ibm cplex optimization studio 12.7 - benders, modeling assistance, etc

© 2016 IBM Corporation

IBM CPLEX Optimizer 12.7Benders decomposition, Modeling Assistance, …

Xavier Nodet, Program Manager, CPLEX Optimization Studio

© 2016 IBM Corporation2

PERFORMANCE


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10.0 (2006)

11.0 (2007)

12.1 (2009)

12.2 (2010)

12.4 (2011)

12.5.0 (2012)

12.6.0 (2013)

12.6.1 (2014)

12.6.3 (2015)

12.7.0 (2016)

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sCPLEX MILP performance evolution

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Date: 6 November 2016Testset: MILP: 3893 modelsMachine: Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads, deterministicTimelimit: 10,000 sec


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CPLEX 12.6.3

CPLEX 12.7.0

Date: 6 November 2016Testset: LP: 2271 modelsMachine: Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads, deterministicTimelimit: 10,000 sec

CPLEX 12.6.3 vs.12.7.0: LP performance improvement

LP barrier(12 threads)

626 models 123 models

Time limits:1 / 0

1.01x 1.07x

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CPLEX 12.6.3

CPLEX 12.7.0

LP barrier with Xover(12 threads)


Time limits:3 / 3

1.05x 1.21x

1.03x

1.09x

299 models

355 models


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CPLEX 12.6.3

CPLEX 12.7.0

Date: 6 November 2016Testset: MILP: 4256 modelsMachine: Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads, deterministicTimelimit: 10,000 sec

Deterministic parallel MILP (12 threads)

2048 models

667 models

Time limits:55 / 24

1.08x 1.18x1.11x

1254 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%

CPLEX 12.6.3 vs.12.7.0: MILP performance improvement


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CPLEX 12.6.3

CPLEX 12.7.0

Date: 6 November 2016Testset: MIQP: 365 models Machine: Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads, deterministicTimelimit: 10,000 sec

CPLEX 12.6.3 vs.12.7.0: MIQP performance improvement

Convex MIQP(12 threads)


Time limits:3 / 0

1.07x 1.18x1.13x

123 models

§ MIQP improvements summary-Node cuts (separation and filtering): 7%-Tuning of node probing effort: 6%


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CPLEX 12.6.3

CPLEX 12.7.0

Date: 6 November 2016Testset: MIQCP: 326 modelsMachine: Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads, deterministicTimelimit: 10,000 sec

CPLEX 12.6.3 vs.12.7.0: MIQCP performance improvement

Convex MIQCP(12 threads)


Time limits:5 / 1

1.09x 1.29x1.15x

154 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%


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CPLEX 12.6.3

CPLEX 12.7.0

Date: 6 November 2016Testset: Non-convex (MI)QP: 742 modelsMachine: Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threads, deterministicTimelimit: 10,000 sec

Non-convex (MI)QP(12 threads)


Time limits:12 / 1

1.27x 2.83x1.65x

157 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, 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.

CPLEX 12.6.3 vs.12.7.0: Global (MI)QP performance improvement


BENDERS DECOMPOSITION


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

Mas

ter

SP2

SP3

SP1


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"# ≤ 𝑥"

These constraints disappear from the master model, and appear in the sub-problem as𝑦"# ≤ 0,1 , depending on the solution for 𝑥" 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.


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.


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


Computational results

§ Benders decomposition is not a silver bullet!

§ Full automatic decomposition, on our MIP library, models solved in 10 sec or more, threshold of 2x ratio, Benders is better on 5% of the models.

§ But it can really shine!

§ Specific benchmark set of 329 models for Benders, some with manual decomposition§ Timeouts: 7 instead of 85§ 3x on all models§ 10x on 100 sec or more


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Benders B&C compared to “regular” B&C

Date: 6 November 2016Testset: 390 modelsMachine: Intel X5650 @ 2.67GHz, 24 GB RAM, 12 threadsTimelimit: 10,000 sec

5.69x 6.40x5.47x

2.53x3.57x

5.41x

Stochastic Non stochastic# models # wins/losses # timeout

(regular B&C / Benders B&C)

# 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 76/17 86 52/32 6/4

> 100 secs 149 117/32 76/17 44 30/13 6/4

1.06x

1.48x

3.07x

Computational results


RUNSEEDS, MODELING ASSISTANCE


Performance variability:seemingly neutral changes lead to a big variation in solution time

By now a well known issue in MIP:l Danna (MIP workshop, 2008)l Koch et al. (Math. Prog. Comp. 2011)l Fischetti and Monaci (Op. Res. 2014)l Achterberg and Wunderling (Facets of Comb. Optimization, 2013)l Lodi and Tramontani (INFORMS 2013 Tutorial)

MIP solver ingredientsl Heuristicsl Cutting planes

l Branching variable selectionl ...

