

14. Performance variability exists and has massive effects
3 instances from MIPLIB2010 test set, run on 5 different random seeds
model
seed 1
seed 2
seed 3
seed 4
seed 5
min
max
max/min
map14
65.41
61.77
62.82
62.35
64.96
1.06
rococoB
811.05
844.39
740.78
3.00
neos
315.22
354.19
1.71
27.40
All solvable instances, run on 5 different seeds, max/min truncated at 1000, log scale
rough classification
≤ 2 ok
131
]2, 4]
shaky 53
> 4
bad 28

15. Performance variability exists and has massive effects
3 instances from MIPLIB2010 test set, run on 5 different random seeds
model
seed 1
seed 2
seed 3
seed 4
seed 5
min
max
max/min
map14
65.41
61.77
62.82
62.35
64.96
1.06
rococoB
811.05
844.39
740.78
3.00
neos
315.22
354.19
1.71
27.40
To assess performance variability with CPLEX:
new command in the interactive: tools→runseeds
run one model with different random seeds
surrogate for random perturbation of the model

16. CPLEX> read beasleyC3.mps
1. Read model

17. parameters set here apply to every run!
CPLEX> read beasleyC3.mps
1. Read model
CPLEX> set mip tolerance mipgap 0
CPLEX> set mip tolerance absmipgap 0
2. Set parameters (if any)
parameters set here apply to every run!

18. runs model with different random seeds default is 30 random seeds
CPLEX> read beasleyC3.mps
1. Read model
CPLEX> set mip tolerance mipgap 0
CPLEX> set mip tolerance absmipgap 0
2. Set parameters (if any)
CPLEX> tools runseeds
3. Launch
runs model with different random seeds
default is 30 random seeds
each solve starts from scratch
CPLEX compiles statistics for each run

19. . . . CPLEX> read beasleyC3.mps
1. Read model
CPLEX> set mip tolerance mipgap 0
CPLEX> set mip tolerance absmipgap 0
2. Set parameters (if any)
CPLEX> tools runseeds
3. Launch
. . .
4. Wait

20. . . . CPLEX> read beasleyC3.mps
1. Read model
CPLEX> set mip tolerance mipgap 0
CPLEX> set mip tolerance absmipgap 0
2. Set parameters (if any)
CPLEX> tools runseeds
3. Launch
. . .
4. Wait
5. Inspect results
====== runseeds statistics of 30 runs
exit sol objective gap iteration node runtime dettime
run code stat value (%) count count seconds ticks
Exit codes:
0 : No error
Optimization status codes:
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average:
std dev: e

21. . . . CPLEX> read beasleyC3.mps
1. Read model
CPLEX> set mip tolerance mipgap 0
CPLEX> set mip tolerance absmipgap 0
2. Set parameters (if any)
CPLEX> tools runseeds
3. Launch
. . .
4. Wait
5. Inspect results
====== runseeds statistics of 30 runs
exit sol objective gap iteration node runtime dettime
run code stat value (%) count count seconds ticks
Exit codes:
0 : No error
Optimization status codes:
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average:
std dev: e

22. CPLEX> read beasleyC3.mps
1. Read model
CPLEX> set mip tolerance mipgap 0
CPLEX> set mip tolerance absmipgap 0
2. Set parameters (if any)
CPLEX> tools runseeds
3. Launch
. . .
4. Wait
5. Inspect results
====== runseeds statistics of 30 runs
exit sol objective gap iteration node runtime dettime
run code stat value (%) count count seconds ticks
Exit codes:
0 : No error
solution information, should be consistent otherwise → modeling assistance
Optimization status codes:
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average:
std dev: e

23. solution effort, indicates variability
CPLEX> read beasleyC3.mps
1. Read model
CPLEX> set mip tolerance mipgap 0
CPLEX> set mip tolerance absmipgap 0
2. Set parameters (if any)
CPLEX> tools runseeds
3. Launch
. . .
4. Wait
5. Inspect results
====== runseeds statistics of 30 runs
exit sol objective gap iteration node runtime dettime
run code stat value (%) count count seconds ticks
Exit codes:
0 : No error
solution effort, indicates variability
Optimization status codes:
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average:
std dev: e

