Solving Optimization Problems with MATLAB (De Gruyter STEM)

344 Pages·2020·5.316 MB·English

by  Dingyü Xue

DingyüXue
SolvingOptimizationProblemswithMATLAB®
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Dingyü Xue
Solving Optimization
Problems with
MATLAB®
|
Author
Prof.DingyüXue
SchoolofInformationScienceandEngineering
NortheasternUniversity
WenhuaRoad3rdStreet
110819Shenyang
China
[email protected]
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softwareorrelatedproductsdoesnotconstituteendorsementorsponsorshipbyTheMathWorksof
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Preface
Scientificcomputingiscommonlyandinevitablyencounteredincourselearning,sci-
entificresearchandengineeringpracticeforeachscientificandengineeringstudent
andresearcher.Forthestudentsandresearchersinthedisciplineswhicharenotpure
mathematics,itisusuallynotawisethingtolearnthoroughlylow-leveldetailsofre-
latedmathematicalproblems,andalsoitisnotasimplethingtofindsolutionsofcom-
plicatedproblemsbyhand.Itisaneffectivewaytotacklescientificproblems,with
highefficiencyandin accurateand creativemanner,withthe mostadvancedcom-
putertools.Thismethodisespeciallyusefulinsatisfyingtheneedsforthoseinthe
areaofscienceandengineering.
Theauthorhadmadesomeefforttowardsthisgoalbyaddressingdirectlytheso-
lutionmethodsforvariousbranchesinmathematicsinasinglebook.Suchabook,
entitled“MATLABbasedsolutionstoadvancedappliedmathematics”,waspublished
firstin2004byTsinghuaUniversityPress.Severalneweditionsarepublishedafter-
wards: in 2015, the second edition in English by CRC Press and in 2018, the fourth
editioninChinesewerepublished.BasedonthelatestChineseedition,abrandnew
MOOCprojectwasreleasedin2018,1 andreceivedsignificantattention.Thenumber
oftheregisteredstudentswasaround14000inthefirstroundoftheMOOCcourse,
andreachedtensofthousandsinlaterrounds.Thetextbookhasbeencitedtensof
thousandstimesbyjournalpapers,books,anddegreetheses.
Theauthorhasover30yearsofextensiveexperienceofusingMATLABinscien-
tificresearchandeducation.Significantamountofmaterialsandfirst-handknowl-
edgehasbeenaccumulated,whichcannotbecoveredinasinglebook.Aseriesenti-
tled“ProfessorXueDingyü’sLectureHall”ofsuchworksarescheduledwithTsinghua
UniversityPress,andtheEnglisheditionsareincludedintheDGSTEMserieswithDe
Gruyter.Thesebooksareintendedtoprovidesystematic,extensiveanddeepexplo-
rationsinscientificcomputingskillswiththeuseofMATLABandrelatedtools.The
authorwantstoexpresshissincerethankstohissupervisor,ProfessorDerekAtherton
ofSussexUniversity,whofirstbroughthimintotheparadiseofMATLAB.
TheMATLABseriesisnotasimplerevisionoftheexistingbooks.Withdecades
ofexperienceandmaterialaccumulation,theideaof“revisiting”isadoptedinau-
thoringthesebooks,incontrasttoothermathematicsandMATLAB-richbooks.The
viewpointofanengineeringprofessorisestablishedandthefocusisonsolvingvar-
iousappliedmathematicalproblemswithtools.Manyinnovativeskillsandgeneral-
purposesolversareprovidedtosolveproblemswithMATLAB,whichisnotpossible
byanyotherexistingsolvers,soastobetterillustratetheapplicationsofcomputer
toolsinsolvingmathematicalproblemsineverymathematicsbranch.Italsohelps
1 MOOC(inChinese)address:https://www.icourse163.org/learn/NEU-1002660001
https://doi.org/10.1515/9783110667011-201
VI | Preface
thereadersbroadentheirviewpointsinscientificcomputing,andevenfindinginno-
vativesolutionsbythemselvestoscientificcomputingwhichcannotbesolvedbyany
otherexistingmethods.
