SCHAUMS OUTLINE OF THEORY AND PROBLEMS OF
newer programming language require a solid background in C. Most of these programming problems require no special mathematical or ... User's Manual.
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M.SC. MATHEMATICS Programming in C
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A Modeling Language for Mathematical Programming
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Bimatrix Equilibrium Points and Mathematical Programming
2 ?.?. 2561 BIMATRIX EQUILIBRIUM POINTS AND. MATHEMATICAL PROGRAMMING*t. C. E. LEMKE. Rensselaer Polytechnic Institute Troy
Rapid Mathematical Programming
For example a linear program like min ?i?I cixi subject to ?i?I xi ? 6 xi ? 0 for all i ? I can be written in as set. I; param c {I};.
Fuzzy Mathematical Programming approach for Solving Fuzzy
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Mathematical programming and especially linear programming
Mathematics programmes of study: key stages 1 and 2 - GOV.UK
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RapidMathematicalProgramming
vorgelegtvonDiplom-MathematikerThorstenKoch
Flensburg
zurErlangungdesakademischenGradesDoktorderNaturwissenschaften
genehmigteDissertationPromotionsausschuß
Berlin2004
D83RapidMathematical
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 x3ImplementingZimpl37
IIApplications63
xi5MOMENTUM95
6SteinerTreePackingRevisited123
7Perspectives147
Appendix149
ANotation149
BZimplInternals151
xiiCZimplPrograms177
DSteinerTreePackingInstances187
ListofFigures193
ListofTables195
Bibliography197
Chapter1
Introduction
Ninety-NinetyRuleofProjectSchedules:
fortune(6)
lifeproblems. arisewithreal-worlddata.Whatmakesgeneraltoolsattractive?
Idonotknowexactlywhattheywant,
Ineeditnextweek,
Ihavenotcollectedyetthedatanecessary,
Iusuallyhavenotreallythedatatheyneed,
Ioftenneedonlyoneshotstudies,
Iareconvincedourproblemisunique".
12RapidMathematicalProgramming
steadilyincreasing.1.1Threestepstosolveaproblem
wecansummarizethistothreetasks: i)Buildamathematicalmodeloftheproblem. iii)Implementthealgorithms. stage: beleftout. beNP-hard. extremelyfastinpractice.Introduction3
AnalyzeReal-
worldProblemIdentify
ModelingGoal
BuildMathe-
maticalModelIdentify
DataSources
Collect&
AnalyzeData
ChooseSolution
Algorithm
TranslateModel
toSolverInputTranslateData
toSolverInputConstruct
DerivedData
RunSolver
AnalyzeOutput
WriteResultReport
Interpret
ResultReport
1.1.1Mathematicalworkbench
2http://www.mathworks.com
3http://www.wolfram.com
4RapidMathematicalProgramming
EffortEffectDo-it-yourself
Workbench
Modeling LanguageFramework
TPFigure1.2:EVortvs.eVect"
hastobechangedtocontriveaneasierproblem.1.1.3Framework
tothebestperformingresultintheend.Introduction5
1.1.4Doingitallyourself
specicimplementations.1.2What'snext?
4http://www.spec.org
PartIDesignandImplementation
7Chapter2
TheZimplModelingLanguage
Whensomeonesays:
onlysaywhatIwishdone," givehimalollipop.fortune(6)
2.1Introduction
Considerthefollowinglinearprogram:
min2x+3y subjecttox+y66 x;y>0 linearprogramlooksasshowninFigure2.1. 910DesignandImplementation
1NAMEex1.mps
2ROWS3NOBJECTIV
4Lc15COLUMNS
6xOBJECTIV2
7xc118yOBJECTIV3
9yc11 10RHS11RHSc16
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 openfreeTable2.1: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 .Aisshownincasenobasiswascomputed3http://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
CPLEXXPress
SoPlex
Clp GLPK lp_solveInstance
Feas.Opti.
Feas.Opti.
Feas.Opti.
Feas.Opti.
Feas.Opti.
Feas.Opti.
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