24. . . . CPLEX> read beasleyC3.mps
1. Read model
CPLEX> set mip tolerance mipgap 0
CPLEX> set mip tolerance absmipgap 0
2. Set parameters (if any)
CPLEX> tools runseeds
3. Launch
. . .
4. Wait
5. Inspect results
====== runseeds statistics of 30 runs
exit sol objective gap iteration node runtime dettime
run code stat value (%) count count seconds ticks
Exit codes:
0 : No error
Optimization status codes:
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average:
std dev: e

25. . . . CPLEX> read beasleyC3.mps
1. Read model
CPLEX> set mip tolerance mipgap 0
CPLEX> set mip tolerance absmipgap 0
2. Set parameters (if any)
CPLEX> tools runseeds
3. Launch
. . .
4. Wait
5. Inspect results
====== runseeds statistics of 30 runs
exit sol objective gap iteration node runtime dettime
run code stat value (%) count count seconds ticks
Exit codes:
0 : No error
Optimization status codes:
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average:
std dev: e

26. . . . CPLEX> read beasleyC3.mps
1. Read model
CPLEX> set mip tolerance mipgap 0
CPLEX> set mip tolerance absmipgap 0
2. Set parameters (if any)
CPLEX> tools runseeds
3. Launch
. . .
4. Wait
Statistics per run
5. Inspect results
====== runseeds statistics of 30 runs
exit sol objective gap iteration node runtime dettime
run code stat value (%) count count seconds ticks
Failed runs
Exit codes:
0 : No error
Aggregated statistics
Optimization status codes:
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average:
std dev: e

27. Some example results A stable model aflow30a
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average:
std dev:

28. A model with high variability
Some example results
A stable model
aflow30a
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average:
std dev:
A model with high variability
gmu-35-40
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average: e
std dev:

29. A model with high variability
Some example results
A stable model
aflow30a
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average:
std dev:
A model with high variability
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average: e
std dev:
A suspicious model
Hurink_edata_mt06_lazy
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average:
std dev:
standard deviation in objective → indicator for numerical problems
→ modeling assistance

30. Modeling asSistance

31. 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)

32. 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
solve

33. 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
solve

34. 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, ...)
modeling assistance
2 – check for common modeling issues
input & check
solve
modeling assistance

35. Modeling assistance
Quick reminder: Assessing numerics in CPLEX (for MIP)

36. Modeling assistance
Quick reminder: Assessing numerics in CPLEX (for MIP)
Solution quality:
CPLEX> display solution quality MILP objective e+05
MILP solution norm |x| (Total, Max) e e+05
MILP solution error (Ax=b) (Total, Max) e e-11
MILP x bound error (Total, Max) e e+00
MILP x integrality error (Total, Max) e e+00
MILP slack bound error (Total, Max) e e-16

37. Modeling assistance
Quick reminder: Assessing numerics in CPLEX (for MIP)
Solution quality:
CPLEX> display solution quality MILP objective e+05
MILP solution norm |x| (Total, Max) e e+05
MILP solution error (Ax=b) (Total, Max) e e-11
MILP x bound error (Total, Max) e e+00
MILP x integrality error (Total, Max) e e+00
MILP slack bound error (Total, Max) e e-16
Kappa statistics: track condition number of basis matrices
CPLEX> set mip strategy kappa 2 CPLEX> optimize … CPLEX> display solution kappa Max condition number: e+13
Percentage (number) of stable bases: % (11908)
Percentage (number) of suspicious bases: % (2966)
Percentage (number) of unstable bases: % (489)
Percentage (number) of ill-posed bases: % (0)
Attention level:

38. Modeling assistance
Quick reminder: Assessing numerics in CPLEX (for MIP)
Solution quality:
CPLEX> display solution quality MILP objective e+05
MILP solution norm |x| (Total, Max) e e+05
MILP solution error (Ax=b) (Total, Max) e e-11
MILP x bound error (Total, Max) e e+00
MILP x integrality error (Total, Max) e e+00
MILP slack bound error (Total, Max) e e-16
Kappa statistics: track condition number of basis matrices
Max condition number:
Percentage (number) of stable bases:
Percentage (number) of suspicious bases:
Percentage (number) of unstable bases:
Percentage (number) of ill-posed bases:
Attention level:
usually no problem
double check
be concerned

