[PDF] Rapid Mathematical Programming





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RapidMathematicalProgramming

vorgelegtvonDiplom-Mathematiker

ThorstenKoch

Flensburg

zurErlangungdesakademischenGrades

DoktorderNaturwissenschaften

genehmigteDissertation

Promotionsausschuß

Berlin2004

D83

RapidMathematical

Programming

ThorstenKoch

Copyright©2004byThorstenKoch

Allrightsreserved.

Allnumeralsinthistextarerecycled.

FürmeineEltern

Preface

Avoidrealityatallcosts

—fortune(6)

reproducible.Isthisthecase? torepeatorreproducetheresultsshown.

Acknowledgements

Ihaveencounteredsofar.

wiaMarkwardt,andTuomoTakkula.

Berlin,October2004

ThorstenKoch

Contents

1Introduction1

IDesignandImplementation7

2TheZimplModelingLanguage9

ix x

3ImplementingZimpl37

IIApplications63

xi

5MOMENTUM95

6SteinerTreePackingRevisited123

7Perspectives147

Appendix149

ANotation149

BZimplInternals151

xii

CZimplPrograms177

DSteinerTreePackingInstances187

ListofFigures193

ListofTables195

Bibliography197

Chapter1

Introduction

Ninety-NinetyRuleofProjectSchedules:

—fortune(6)

lifeproblems. arisewithreal-worlddata.

Whatmakesgeneraltoolsattractive?

Idonotknowexactlywhattheywant,

Ineeditnextweek,

Ihavenotcollectedyetthedatanecessary,

Iusuallyhavenotreallythedatatheyneed,

Ioftenneedonlyoneshotstudies,

Iareconvinced“ourproblemisunique".

1

2RapidMathematicalProgramming

steadilyincreasing.

1.1Threestepstosolveaproblem

wecansummarizethistothreetasks: i)Buildamathematicalmodeloftheproblem. iii)Implementthealgorithms. stage: beleftout. beNP-hard. extremelyfastinpractice.

Introduction3

AnalyzeReal-

worldProblem

Identify

ModelingGoal

BuildMathe-

maticalModel

Identify

DataSources

Collect&

AnalyzeData

ChooseSolution

Algorithm

TranslateModel

toSolverInput

TranslateData

toSolverInput

Construct

DerivedData

RunSolver

AnalyzeOutput

Write

ResultReport

Interpret

ResultReport

1.1.1Mathematicalworkbench

2http://www.mathworks.com

3http://www.wolfram.com

4RapidMathematicalProgramming

EffortEffectDo-it-yourself

Workbench

Modeling LanguageFramework

TP

Figure1.2:“EVortvs.eVect"

hastobechangedtocontriveaneasierproblem.

1.1.3Framework

tothebestperformingresultintheend.

Introduction5

1.1.4Doingitallyourself

specicimplementations.

1.2What'snext?

4http://www.spec.org

PartI

DesignandImplementation

7

Chapter2

TheZimplModelingLanguage

Whensomeonesays:

onlysaywhatIwishdone," givehimalollipop.

—fortune(6)

2.1Introduction

Considerthefollowinglinearprogram:

min2x+3y subjecttox+y66 x;y>0 linearprogramlooksasshowninFigure2.1. 9

10DesignandImplementation

1NAMEex1.mps

2ROWS

3NOBJECTIV

4Lc1

5COLUMNS

6xOBJECTIV2

7xc11

8yOBJECTIV3

9yc11 10RHS

11RHSc16

12BOUNDS

13LOBNDx0

14LOBNDy0

15ENDATA

Figure2.1:mpsleformatexample

min P i2Icixi subjecttoP i2Ixi66 x i>0foralli2I canbewritteninamplas setI; paramc{I}; varx{iinI}>=0; minimizecost:sum{iinI}c[i]*x[i]; subjecttocons:sum{iinI}x[i]<=6; linearprogrammingkit.

TheZimplModelingLanguage11

www.aimms.com opencommercial www.ampl.com opencommercial gamsGeneralAlgebraicModelingSystem www.gams.com opencommercial lingoLingo www.lindo.com xedcommercial www.virtual-optima.com opencommercial minoptMixedIntegerNon-linearOptimizer titan.princeton.edu/MINOPT openmixed moselMosel www.dashoptimization.com xedcommercial mplMathematicalProgrammingLanguage www.maximalsoftware.com opencommercial omniOmni www.haverly.com opencommercial oplOptimizationProgrammingLanguage www.ilog.com xedcommercial gnu-mpGNUMathematicalProgrammingLanguage www.gnu.org/software/glpk xedfree www.zib.de/koch/zimpl openfree

Table2.1:Modelinglanguages

12DesignandImplementation

aswewillseeinthesecondpartofthisthesis. mpslestodisk.

UniquefeaturesinZimpl

thatnoroundingerrorscanoccur. thefollowinglinearprogram: minx subjectto10-14x-10-14y>0 x+y>4 x;y>0 cplex ticallyvalidtransformations.

2http://www.ilog.com/products/cplex

TheZimplModelingLanguage13

maxx+2y subjecttox+y60:0000001 y>0:0000002 x>0 .A—isshownincasenobasiswascomputed

3http://www.zib.de/koch/perplex

4http://miplib.zib.de

5http://www.ilog.com/products/cplex

6http://www.dashoptimization.com

8http://www.coin-or.org

9http://www.gnu.org/software/glpk

10http://groups.yahoo.com/group/lp_solve

14DesignandImplementation

CPLEX

XPress

SoPlex

Clp GLPK lp_solve

Instance

Feas.Opti.

Feas.Opti.

Feas.Opti.

Feas.Opti.

Feas.Opti.

Feas.Opti.

C arki001 p p p p p p 7 atlanta-ip pp p p p p p 6 dano3mip pp p p pp pp pp 7 harp2 pp p pp pp pp p 5 momentum1 pp pp pp pp p 6 momentum2 p p pp p 7 momentum3 p p 7 msc98-ip p p p p 4 mzzv11 pp pp pp pp 5 qiu p p p p pp pp 4 roll3000 pp pp pp pp ppquotesdbs_dbs47.pdfusesText_47
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