Changes in platform, code, parameter settings …→ may have (big) impact on any ingredient→ on the whole solution process→ (sometimes) hard to get robust performance


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 61.77 65.41 1.06

rococoB10-011000

1582.72 811.05 844.39 2223.25 740.78 740.78 2223.25 3.00

neos-911880

315.22 10000.14 354.19 1.71 27.40 1.71 10000.14 5848.04

3 instances from MIPLIB2010 test set, run on 5 different random seedsPerformance variability exists and has massive effects


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10

100

1000


map14 65.41 61.77 62.82 62.35 64.96 61.77 65.41 1.06

rococoB10-011000

1582.72 811.05 844.39 2223.25 740.78 740.78 2223.25 3.00

neos-911880

315.22 10000.14 354.19 1.71 27.40 1.71 10000.14 5848.04

3 instances from MIPLIB2010 test set, run on 5 different random seedsPerformance variability exists and has massive effects

All solvable instances, run on 5 different seeds, max/min truncated at 1000, log scale


1

10

100

1000


map14 65.41 61.77 62.82 62.35 64.96 61.77 65.41 1.06

rococoB10-011000

1582.72 811.05 844.39 2223.25 740.78 740.78 2223.25 3.00

neos-911880

315.22 10000.14 354.19 1.71 27.40 1.71 10000.14 5848.04

Performance variability exists and has massive effects

≤ 2 ok 131

]2, 4] shaky 53

> 4 bad 28

rough classification

All solvable instances, run on 5 different seeds, max/min truncated at 1000, log scale

3 instances from MIPLIB2010 test set, run on 5 different random seeds


To assess performance variability with CPLEX:


map14 65.41 61.77 62.82 62.35 64.96 61.77 65.41 1.06

rococoB10-011000

1582.72 811.05 844.39 2223.25 740.78 740.78 2223.25 3.00

neos-911880

315.22 10000.14 354.19 1.71 27.40 1.71 10000.14 5848.04

3 instances from MIPLIB2010 test set, run on 5 different random seeds

l run one model with different random seeds

l new command in the interactive: tools→runseeds

Performance variability exists and has massive effects

l surrogate for random perturbation of the model


CPLEX> read beasleyC3.mps 1. Read model


CPLEX> read beasleyC3.mpsCPLEX> set mip tolerance mipgap 0CPLEX> set mip tolerance absmipgap 0

1. Read model

2. Set parameters (if any)

l parameters set here apply to every run!


CPLEX> read beasleyC3.mpsCPLEX> set mip tolerance mipgap 0CPLEX> set mip tolerance absmipgap 0CPLEX> tools runseeds

1. Read model


3. Launch

l runs model with different random seedsl default is 30 random seedsl each solve starts from scratchl CPLEX compiles statistics for each run



1. Read model


3. Launch

4. Wait . . .



1. Read model


3. Launch

4. Wait . . .

====== runseeds statistics of 30 runs

exit sol objective gap iteration node runtime dettimerun code stat value (%) count count seconds ticks

1 0 101 754 0.00 478311 4051 10.68 6941.202 0 101 754 0.00 378060 3991 8.91 5653.563 0 101 754 0.00 421786 4021 10.04 6489.83

. . .30 0 101 754 0.00 413233 3969 9.06 5775.45

Optimization status codes:objective gap iteration node runtime dettime

value (%) count count seconds ticks101 : integer optimal solution (30 times)average: 754 0.00 459225 4009 10.84 7021.98std dev: 1.87205e-10 0.00 66658 43 1.07 779.37

Exit codes:0 : No error

5. Inspect results



1. Read model


3. Launch

4. Wait . . .



1 0 101 754 0.00 478311 4051 10.68 6941.202 0 101 754 0.00 378060 3991 8.91 5653.563 0 101 754 0.00 421786 4021 10.04 6489.83