39. Issues with problem formulation: big-M constraints
aTx ≤ My y binary, M huge
y used to model availability of a resource
y = 0 → aTx ≤ 0 y = 1 → aTx ≤ M
resource not available resource available

40. Issues with problem formulation: big-M constraints
aTx ≤ My y binary, M huge
y used to model availability of a resource
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

41. Issues with problem formulation: big-M constraints
aTx ≤ My y binary, M huge
y used to model availability of a resource
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

42. Issues with problem formulation: big-M constraints
A model with big-M constraints
FALL_12_3
Max condition number: e+13
Percentage (number) of stable bases: %
Percentage (number) of suspicious bases: %
Percentage (number) of unstable bases: %
Percentage (number) of ill-posed bases: %
Attention level:

43. Issues with problem formulation: big-M contraints
A model with big-M constraints
FALL_12_3
Max condition number: e+13
Percentage (number) of stable bases: %
Percentage (number) of suspicious bases: %
Percentage (number) of unstable bases: %
Percentage (number) of ill-posed bases: %
Attention level:
CPLEX Warning 1042: In constraint 'c1109', binary variable 'x257' is linked with variable 'x2079' having ratio 1e+06.
c1109: x257 + x2079 ≤ 0
x257 binary, x2079 ≥ 0

44. Issues with problem formulation: big-M constraints
A model with big-M constraints
FALL_12_3
Max condition number: e+13
Percentage (number) of stable bases: %
Percentage (number) of suspicious bases: %
Percentage (number) of unstable bases: %
Percentage (number) of ill-posed bases: %
Attention level:
CPLEX Warning 1042: In constraint 'c1109', binary variable 'x257' is linked with variable 'x2079' having ratio 1e+06.
c1109: x257 + x2079 ≤ 0
x257 = 0 → x2079 = 0 x257 = 1 → x2079 ≤
x257 binary, x2079 ≥ 0

45. Issues with problem formulation: big-M constraints
A model with big-M constraints
FALL_12_3
Max condition number: e+13
Percentage (number) of stable bases: %
Percentage (number) of suspicious bases: %
Percentage (number) of unstable bases: %
Percentage (number) of ill-posed bases: %
Attention level:
CPLEX Warning 1042: In constraint 'c1109', binary variable 'x257' is linked with variable 'x2079' having ratio 1e+06.
c1109: x257 + x2079 ≤ 0
x257 = 0 → x2079 = 0 x2079 ≤
x257 binary, x2079 ≥ 0

46. Issues with problem formulation: big-M constraints
A model with big-M constraints
FALL_12_3
Max condition number: e+13
Percentage (number) of stable bases: %
Percentage (number) of suspicious bases: %
Percentage (number) of unstable bases: %
Percentage (number) of ill-posed bases: %
Attention level:
CPLEX Warning 1042: In constraint 'c1109', binary variable 'x257' is linked with variable 'x2079' having ratio 1e+06.
c1109: x257 + x2079 ≤ 0
x257 = 0 → x2079 = 0 x2079 ≤
x257 binary, x2079 ≥ 0
Max condition number: e+13
Percentage (number) of stable bases: %
Percentage (number) of suspicious bases: %
Percentage (number) of unstable bases: %
Percentage (number) of ill-posed bases: %
Attention level:
2.2289e+13
80.15%
19.83%
0.02%
0.00%
→ numerics improve significantly

47. 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

48. 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
single
double
1/k 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
1/6
5.0e-9
9.3e-18
only 1/k in single
rest in double!
xi exact integer
→infeasible by at least 1e-8 with single precision data
→in practice often a lot worse

49. 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

50. 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 →
∑i in {1, ..., k} xi = k

51. Issues with problem data: Single precision input data
A model with single precision data
ns mps.gz
Max condition number: e+14
Percentage (number) of stable bases: %
Percentage (number) of suspicious bases: %
Percentage (number) of unstable bases: %
Percentage (number) of ill-posed bases: %
Attention level:

52. definitely looks truncated/rounded
Issues with problem data: Single precision input data
A model with single precision data
Max condition number: e+14
Percentage (number) of stable bases: %
Percentage (number) of suspicious bases: %
Percentage (number) of unstable bases: %
Percentage (number) of ill-posed bases: %
Attention level:
CPLEX Warning 1036: Coefficient for variable 'C7932' in constraint 'R1024' looks like 1/24 in single precision.
R1024: C C C C C C C7932 = 0
definitely looks truncated/rounded

53. definitely looks truncated/rounded
Issues with problem data: Single precision input data
A model with single precision data
Max condition number: e+14
Percentage (number) of stable bases: %
Percentage (number) of suspicious bases: %
Percentage (number) of unstable bases: %
Percentage (number) of ill-posed bases: %
Attention level:
CPLEX Warning 1036: Coefficient for variable 'C7932' in constraint 'R1024' looks like 1/24 in single precision.
R1024: C C C C C C C7932 = 0
definitely looks truncated/rounded
Max condition number: e+14
Percentage (number) of stable bases: %
Percentage (number) of suspicious bases: %
Percentage (number) of unstable bases: %
Percentage (number) of ill-posed bases: %
Attention level:
8.7299e+06
100.00%
0.00%

54. definitely looks truncated/rounded
Issues with problem data: Single precision input data
A model with single precision data
Max condition number: e+14
Percentage (number) of stable bases: %
Percentage (number) of suspicious bases: %
Percentage (number) of unstable bases: %
Percentage (number) of ill-posed bases: %
Attention level:
CPLEX Warning 1036: Coefficient for variable 'C7932' in constraint 'R1024' looks like 1/24 in single precision.
R1024: C C C C C C C7932 = 0
definitely looks truncated/rounded
Max condition number: e+14
Percentage (number) of stable bases: %
Percentage (number) of suspicious bases: %
Percentage (number) of unstable bases: %
Percentage (number) of ill-posed bases: %
Attention level:
8.7299e+06
100.00%
0.00%
solution time from 120 seconds
90 seconds

55. definitely looks truncated/rounded
Issues with problem data: Single precision input data
A model with single precision data
Max condition number: e+14
Percentage (number) of stable bases: %
Percentage (number) of suspicious bases: %
Percentage (number) of unstable bases: %
Percentage (number) of ill-posed bases: %
Attention level:
CPLEX Warning 1036: Coefficient for variable 'C7932' in constraint 'R1024' looks like 1/24 in single precision.
Attention
R1024: C C C C C C C7932 = 0
blindly changing coefficients may change feasible region
→ may end up solving a different model
→ change model generation, not model
definitely looks truncated/rounded
Max condition number: e+14
Percentage (number) of stable bases: %
Percentage (number) of suspicious bases: %
Percentage (number) of unstable bases: %
Percentage (number) of ill-posed bases: %
Attention level:
8.7299e+06
100.00%
0.00%
solution time from 120 seconds
90 seconds

56. 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

57. Issues with parameter settings: relative MIP gap
best integer – best objective
relative MIP gap (in CPLEX):
1e-10 + |best integer|
CPX_PARAM_EPGAP: stop if relative MIP gap drops below a value

58. Issues with parameter settings: relative MIP gap
best integer – best objective
relative MIP gap (in CPLEX):
1e-10 + |best integer|
CPX_PARAM_EPGAP: stop if relative MIP gap drops below a value (default 1e-4)
aTx* + K – (aTx' + K)
aTx* – aTx'
Objective:
min aTx + K
gap: →
1e-10 + |aTx* + K|
1e-10 + |aTx* + K|

59. Issues with parameter settings: relative MIP gap
best integer – best objective
relative MIP gap (in CPLEX):
1e-10 + |best integer|
CPX_PARAM_EPGAP: stop if relative MIP gap drops below a value
aTx* + K – (aTx' + K)
aTx* – aTx'
Objective:
min aTx + K
gap: →
1e-10 + |aTx* + K|
1e-10 + |aTx* + K|
K huge → gap small, independent of aTx* and aTx'

