Table Of ContentSPRINGER BRIEFS IN OPTIMIZATION
Remigijus Paulavičius
Julius Žilinskas
Simplicial Global
Optimization
SpringerBriefs in Optimization
SeriesEditors
PanosM.Pardalos
JánosD.Pintér
StephenM.Robinson
TamásTerlaky
MyT.Thai
SpringerBriefs in Optimization showcases algorithmic and theoretical tech-
niques,casestudies,andapplicationswithinthebroad-basedfieldofoptimization.
Manuscripts related to the ever-growing applications of optimization in applied
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Forfurthervolumes:
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Remigijus Paulavicˇius (cid:129) Julius Žilinskas
Simplicial Global
Optimization
123
RemigijusPaulavicˇius JuliusŽilinskas
InstituteofMathematicsandInformatics InstituteofMathematicsandInformatics
VilniusUniversity VilniusUniversity
Vilnius,Lithuania Vilnius,Lithuania
ISSN2190-8354 ISSN2191-575X(electronic)
ISBN978-1-4614-9092-0 ISBN978-1-4614-9093-7(eBook)
DOI10.1007/978-1-4614-9093-7
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013949407
MathematicsSubjectClassification(2010):90C26,90C57,52B11,26B35,90C90
©RemigijusPaulavicˇius,JuliusŽilinskas2014
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Preface
Simplicialglobaloptimizationfocusesondeterministiccoveringmethodsforglobal
optimization partitioning the feasible region by simplices. Although rectangular
partitioning is used most often in global optimization, simplicial covering has
advantages shown in this book. The purpose of the book is to present global
optimization methods based on simplicial partitioning in one volume. The book
describes features of simplicial partitioning and demonstrates its advantages in
globaloptimization.
Asimplexisapolyhedroninamultidimensionalspace,whichhastheminimal
numberofvertices.Thereforesimplicialpartitionsarepreferableinglobaloptimiza-
tionwhenthevaluesoftheobjectivefunctionatallverticesofpartitionsareusedto
evaluatesubregions.
The feasible region defined by linear constraints may be covered by simplices
and therefore simplicial optimization algorithms may cope with linear constraints
inadelicatewaybyinitialcovering.Thismakessimplicialpartitionsveryattractive
foroptimizationproblemswithlinearconstraints.
Thereareoptimizationproblemswheretheobjectivefunctionshavesymmetries
which may be taken into account for reducing the search space significantly by
settinglinearinequalityconstraints.Theresultedsearchregionmaybecoveredby
simplices.
Applications benefiting from simplicial partitioning are examined in the book:
nonlinear least squares regression, center-based clustering of data having one
feature, and pile placement in grillage-type foundations. In the examples shown,
thesearchregionreducedtakingintoaccountsymmetriesoftheobjectivefunctions
isasimplexthussimplicialglobaloptimizationalgorithmsmayuseitasastarting
partition.
The book provides exhaustive experimental investigation and shows theimpact
of various bounds, types of subdivision, and strategies of candidate selection on
theperformanceofglobaloptimizationalgorithms.Researchersandengineerswill
benefit from simplicial partitioning algorithms presented in the book: Lipschitz
branch-and-bound,LipschitzoptimizationwithouttheLipschitzconstant.Wehope
v
vi Preface
the readers will be inspired to develop simplicial versions of other algorithms
for global optimization and even use other non-rectangular partitions for special
applications.
The book deals with theoretical, computational, and application aspects of
simplicial global optimization. It is intended for scientists and researchers in
optimizationandmayalsoserveasausefulresearchsupplementforPh.D.students
inmathematics,computerscience,andoperationsresearch.
TheauthorsareverygratefultoProf.PanosPardalos,DistinguishedProfessorat
theUniversityofFloridaandDirectoroftheCenterforAppliedOptimization,for
hiscontinuingencouragementandsupport.TheauthorshighlyappreciateSpringer’s
initiative to publish SpringerBriefs on Optimization and the given opportunity
to publish their book in this series. The authors would like to thank Springer’s
publishingeditorRaziaAmzadforguidingustopublicationofthebook.
Postdoctoral fellowship of R. Paulavicˇius is being funded by European Union
Structural Funds project “Postdoctoral Fellowship Implementation in Lithuania”
withintheframeworkoftheMeasureforEnhancingMobilityofScholarsandOther
Researchers and the Promotion of Student Research (VP1-3.1-ŠMM-01) of the
ProgramofHumanResourcesDevelopmentActionPlan.
Vilnius,Lithuania RemigijusPaulavicˇius
JuliusŽilinskas
Contents
1 SimplicialPartitionsinGlobalOptimization............................. 1
1.1 CoveringMethodsforGlobalOptimization........................... 1
1.2 SimplicialPartitioning.................................................. 5
1.3 CoveringaHyper-RectanglebySimplices ............................ 9
1.4 CoveringofFeasibleRegionDefinedbyLinearConstraints......... 16
2 LipschitzOptimizationwithDifferentBoundsoverSimplices ......... 21
2.1 LipschitzOptimization ................................................. 21
2.2 ClassicalLipschitzBounds............................................. 23
2.3 ImpactofNormsonLipschitzBounds................................. 28
2.4 LipschitzBoundBasedonCircumscribedSpheres ................... 34
2.5 TightLipschitzBoundoverSimpliceswiththe1-norm .............. 37
2.6 Branch-and-BoundwithSimplicialPartitionsandVarious
LipschitzBounds........................................................ 42
2.7 ParallelBranch-and-BoundwithSimplicialPartitions................ 49
2.8 ExperimentalComparisonofSelectionStrategies .................... 56
3 SimplicialLipschitzOptimizationWithoutLipschitzConstant........ 61
3.1 DIRECTAlgorithm...................................................... 62
3.2 ModificationsofDIRECTAlgorithm................................... 66
3.2.1 ModificationsofDIRECTforProblemswithConstraints..... 68
3.2.2 SymDIRECTAlgorithm ........................................ 69
3.3 DISIMPLAlgorithm..................................................... 71
3.3.1 DISIMPL forLipschitzOptimizationProblems
withLinearConstraints......................................... 77
3.3.2 ParallelDISIMPLAlgorithm ................................... 79
3.4 ExperimentalInvestigations............................................ 81
4 ApplicationsofGlobalOptimizationBenefiting
fromSimplicialPartitions................................................... 87
4.1 GlobalOptimizationinNonlinearLeastSquaresRegression......... 87
vii
viii Contents
4.2 Center-BasedClusteringProblemforData
HavingOnlyOneFeature .............................................. 96
4.3 PilePlacementinGrillage-TypeFoundations ......................... 101
AppendixA DescriptionofTestProblems................................... 107
References......................................................................... 131
Acronyms
n Numberofvariables
Rn n-dimensionalEuclideanspace
D Feasibleregion
" Tolerance
x;y;z Variables
x;y;z Vectorsofvariables
f.x/ Objectivefunction
rf.x/ Gradientofobjectivefunctionf.x/
F.x/ Lowerboundingfunction
f(cid:2) Globaloptimumfunctionvalue
f.x / "-globaloptimum
opt
x(cid:2) Globaloptimumvector
x "-globaloptimumvector
opt
S Solution(subregion,optimumpoint)
T Finite set of points where the objective function value has been
evaluated
I Subregionoffeasibleregion
L Candidateset
jLj Cardinalityofacandidateset
v Vertexofsubregion
V.I/ Setofverticesofsubregion
O n-dimensionalball
LB Lowerboundforminimum
UB Upperboundforminimum
R Circumradius
D Determinant
p Numberofprocessors
p;q Normindex
s Speedup
p
e Efficiency
p
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