. . .30 0 101 754 0.00 413233 3969 9.06 5775.45




5. Inspect results



1. Read model


3. Launch

4. Wait . . .



1 0 101 754 0.00 478311 4051 10.68 6941.202 0 101 754 0.00 378060 3991 8.91 5653.563 0 101 754 0.00 421786 4021 10.04 6489.83

. . .30 0 101 754 0.00 413233 3969 9.06 5775.45




5. Inspect results

solution information, should be consistentotherwise → modeling assistance



1. Read model


3. Launch

4. Wait . . .



1 0 101 754 0.00 478311 4051 10.68 6941.202 0 101 754 0.00 378060 3991 8.91 5653.563 0 101 754 0.00 421786 4021 10.04 6489.83

. . .30 0 101 754 0.00 413233 3969 9.06 5775.45




5. Inspect results

solution effort,indicates variability



1. Read model


3. Launch

4. Wait . . .



1 0 101 754 0.00 478311 4051 10.68 6941.202 0 101 754 0.00 378060 3991 8.91 5653.563 0 101 754 0.00 421786 4021 10.04 6489.83

. . .30 0 101 754 0.00 413233 3969 9.06 5775.45




5. Inspect results



1. Read model


3. Launch

4. Wait . . .



1 0 101 754 0.00 478311 4051 10.68 6941.202 0 101 754 0.00 378060 3991 8.91 5653.563 0 101 754 0.00 421786 4021 10.04 6489.83

. . .30 0 101 754 0.00 413233 3969 9.06 5775.45




5. Inspect resultsStatistics per run

Failed runs

Aggregated statistics


aflow30a

Some example results

objective gap iteration node runtime dettimevalue (%) count count seconds ticks

101 : integer optimal solution (30 times)average: 1158 0.00 110298 3980 3.19 1883.01std dev: 0 0.00 24585 779 0.30 181.44

l A stable model


gmu-35-40

aflow30a




l A stable model


101 : integer optimal solution (30 times)average: -2.40673e+06 0.00 36824708 4624403 566.02 339839.59std dev: 0 0.00 94077727 12739496 1607.25 935535.50

l A model with high variability


Hurink_edata_mt06_lazy

aflow30a


101 : integer optimal solution (30 times)average: 49.2667 0.00 499151265 72138619 5946.96 3269407.19std dev: 5.52695 0.00 532709392 107808642 7681.00 4400455.20




l A stable model

l A suspicious model

l A model with high variability

standard deviation in objective → indicator for numerical problems→ modeling assistance


101 : integer optimal solution (30 times)average: -2.40673e+06 0.00 36824708 4624403 566.02 339839.59std dev: 0 0.00 94077727 12739496 1607.25 935535.50


Modeling assistance


Modeling assistancel detect problems in model data (small numbers, …)l detect bad modeling constructs (big-M, ...)

→ improve numerical robustness

l issues collected from years of CPLEX support and developmentl issues with problem datal issues in the model formulationl issues with parameter settingsl issues during the solution process

l CPLEX scans your model and looks for these issuesl for each issue found a warning is printedl enabled by parameter setting (disabled by default)




l CPLEX scans your model and looks for these issuesl for each issue found a warning is printedl enabled by parameter setting (disabled by default)l CPX_PARAM_DATACHECK 0 – off

solveinput






solveinput &check

1 – check for invalid data (dup indices, NaNs, ...)






2 – check for common modeling issuessolve

modeling assistance

modeling assistance

input &check

1 – check for invalid data (dup indices, NaNs, ...)



Modeling assistance

Quick reminder: Assessing numerics in CPLEX (for MIP)


Modeling assistance


l Solution quality: CPLEX> display solution qualityMILP objective 2.1961300000e+05MILP solution norm |x| (Total, Max) 4.22897e+06 5.34130e+05MILP solution error (Ax=b) (Total, Max) 1.02709e-09 4.36557e-11MILP x bound error (Total, Max) 0.00000e+00 0.00000e+00MILP x integrality error (Total, Max) 0.00000e+00 0.00000e+00MILP slack bound error (Total, Max) 3.55271e-15 2.22045e-16