60. Issues with parameter settings: relative MIP gap
best integer – best objective
relative MIP gap (in CPLEX):
1e-10 + |best integer|
CPX_PARAM_EPGAP: stop if relative MIP gap drops below a value
aTx* + K – (aTx' + K)
aTx* – aTx'
Objective:
min aTx + K
gap: →
1e-10 + |aTx* + K|
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

61. 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 ns
CPLEX Warning 1033: Detected 1277 suspicious condition number(s) >= 1e+07.
CPLEX Warning 1034: Detected 30 unstable condition number(s) >= 1e+10.
→ Check/fix data, check solution quality, use numerical emphasis, tighten tolerances

62. Overview of modeling assistance warnings
Potential issues in problem data
Potential issues in parameter settings
Potential issues in problem formulation
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
█ █ █ █

63. Combining modeling assistance and runseeds
momentum1.mps.gz (absolute and relative MIP gap 0):
runseeds on original model:
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average:
std dev:

64. Combining modeling assistance and runseeds
momentum1.mps.gz (absolute and relative MIP gap 0):
runseeds on original model:
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average:
std dev:
fix modeling assistance warnings:
big-M constraints
Single precision data
scaled rows
wide coefficient ranges

65. Combining modeling assistance and runseeds
momentum1.mps.gz (absolute and relative MIP gap 0):
runseeds on original model:
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average:
std dev:
fix modeling assistance warnings
runseeds on model with all warnings fixed:
objective gap iteration node runtime dettime
value (%) count count seconds ticks
101 : integer optimal solution (30 times)
average:
std dev:
→ iteration count, solution time, number of nodes reduced
→ variability in iteration count, solution time, number of nodes reduced

66. Be careful when auto-fixing warnings
Model ns 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.

67. Be careful when auto-fixing warnings
Model ns 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: e+20
Percentage (number) of stable bases: % (0)
Percentage (number) of suspicious bases: % (0)
Percentage (number) of unstable bases: % (0)
Percentage (number) of ill-posed bases: % (1)
Attention level:
CPLEX stops at root with sub-optimal solution

68. Be careful when auto-fixing warnings
Model ns 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:
Solve fixed
Max condition number: e+20
Percentage (number) of stable bases: % (0)
Percentage (number) of suspicious bases: % (0)
Percentage (number) of unstable bases: % (0)
Percentage (number) of ill-posed bases: % (1)
Attention level:
3.4594e+15
0.55% (17)
93.47% (2906)
5.92% (184)
0.06% (2)
CPLEX stops at root with sub-optimal solution
not too bad

69. Be careful when auto-fixing warnings
Model ns 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:
Solve fixed
Max condition number: e+20
Percentage (number) of stable bases: % (0)
Percentage (number) of suspicious bases: % (0)
Percentage (number) of unstable bases: % (0)
Percentage (number) of ill-posed bases: % (1)
Attention level:
3.4594e+15
0.55% (17)
93.47% (2906)
5.92% (184)
0.06% (2)
CPLEX stops at root with sub-optimal solution
not too bad
but:
fixed model gives super-optimal solutions!
solutions are infeasible in original model!

70. careful! Be careful when auto-fixing warnings
Model ns 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:
Solve fixed
Max condition number: e+20
Percentage (number) of stable bases: % (0)
Percentage (number) of suspicious bases: % (0)
Percentage (number) of unstable bases: % (0)
Percentage (number) of ill-posed bases: % (1)
Attention level:
3.4594e+15
0.55% (17)
93.47% (2906)
5.92% (184)
0.06% (2)
CPLEX stops at root with sub-optimal solution
not too bad
but:
fixed model gives super-optimal solutions!
solutions are infeasible in original model!
careful!

71. Piecewise linear

72. 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
Version 12.7 fixes that
Example of LP file format:
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. 2
400
240
Minimize
[...]
Subject To
Pwl f: y = x 0.5 (20,240) (40,400) 2.0
End
0.5 20 40

73. ‘no check’ MIP starts

74. ‘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.

