2009 : Solving linear optimization problems with MOSEK

http://www.mosek.com

Solving Linear OptimizationProblems with MOSEK.

Bo Jensen MOSEK ApS,Fruebjergvej 3, Box 16, 2100 Copenhagen,

Denmark.Email: [email protected]

INFORMS Annual San Diego Tuesday Oct 13, 13:30 - 15:00, 2009

Introduction

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What is MOSEK

Introduction

Aim of talk

Topic

Introduction of theoptimizers

Introduction of theoptimizers(Continued)

Computational testsetup

Improvements in thepresolve

Improvements in thesimplex optimizers

Improvements in theinterior pointoptimizer

Conclusions

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■ A software package for solving large-scale optimizationproblems.

■ Solves linear, conic, and nonlinear convex problems.Mixed integer optimizer.

■ Used to solve problems with up to millions of constraintsand variables.

■ Stand-alone as well as embedded :

◆ C◆ Java◆ MATLAB◆ Python◆ .NET

What is MOSEK (Continued)

Introduction

Aim of talk

Topic







Conclusions

4 / 26

■ Third party interfaces :

◆ AIMMS◆ COIN-OR◆ Frontline Systems◆ GAMS◆ Microsoft Solver Foundation◆ OptimJ

■ Version 1 released 1999.■ Version 6 released October 2009 (Next week).

For further information see www.MOSEK.com.Prices public available directly on our website !

Aim of talk

Introduction

Aim of talk

Topic







Conclusions

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■ Highlight recent key improvements in the linearoptimizers.

■ Show computational results on a larger set of benchmarkproblems.

Topic

Introduction

Aim of talk

Topic







Conclusions

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■ The problem(P ) min cT x

st Ax = b,

x ≥ 0.

■ The linear optimizers.

◆ Interior-point based.◆ Primal and dual simplex based.◆ Primal network flow simplex.

■ What are the recent improvements?

Introduction of the optimizers

Introduction

Aim of talk

Topic







Conclusions

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■ Interior point.

◆ Based on the homogeneous and self-dual model.◆ Basis identification.◆ Highly tuned implementation.◆ Can exploit multiple processors.

■ Primal and Dual simplex.

◆ Exploits hyper sparsity (LU, solves and loops).◆ Multiple pricing options : Dantzig, partial, devex,

exact steepest edge, approximate steepest edge andhybrid pricing.

◆ Crashes an advanced initial basis.◆ Aggressive anti-degeneracy schemes.

Introduction of the optimizers (Continued)

Introduction

Aim of talk

Topic







Conclusions

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■ Network optimizer.

◆ Specialized primal simplex implementation.◆ Exploits the tree structure of the basis and other

tricks.◆ Can extract embedded network structure.

The network optimizer has not been changed since mosek 5and will be left out of benchmark.

Computational test setup

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Computational test setup

Introduction



How do we measureimprovements ?




Conclusions

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All computational results are run on the same machine :

■ Intel Xeon 2.27 GHz 16 Cores.■ 16 GB RAM.■ Running Linux.

Selected about 1000 problems from our clients and public

ressources

How do we measure improvement ?

Introduction



How do we measureimprovements ?




Conclusions

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■ Maximum solution time allowed 10000 sec, timeouts areexcluded.

■ Group problems by solution times :

◆ Small problems => best solution time ≤ 6 sek.◆ Medium problems => 6 sek < best solution time ≤

60 sek.◆ Large problems => best solution time > 60

■ Sum of solution time Bad benchmarking criteria!.■ Geometric mean of ratios of optimizer time i.e

MOSEK6/MOSEK5.■ Number of wins.

Improvements in the presolve

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Presolve overview

Introduction



Presolve overview

Presolving hotstarts


Numerical examples



Conclusions

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■ MOSEK has had an extensive presolve since version 1.See [AND:95].

■ It was only employed in cold start cases.■ Since MOSEK 5 we can also presolve hotstart.■ Presolving hotstarts has been improved in MOSEK 6.


Introduction



Presolve overview



Numerical examples



Conclusions

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■ Assume we have a ”good” (optimal?) primal x∗ or dualy∗ solution to P .But now the problem has been modified by the user into :

(P ) min cT x

st Ax = b,

x ≥ 0.