Modeling assistance



l Kappa statistics: track condition number of basis matricesCPLEX> set mip strategy kappa 2CPLEX> optimize…CPLEX> display solution kappaMax condition number: 2.7435e+13Percentage (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


Modeling assistance



l Kappa statistics: track condition number of basis matricesMax 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 problemdouble checkbe concerned


aTx ≤ My y binary, M huge

y = 0 → aTx ≤ 0y = 1 → aTx ≤ M

y used to model availabilityof a resource

Issues with problem formulation: big-M constraints

resource not availableresource available



y = 0 → aTx ≤ 0y = 1 → aTx ≤ M


Typically M is much larger than ai→ wide range of coefficients in the coefficient matrix

causes numerical trouble





y = 0 → aTx ≤ 0y = 1 → aTx ≤ M


Typically M is much larger than ai→ wide range of coefficients in the coefficient matrix

causes numerical trouble

→ use indicator constraints y = 0 → aTx ≤ 0y = 1 → aTx ≤ M



often one of the two can be expressed as bound on a variable


FALL_12_3

Max condition number: 1.6814e+13Percentage (number) of stable bases: 8.06% Percentage (number) of suspicious bases: 89.66% Percentage (number) of unstable bases: 2.28% Percentage (number) of ill-posed bases: 0.00% Attention level: 0.015804


A model with big-M constraints


FALL_12_3


Issues with problem formulation: big-M contraints

CPLEX Warning 1042: In constraint 'c1109', binary variable 'x257' is linked with variable 'x2079' having ratio 1e+06.

c1109: - 1000000 x257 + x2079 ≤ 0

x257 binary, x2079 ≥ 0



FALL_12_3




c1109: - 1000000 x257 + x2079 ≤ 0

x257 binary, x2079 ≥ 0x257 = 0 → x2079 = 0x257 = 1 → x2079 ≤ 1000000



FALL_12_3




c1109: - 1000000 x257 + x2079 ≤ 0

x257 binary, x2079 ≥ 0x257 = 0 → x2079 = 0x2079 ≤ 1000000



FALL_12_3




c1109: - 1000000 x257 + x2079 ≤ 0

x257 binary, x2079 ≥ 0x257 = 0 → x2079 = 0x2079 ≤ 1000000


2.2289e+1380.15% 19.83% 0.02% 0.00% 0.002048

→ numerics improve significantly



l CPLEX uses IEEE double precision arithmetic (quad precision in some places)l not every rational number has an exact representation→ round-off error will occur

∑i in {1, ..., k} 1/k xi = 1

Issues with problem data: Single precision input data

Consider fraction 1/k and a simple constraint


l CPLEX uses IEEE double precision arithmetic (quad precision in some places)l not every rational number has an exact representation→ round-off error will occur

∑i in {1, ..., k} 1/k xi = 1

single double1/k Δ Δ(∑) Δ Δ(∑)1/3 9.9e-9 3.0e-8 1.9e-17 5.6e-171/5 3.0e-9 1.5e-8 1.1e-17 5.6e-171/6 5.0e-9 3.0e-8 9.3e-18 5.6e-17

only 1/k in singlerest in double!

xi exact integer→infeasible by at least 1e-8 with single precision data→in practice often a lot worse


Consider fraction 1/k and a simple constraint

∆ to exact value


l Obvious warning

→ Fix data generation



CPLEX Warning 1036: Coefficient for %s in %s looks like %d/%d in single precision


CPLEX Warning 1036: Coefficient for %s in %s looks like %d/%d in single precision

l Obvious warning

→ Fix data generation

l But alsoCPLEX Warning 1047: Coefficients in %s are fractions and can be scaled with %d/%d

→ scale row to get exact values

∑i in {1, ..., k} xi = k∑i in {1, ..., k} 1/k xi = 1 →





ns2118727.mps.gz


A model with single precision data




CPLEX Warning 1036: Coefficient for variable 'C7932' in constraint 'R1024' looks like 1/24 in single precision.