75. 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

76. CPLEX MILP performance evolution
 1000 sec
 100 sec
A = cplex100
B = cplex110
C = cplex-cplex121f0
D = cplex-cplex122f0rc3
E = cplex-cplex124f0-bench
F = cplex-cplex125f0rc7
G = cplex-cplex126f0rc2
H = cplex R0-13-g1c434b9
I = cplex R0
J = cplex R5
| A | B | C | D | E | F | G | H | I | J |
| # of # of| # of Time| # of Time| # of Time| # of Time| # of Time| # of Time| # of Time| # of Time| # of Time|
| eval Time| Time opt| Time opt| Time opt| Time opt| Time opt| Time opt| Time opt| Time opt| Time opt|
suptime |model limit|limit geom|limit geom|limit geom|limit geom|limit geom|limit geom|limit geom|limit geom|limit geom|
| hits| hits ratio| hits ratio| hits ratio| hits ratio| hits ratio| hits ratio| hits ratio| hits ratio| hits ratio|
all | | | | | | | | | | |
[0,10k} | | | | | | | | | | |
[1,10k} | | | | | | | | | | |
[10,10k}| | | | | | | | | | |
100,10k}| | | | | | | | | | |
[1k,10k}| | | | | | | | | | |
 10 sec
Date: 6 November 2016 Testset: MILP: 3893 models
Machine: Intel 2.67GHz, 24 GB RAM, 12 threads, deterministic
Timelimit: 10,000 sec
IBM Analytics © 2015 IBM Corporation

77. CPLEX 12.6.3 vs.12.7.0: LP performance improvement
LP barrier
(12 threads)
Time limits:
1 / 0
626 models
299 models
123 models
1.05x
1.09x
1.21x
LP barrier with Xover
(12 threads)
Used cmpseeds.sh
Time limits:
3 / 3
696 models
355 models
165 models
Date: 6 November 2016 Testset: LP: 2271 models
Machine: Intel 2.67GHz, 24 GB RAM, 12 threads, deterministic
Timelimit: 10,000 sec
IBM Analytics © 2015 IBM Corporation

78. CPLEX 12.6.3 vs.12.7.0: MILP performance improvement
Deterministic parallel MILP (12 threads)
Time limits:
55 / 24
2048 models
1254 models
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%
Used cmpseeds.sh
Date: 6 November 2016 Testset: MILP: 4256 models
Machine: Intel 2.67GHz, 24 GB RAM, 12 threads, deterministic
Timelimit: 10,000 sec
IBM Analytics © 2015 IBM Corporation

79. CPLEX 12.6.3 vs.12.7.0: MIQP performance improvement
Convex MIQP
(12 threads)
Time limits:
3 / 0
263 models
123 models
84 models
MIQP improvements summary
Node cuts (separation and filtering): 7%
Tuning of node probing effort: 6%
Used cmpseeds.sh
Date: 6 November 2016 Testset: MIQP: 365 models
Machine: Intel 2.67GHz, 24 GB RAM, 12 threads, deterministic
Timelimit: 10,000 sec
IBM Analytics © 2015 IBM Corporation

80. CPLEX 12.6.3 vs.12.7.0: MIQCP performance improvement
Convex MIQCP
(12 threads)
Time limits:
5 / 1
259 models
154 models
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%
Used cmpseeds.sh
Date: 6 November 2016 Testset: MIQCP: 326 models
Machine: Intel 2.67GHz, 24 GB RAM, 12 threads, deterministic
Timelimit: 10,000 sec
IBM Analytics © 2015 IBM Corporation

81. CPLEX 12.6.3 vs.12.7.0: Global (MI)QP performance improvement
Non-convex (MI)QP
(12 threads)
Time limits:
12 / 1
447 models
157 models
79 models
All the improvement comes from newly added cutting planes: Cuts from Reformulation-Linearization Techniques (RLT-cuts)
H. D. Sherali and A. Alameddine, “A new reformulation-linearization technique for bilinear programming problems”, Journal of Global Optimization 2, , 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.
Used cmpseeds.sh
Date: 6 November 2016 Testset: Non-convex (MI)QP: 742 models
Machine: Intel 2.67GHz, 24 GB RAM, 12 threads, deterministic
Timelimit: 10,000 sec
IBM Analytics © 2015 IBM Corporation

82. CPO performance improvements
On integer constraint programming models: +10%
On Scheduling constraint programming models: +20%

83. IBM Big Data & Analytics © 2014 IBM Corporation