ThefirsttitleintheMATLABseries:“MATLABProgramming”,canbeusedasan
entry-level textbook or reference book to MATLAB programming, so as to establish
asolidfoundationanddeepunderstandingfortheapplicationofMATLABinscien-
tificcomputing.Eachsubsequentvolumestriestocoverabranchortopicinmathe-
maticalcourses.Bearinginmindthe“computationalthinking”inauthoringthese-
ries, deep understanding and explorations are made for each mathematics branch
involved.TheseMATLABbooksaresuitableforthereaderswhohavealreadylearnt
therelatedmathematicalcourses,andwanttorevisitthecoursestolearnhowtosolve
theproblemsbyusingcomputertools.Itcanalsobeusedasacompanioninsynchro-
nizingthelearningofrelatedmathematicscourses,andviewingthecoursefromadif-
ferentangle,sothatthereadersmayexpandtheirknowledgeinlearningtherelated
courses,soastobetterlearn,understandandpracticethematerialsinthecourses.
ThisbookisthefourthoneintheMATLABseries.Twomaintopics–nonlinear
equationsolutionsandoptimizationtechniques–arecoveredinthisbook.Wecon-
centrateonsolvingtheproblemsinthesetwofields.Analyticalandnumericalsolu-
tions of various nonlinear algebraic equations are discussed first, and solution ap-
proachesareprovidedtoequationswithmultiplesolutions.Thesubsequentchapters
are devoted to introducing unconstrained optimization, linear and quadratic, con-
strainednonlinear,mixed-integer,multiobjectiveanddynamicalprogramming.Prac-
ticalattemptsaremadeforfindingglobaloptimumsolutions.Someintelligentopti-
mizationmethodsareintroduced,andstrictcomparisonsaremadetoassessthebe-
haviorsofconventionalandintelligentsolvers,andusefulconclusionsarereached.
Atthetimethebooksarepublished,theauthorwishestoexpresshissinceregrat-
itudetohiswife,ProfessorYangJun.Herloveandselflesscareoverthedecadespro-
vided the author immense power, which supports the authors’ academic research,
teaching,andwriting.
September2019 XueDingyü
Contents
Preface|V
1 Anintroductiontoequationsandoptimizationproblems|1
1.1 Equationsandtheirsolutions|1
1.2 Originsanddevelopmentofoptimizationproblems|2
1.3 Structureofthebook|3
1.4 Exercises|4
2 Solutionsofalgebraicequations |7
2.1 Solutionsofpolynomialequations|7
2.1.1 Polynomialequationsofdegrees1and2|8
2.1.2 Analyticalsolutionsofcubicequations|9
2.1.3 Analyticalsolutionsofquarticequation|10
2.1.4 Higher-degreeequationsandAbel–Ruffinitheorem|13
2.2 Graphicalmethodsfornonlinearequations|13
2.2.1 Smoothgraphicsforimplicitfunctions|13
2.2.2 Univariateequations|15
2.2.3 Equationswithtwounknowns|17
2.2.4 Isolatedequationsolutions|20
2.3 Numericalsolutionsofalgebraicequations|20
2.3.1 Newton–Raphsoniterativealgorithm|20
2.3.2 DirectsolutionmethodswithMATLAB|25
2.3.3 Accuracyspecifications|28
2.3.4 Complexdomainsolutions|29
2.4 Accuratesolutionsofsimultaneousequations|31
2.4.1 Analyticalsolutionsoflow-degreepolynomialequations|32
2.4.2 Quasianalyticalsolutionsofpolynomial-typeequations|35
2.4.3 Quasianalyticalsolutionsofpolynomialmatrixequations|37
2.4.4 Quasianalyticalsolutionsofnonlinearequations|40
2.5 Nonlinearmatrixequationswithmultiplesolutions|41
2.5.1 Anequationsolutionideaanditsimplementation|41
2.5.2 Pseudopolynomialequations|46
2.5.3 Aquasianalyticalsolver|48
2.6 Underdeterminedalgebraicequations|49
2.7 Exercises|51
3 Unconstrainedoptimizationproblems|55
3.1 Introductiontounconstrainedoptimizationproblems|55
3.1.1 Themathematicalmodelofunconstrainedoptimizationproblems|55
3.1.2 Analyticalsolutionsofunconstrainedminimizationproblems|56
VIII | Contents
3.1.3 Graphicalsolutions|56