R1024: C0017 - C0700 - C0926 + C2957 + C5189 - C5190 - 0.0416667 C7932 = 0

definitely looks truncated/rounded






R1024: C0017 - C0700 - C0926 + C2957 + C5189 - C5190 - 0.0416667 C7932 = 0

8.7299e+06100.00%

0.00% 0.00% 0.00%








R1024: C0017 - C0700 - C0926 + C2957 + C5189 - C5190 - 0.0416667 C7932 = 0

8.7299e+06100.00%

0.00% 0.00% 0.00%

solution time from 120 seconds



90 seconds






R1024: C0017 - C0700 - C0926 + C2957 + C5189 - C5190 - 0.0416667 C7932 = 0

8.7299e+06100.00%

0.00% 0.00% 0.00%

solution time from 120 seconds



90 seconds


Attention

blindly changing coefficients may change feasible region

→ may end up solving a different model

→ change model generation, not model


l Numbers below CPLEX default tolerances→ essentially 0

→ Fix numbers or tighten tolerance parameters

l Numbers too small for the solver (smaller than 1e-13)→ essentially 0

→ Fix numbers

l Numbers larger than CPX_INFBOUND→ CPLEX will consider them as infinity

→ Fix numbers or, e.g., remove constraints with right-hand side > CPX_INFBOUND

Issues with problem data: extreme numbers

l feasibility tolerancel optimality tolerancel numerical emphasisl ...


l CPX_PARAM_EPGAP: stop if relative MIP gap drops below a value

l relative MIP gap (in CPLEX): best integer – best objective1e-10 + |best integer|

Issues with parameter settings: relative MIP gap


l CPX_PARAM_EPGAP: stop if relative MIP gap drops below a value (default 1e-4)


l Objective: min aTx + KaTx* + K – (aTx' + K) aTx* – aTx'1e-10 + |aTx* + K| 1e-10 + |aTx* + K|


→ gap:





l K huge → gap small, independent of aTx* and aTx'


→ gap:





l K huge → gap small, independent of aTx* and aTx'

CPLEX Warning 1037: Solution optimal within relative optimality tolerance of %g, but not optimal if objective offset of value %g is removed

l related

l modeling assistance warning

CPLEX Warning 1038: Solution optimal within relative optimality tolerance of %g, but not optimal within absolute optimality tolerance of value %g


→ gap:

→ fix K or adapt CPX_PARAM_EPGAP


l CPLEX tracks condition number of basis matrices in LP solve.l Large number of bases with large condition number

→ potential numerical problems→ inexact/incorrect answers

l Numerical trouble while solving the problem→ Can you trust the results? How much can you trust them?

→ Check/fix data, check solution quality, use numericalemphasis, tighten tolerances

Issues during solve: big condition numbers

CPLEX Warning 1033: Detected 1277 suspicious condition number(s) >= 1e+07.CPLEX Warning 1034: Detected 30 unstable condition number(s) >= 1e+10.

l Solving ns2034125


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 █ █ █ █

Potential issues in problem dataPotential issues in problem formulation

Potential issues in parameter settingsPotential issues in solve

Overview of modeling assistance warnings


l momentum1.mps.gz (absolute and relative MIP gap 0):

Combining modeling assistance and runseeds

l runseeds on original model:objective gap iteration node runtime dettime

value (%) count count seconds ticks101 : integer optimal solution (30 times)average: 109143 0.00 40986027 1128777 2881.88 1725072.05std dev: 0.000312912 0.00 27028241 726250 1807.30 1071736.66






l big-M constraintsl Single precision datal scaled rowsl wide coefficient ranges

l fix modeling assistance warnings:




→ iteration count, solution time, number of nodes reduced→ variability in iteration count, solution time, number of nodes reduced



l runseeds on model with all warnings fixed:objective gap iteration node runtime dettime


l fix modeling assistance warnings


Be careful when auto-fixing warningsl Model ns2122603 from “unstable” MIPLIB 2010 testsetl 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.





l Solve original model:Max condition number: 1.7025e+20Percentage (number) of stable bases: 0.00% (0)Percentage (number) of suspicious bases: 0.00% (0)Percentage (number) of unstable bases: 0.00% (0)Percentage (number) of ill-posed bases: 100.00% (1)Attention level: 1.000000

CPLEX stops at root with sub-optimal solution





→ replace by indicator

→ recompute coefficients



3.4594e+150.55% (17)

93.47% (2906)5.92% (184)0.06% (2)0.027745

l Solve fixed

not too bad









3.4594e+150.55% (17)

93.47% (2906)5.92% (184)0.06% (2)0.027745

l Solve fixed

not too bad

l but: l fixed model gives super-optimal solutions!l solutions are infeasible in original model!