Introduction



Presolve overview



Numerical examples



Conclusions

14 / 26

■ Assume we have a ”good” (optimal?) primal x∗ or dualy∗ solution to P .But now the problem has been modified by the user into :

(P ) min cT x

st Ax = b,

x ≥ 0.

■ Challenges :

◆ The problem may have been modified in any way.◆ Some presolve techniques destroy primal and dual

feasibility.

■ Ex : Assume x∗ is primal feasible and x∗

k = uk,but dual presolve says x∗

k = lk ?

Presolving hotstarts (Continued)

Introduction



Presolve overview



Numerical examples



Conclusions

15 / 26

■ The MOSEK presolve strategy:

◆ Aimed at ”hard” hotstarts where presolve reallymakes a difference.


Introduction



Presolve overview



Numerical examples



Conclusions

15 / 26



◆ Preserve either primal (primal simplex) or dualfeasibility (dual simplex).


Introduction



Presolve overview



Numerical examples



Conclusions

15 / 26




◆ Also try to keep the basis intact.


Introduction



Presolve overview



Numerical examples



Conclusions

15 / 26




◆ Also try to keep the basis intact.◆ If your problem takes few iterations in reoptimizing,

then you should switch off presolve.

Numerical examples

Introduction



Presolve overview



Numerical examples



Conclusions

16 / 26

Selected 82 hotstart instances

Warning biased problem selection ahead!

small medium6-nopre 6 5 6-nopre 6 5

Num. 74 74 74 8 8 8Firsts 9 51 18 1 6 1Total time 1058.49 170.90 323.95 602.72 98.20 618.54G. avg. 5.79 1.77 2.39 41.68 11.21 44.37G. avg. r 2.42 0.74 0.94 0.25

Table 1: Presolve hotstart effect on client problems (Dual sim-plex).

Improvements in the simplex optimizers

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Key areas simplex optimizers

Introduction




Computationalresults-primalsimplex

Computationalresults-dual simplex


Conclusions

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■ Speed and stability has in general been greatly improved.


Introduction







Conclusions

18 / 26

■ Speed and stability has in general been greatly improved.■ Larger parts of the code has been rewritten.


Introduction







Conclusions

18 / 26

■ Speed and stability has in general been greatly improved.■ Larger parts of the code has been rewritten.■ Parameter tuning (i.e running many test with different

internal settings).


Introduction







Conclusions

18 / 26


internal settings).■ Hybrid pricing in primal simplex has been completely

rewritten.

◆ Improved the automatic switch between partial andapproximate steepest-edge pricing in primal simplex.

◆ Restricted pricing (i.e optimizing over a subset ofcolumns) is now used more intensively.


Introduction







Conclusions

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rewritten.



■ Better crash module in dual simplex.


Introduction







Conclusions

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rewritten.



■ Better crash module in dual simplex.■ The automatic refactor frequency in the LU module has

been improved.

Computational results-primal simplex

Introduction







Conclusions

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■ Small problems are misleading, since time measurementin MOSEK 5 and 6 are different !.

■ At least 25% speed up on average.

Computational results-dual simplex

Introduction







Conclusions

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■ Small problems are misleading, since time measurementin MOSEK 5 and 6 are different !.

■ At least 25% speed up on average.

Improvements in the interior pointoptimizer

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Key areas interior point optimizers

Introduction





Computationalresults-interior point

Conclusions

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■ Improved speed in general.■ Improved numerical stability on hard problems.

Computational results-interior point

Introduction





Computationalresults-interior point

Conclusions

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■ The testset is too easy for interior point !■ Interior point was 3-4 times faster than dual simplex !■ About 5-10% speed up (more for hard problems).

Conclusions

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Conclusions

Introduction





Conclusions

Conclusions

References

25 / 26

■ Much improved presolve performance for hotstarts.■ All optimizers more numerical stable.■ Primal simplex optimizer 25% faster on average.■ Dual simplex optimizer 25% faster on average.■ Interior point optimizer 5-10% faster on average.

References

Introduction





Conclusions

Conclusions

References

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[AND:95] E. Andersen and K. Andersen, ”Presolve in linear

programming”, Mathematical Programming, Volume 71 ,

1995

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2009 : Solving linear optimization problems with MOSEK

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