3.1.4 Localandglobaloptimumsolutions|58
3.1.5 MATLABimplementationofoptimizationalgorithms|60
3.2 Directsolutionsofunconstrainedoptimizationproblemwith
MATLAB|62
3.2.1 Directsolutionmethods|62
3.2.2 Controloptionsinoptimization|65
3.2.3 Additionalparameters|69
3.2.4 Intermediatesolutionprocess|70
3.2.5 Structuredvariabledescriptionofoptimizationproblems|72
3.2.6 Gradientinformation|73
3.2.7 Optimizationsolutionsfromscattereddata|77
3.2.8 Parallelcomputationinoptimizationproblems|78
3.3 Towardsglobaloptimumsolutions|79
3.4 Optimizationwithdecisionvariablebounds|83
3.4.1 Univariateoptimizationproblem|83
3.4.2 Multivariateoptimizationproblems|85
3.4.3 Globaloptimumsolutions|87
3.5 Applicationexamplesofoptimizationproblems|87
3.5.1 Solutionsoflinearregressionproblems|88
3.5.2 Least-squarescurvefitting|89
3.5.3 Shootingmethodinboundaryvaluedifferentialequations|93
3.5.4 Convertingalgebraicequationsintooptimizationproblems|96
3.6 Exercises|97
4 Linearandquadraticprogramming|103
4.1 Anintroductiontolinearprogramming|104
4.1.1 Mathematicalmodeloflinearprogrammingproblems|104
4.1.2 Graphicalsolutionsoflinearprogrammingproblems|105
4.1.3 Introductiontothesimplexmethod|106
4.2 Directsolutionsoflinearprogrammingproblems|110
4.2.1 Alinearprogrammingproblemsolver|110
4.2.2 Linearprogrammingproblemswithmultipledecisionvectors|116
4.2.3 Linearprogrammingwithdoublesubscripts|117
4.2.4 Transportationproblem|118
4.3 Problem-baseddescriptionandsolutionoflinearprogramming
problems|122
4.3.1 MPSfileforlinearprogrammingproblems|122
4.3.2 Problem-baseddescriptionoflinearprogrammingproblems|124
4.3.3 Conversionsinlinearprogrammingproblems|129
4.4 Quadraticprogramming|130
4.4.1 Mathematicalquadraticprogrammingmodels|131
Contents | IX
4.4.2 Directsolutionsofquadraticprogrammingproblems|131
4.4.3 Problem-basedquadraticprogrammingproblemdescription|132
4.4.4 Quadraticprogrammingproblemwithdoublesubscripts|136
4.5 Linearmatrixinequalities|138
4.5.1 Descriptionoflinearmatrixinequalityproblems|138
4.5.2 Lyapunovinequalities|139
4.5.3 ClassificationsofLMIproblems|141
4.5.4 MATLABsolutionsofLMIproblems|142
4.5.5 OptimizationsolutionswithYALMIPtoolbox|144
4.5.6 Trialsonnonconvexproblems|146
4.5.7 Problemswithquadraticconstraints|147
4.6 Exercises|149
5 Nonlinearprogramming|153
5.1 Introductiontononlinearprogramming|153
5.1.1 Mathematicalmodelsofnonlinearprogrammingproblems|154
5.1.2 Feasibleregionsandgraphicalmethods|154
5.1.3 Examplesofnumericalmethods|157
5.2 Directsolutionsofnonlinearprogrammingproblems|159
5.2.1 DirectsolutionusingMATLAB|159
5.2.2 Handlingofearlierterminationphenomenon|165
5.2.3 Gradientinformation|166
5.2.4 Solvingproblemswithmultipledecisionvectors|168
5.2.5 Complicatednonlinearprogrammingproblems|169
5.3 Trialswithglobalnonlinearprogrammingsolver|171
5.3.1 Trialsonglobaloptimumsolutions|171
5.3.2 Nonconvexquadraticprogrammingproblems|174
5.3.3 Concave-costtransportationproblem|176
5.3.4 Testingoftheglobaloptimumproblemsolver|178
5.3.5 Handlingpiecewiseobjectivefunctions|179
5.4 Bilevelprogrammingproblems|181
5.4.1 Bilevellinearprogrammingproblems|182
5.4.2 Bilevelquadraticprogrammingproblem|183
5.4.3 BilevelprogramsolutionswithYALMIPToolbox|184
5.5 Nonlinearprogrammingapplications|185
5.5.1 Maximuminnerpolygoninsideacircle|185
5.5.2 Semiinfiniteprogrammingproblems|189
5.5.3 Poolingandblendingproblem|193
5.5.4 Optimizationdesignofheatexchangenetwork|196
5.5.5 Solvingnonlinearequationswithoptimizationtechniques|199
5.6 Exercises|201