3.4594e+150.55% (17)

93.47% (2906)5.92% (184)0.06% (2)0.027745

l Solve fixed

not too bad

l but: l fixed model gives super-optimal solutions!l solutions are infeasible in original model! careful!


PIECEWISE LINEAR


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.

Minimize [...]

Subject To [...]

Pwlf: y = x 0.5 (20,240) (40,400) 2.0

End


‘NO CHECK’ MIP STARTS


‘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.


CP Optimizer performance enhancements

V12.5.1 06/2013 V12.6 10/2013 V12.6.1 9/2014 V12.6.2 5/2015 V12.6.3 11/2015 V12.7 10/20160%

10%

20%

30%

40%

50%

60%

70%

Progress factor between CP Optimizer releases

Integer

Scheduling


CP Optimizer Interactive

■Equivalent of CPLEX interative for CP Optimizer

> cpoptimizer

Welcome to IBM(R) ILOG(R) CP Interactive Optimizer 12.7.0.05724-Y48 5724-Y49 5724-Y54 5724-Y55 5725-A06 5725-A29Copyright IBM Corp. 1990, 2016. All Rights Reserved

Type 'help' for a list of available commandsType 'help' followed by a command name for moreinformation on commands

CP-Optimizer>



■Main functionalitieso Read a problem instance as a .cpo file

CP-Optimizer> read myProblem.cpo

o Change parameters of the current problemCP-Optimizer> set Workers 1CP-Optimizer> set TemporalRelaxation Off

o Write the current problem (including parameter changes)CP-Optimizer> write model myModifiedProblem.cpo



■Main functionalitieso Launch CP Optimizer optimization on the current problem

CP-Optimizer> optimize

! ---------------------------------------------------------------! Minimization problem - 465 variables, 435 constraints

...! ---------------------------------------------------------------! Best Branches Non-fixed W Branch decision* 2145 901 0.15s 1 -

...! ---------------------------------------------------------------! Search terminated by abort, 6 solutions found.! Best objective : 2075

...! Search speed (br. / s) : 123617.5! ----------------------------------------------------------------

Solution found with objective 2075, search terminated by abort.



■Main functionalitieso Display all or some variables in a solution

CP-Optimizer> display solution J27O7J27O7[1: 809 -- 57 --> 866]



■Main functionalitieso Launch initial propagation at root node

CP-Optimizer> propagatePropagation succeeded.

o Display all or some variables domain after propagationCP-Optimizer> display solution J27O7J27O7[1: 431..IloIntervalMax-355 -- 57 --> 488..IloIntervalMax-298]



■Main functionalitieso Launch conflict refiner on an infeasible problem

CP-Optimizer> tools conflict

! ----------------------------------------------------------------------------! Conflict refining - 64 constraints, 60 variables! ----------------------------------------------------------------------------! Iteration Number of constraints* 1 64...* 9 1! Conflict refining terminated! ----------------------------------------------------------------------------! Conflict status : Terminated normally, conflict found! Conflict size : 1 constraint! Number of iterations : 9! Total memory usage : 965.3 kB! Conflict computation time : 0.15s! ----------------------------------------------------------------------------

Conflict found.



■Main functionalitieso Display current conflict

CP-Optimizer> display conflict

// ------ Conflict members: ---------------------------------------------------#line 116 "../../../examples/src/cpp/sched_conflict.cpp"12 >= RES1;

o Write current conflict as a .cpo fileCP-Optimizer> write conflict conflict.cpo



■Main functionalitieso Optimizes the current problem using different seeds and displays

some statisticsCP-Optimizer> tools runseeds 5

Benchmarking current problem on 5 runs...Run Solution Proof Time (s) Objective---------------------------------------------------------1 1 1 5.727 9302 1 1 4.94 9303 1 1 6.39 9304 1 1 5.59 9305 1 1 4.068 930---------------------------------------------------------All runs completeAverage 100 % 100 % 5.343 930Std dev 0.8793 0Min 4.068 930Max 6.39 930


CPO file format

■CPO file format is low-level specification of optimization problem.■CP Optimizer can both read and write .cpo files.

// Decision variables:

Belgium = intVar(1..4);Denmark = intVar(1..4);France = intVar(1..4);Germany = intVar(1..4);Luxembourg = intVar(1..4);Netherlands = intVar(1..4);

/* Constraints: */

Belgium != France;

Belgium != Germany;

Belgium != Netherlands;

Belgium != Luxembourg;

Denmark != Germany;

France != Germany;

France != Luxembourg;

Germany != Luxembourg;

Germany != Netherlands;

parameters {

SearchType = DepthFirst;}


New: Starting point in CPO file format

■User can specify an initial solution to speed up search.■The solution can specify ranges and can omit some variables.

// Decision variables:

Belgium = intVar(1..4);Denmark = intVar(1..4);France = intVar(1..4);Germany = intVar(1..4);Luxembourg = intVar(1..4);Netherlands = intVar(1..4);

/* Constraints: */

Belgium != France;

Belgium != Germany;

Belgium != Netherlands;

Belgium != Luxembourg;

Denmark != Germany;

France != Germany;

France != Luxembourg;

Germany != Luxembourg;

Germany != Netherlands;parameters {

SearchType = DepthFirst;}

startingPoint {

France = 1..3;

Belgium = 2;

}


Syntax of starting point in CPO file format

Integer variables

x = intVar(1..10);y = intVar(1..10);z = intVar(1..10);

startingPoint {

x = 1;

y = 2..5;

z = intVar(2..5);}

Interval variables (scheduling)

a = intervalVar(optional, length=5);b = intervalVar(optional, length=2);c = intervalVar(optional);d = intervalVar(optional);

startingPoint {

a = (present, start=1, end=6);b = (present, start=10..15);c = intervalVar(absent);d = absent;

}


New warnings for starting point (examples)

§ Starting point value 1 is not in the domain of integer variable 'x'.§ Starting point range 1..5 is partly outside of the domain of integer variable

'y'.§ Starting point specifies attributes of absent interval variable 'a'.§ Starting point for interval variable 'a' is inconsistent with its initial domain.


Modeling with Python for CP Optimizer 12.6.3

§ All CPO modeling function are available from Python§ Python operators overloaded for most common arithmetic and logical

operations.§ Solving is done transparently by submitting the model to a CPO solver

available on DOcloud:o No CPO solver to install locally.o Free trial available at https://developer.ibm.com/docloud/try-docloud-free/

§ Single Python package docplex for both CPO and CPLEX.§ try “pip install docplex” !


Python modeling example: Golomb rulerA Golomb ruler is a set of marks at integer positions along an imaginary ruler such that no two pairs of marks are the same dist

from docplex.cp.model import CpoModel

# Create modelmdl = CpoModel()

# Create an array of variables for positions of ruler marksORDER = 6marks = mdl.integer_var_list(ORDER, 0, (ORDER - 1) ** 2, "M")

# Force all marks distances to be differentmdl.add(mdl.all_diff([marks[i] - marks[j] for i in range(1, ORDER) for j in range(0, i)]))

# Avoid symmetric and mirror solutionsfor i in range(1, ORDER):

mdl.add(marks[i] > marks[i - 1])mdl.add((marks[1] - marks[0]) < (marks[ORDER - 1] - marks[ORDER - 2]))

# Minimize ruler size (position of the last mark)mdl.add(mdl.minimize(marks[ORDER - 1]))

# Solve modelmsol = mdl.solve()print("Solution: " + ' '.join(str(msol[v]) for v in marks))

> python golomb_ruler.pySolution: 0 1 4 10 12 17


Improvements to the Python API in CP Optimizer 12.7.0

§ Local solve, using CP Optimizer interactive as back-end§ With local solve only:

o propagation (available soon also on DOcloud),o conflict refiner,o retrieve multiple solutions with start/next implemented as iterator.

§ With local and DOcloud solve:o warm start

§ Support of numpy types and arrayso support of numpy arrays and numeric types

§ Integration in DOcloud Python workero allows to submit directly a Python source to DOcloudo all functions become available as for local solve.


Python new feature: Propagation

§ Constraint propagation is the process of communicating the domain reduction of a decision variable to all of the constraints that are stated over this variable.

§ Result is a partial solution with a reduced domain for each variable.



# Create an array of variables for positions of ruler marksORDER = 6marks = mdl.integer_var_list(ORDER, 0, (ORDER - 1) ** 2, "M")





# Propagate constraintsmsol = mdl.propagate()print("Solution: " + ' '.join(str(msol[v]) for v in marks))

> python golomb_ruler.pySolution: ((0, 20),) ((1, 21),) ((2, 22),) ((3, 23),) ((4, 24),) ((5, 25),)


Python new feature: Conflict refiner

§ The conflict refiner is used to find an explanation for the infeasibility of a model.§ Result is a list of constraints and/or variable that are a primer failing cause.



# Create an array of variables for positions of ruler marksORDER = 4 # Number of marksMAX_LENGTH = 4 # Max rule length, voluntarily TOO SHORT to have an infeasible modelmarks = mdl.integer_var_list(ORDER, 0, MAX_LENGTH, "M")





# Call conflict refiner to identify failing causesrsol = mdl.refine_conflict()rsol.print_conflict()

> python golomb_ruler.pyConflict refiner result:Member constraints:

_CTR_3 = M2 > M1_CTR_2 = M1 > M0_CTR_1 = alldiff([M1 - M0, M2 - M0, M2 - M1, M3 - M0, M3 - M1, M3 - M2])_CTR_4 = M3 > M2


Python new feature: Solution iteration

from docplex.cp.model import CpoModelfrom docplex.cp.solver.solver import CpoSolver

# Create model. . . . . (see above) . . . . .

# Minimize ruler size (position of the last mark)minexpr = mdl.minimize(marks[ORDER - 1])mdl.add(minexpr)

# First solve the model to find the smallest ruler lengthmsol = mdl.solve()rsize = msol[marks[ORDER - 1]]print("Shortest ruler for order {} has length {}".format(ORDER, rsize))

# Update the model to remove minimization and force last mark positionmdl.remove(minexpr)mdl.add(marks[ORDER - 1] == rsize)

# Request all solutionsprint("List of all possible rulers:")for msol in CpoSolver(mdl, SearchType='DepthFirst', Workers=1):

print(' '.join(str(msol[v]) for v in marks))

> python golomb_ruler.pyShortest ruler for order 6 has length 17List of all possible rulers for length 17:0 1 4 10 12 170 1 8 11 13 170 1 8 12 14 170 1 4 10 15 17

§ With our Golomb ruler example, suppose we would like to retrieve ALL possible rulers that have a minimal length.

§ We first solve the initial model to get the minimal ruler size, force the last mark to this size, and then iterate on all solutions :


Python new feature: Warm start

§ Warm start allows to initialize the solver with a partial solution that can speed-up the search of the solution.

§ This function is available when solving locally or with DOcloud.

from docplex.cp.model import CpoModelfrom docplex.cp.solution import CpoModelSolution

# Create model. . . . . (as above) . . . . .

# Add starting pointsp = CpoModelSolution()sp.add_integer_var_solution(marks[1], 1)sp.add_integer_var_solution(marks[2], 4)mdl.set_starting_point(sp)

# Solve modelmsol = mdl.solve()print("Solution: " + ' '.join(str(msol[v]) for v in marks))

> python golomb_ruler.pySolution: 0 1 4 10 12 17


V12.5.1 06/2013

V12.6 10/2013

V12.6.1 9/2014

V12.6.2 5/2015

V12.6.3 11/2015

V12.7 10/2016

0%10%20%30%40%50%60%70%

Progress factor between CP Optimizer releases

Integer

Scheduling


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• © IBM Corporation 2015. All Rights Reserved.• The information contained in this publication is provided for informational purposes only. While efforts were made to verify

the completeness and accuracy of the information contained in this publication, it is provided AS IS without warranty of any kind, express or implied. In addition, this information is based on IBM’s current product plans and strategy, which are subject to change by IBM without notice. IBM shall not be responsible for any damages arising out of the use of, or otherwise related to, this publication or any other materials. Nothing contained in this publication is intended to, nor shall have the effect of, creating any warranties or representations from IBM or its suppliers or licensors, or altering the terms and conditions of the applicable license agreement governing the use of IBM software.

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• Performance is based on measurements and projections using standard IBM benchmarks in a controlled environment. The actual throughput or performance that any user will experience will vary depending upon many factors, including considerations such as the amount of multiprogramming in the user's job stream, the I/O configuration, the storage configuration, and the workload processed. Therefore, no assurance can be given that an individual user will achieve results similar to those stated here.

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