LARGE-SCALE OPTIMIZATION FOR DATA PLACEMENT PROBLEM LAZIMA ANSARI Bachelor of

82 Pages·2017·0.27 MB·English

LARGE-SCALEOPTIMIZATIONFORDATAPLACEMENTPROBLEM
LAZIMAANSARI
BachelorofScience,MilitaryInstituteofScienceandTechnology,2010
AThesis
SubmittedtotheSchoolofGraduateStudies
oftheUniversityofLethbridge
inPartialFulfillmentofthe
RequirementsfortheDegree
MASTEROFSCIENCE
DepartmentofMathematicsandComputerScience
UniversityofLethbridge
LETHBRIDGE,ALBERTA,CANADA
(cid:2)c LazimaAnsari,2017
LARGE-SCALEOPTIMIZATIONFORDATAPLACEMENTPROBLEM
LAZIMAANSARI
DateofDefence: August17,2017
Dr. DayaGaur
Supervisor Professor Ph.D.
Dr. ShahadatHossain
CommitteeMember Professor Ph.D.
Dr. RobertBenkoczi
CommitteeMember AssociateProfessor Ph.D.
Dr. HowardCheng
Chair, Thesis Examination Com- AssociateProfessor Ph.D.
mittee
Dedication
Idedicatethisthesistomylovingparents. IcannotexpresshowluckyIamtohave
parentswholoveendlessly. Thankyousomuchforbelievinginme.
iii
Abstract
Large-scale optimization of combinatorial problems is one of the most challenging areas.
These problems are characterized by large sets of data (variables and constraints). In this
thesis, we study large-scale optimization of the data placement problem with zero storage
cost. The goal in the data placement problem is to find the placement of data objects
in a set of fixed capacity caches in a network to optimize the latency of access. Data
placementproblemarisesnaturallyinthedesignofcontentdistributionnetworks. Wereport
on an empirical study of the upper bound and the lower bound of this problem for large
sized instances. We also study a semi-Lagrangean relaxation of a closely related k-median
problem. Inthisthesis,westudythetheoryandpracticeofapproximationalgorithmforthe
dataplacementproblemandthek-medianproblem.
iv
Acknowledgments
I would like to express my sincere gratitude to my supervisor Dr. Daya Gaur for his guid-
ance and support throughout the whole learning process. I would also like to thank my
supervisory committee members Dr. Robert Benkoczi and Dr. Shahadat Hossain for their
preciousadviceandinspiration.
I am thankful to all the members of optimization research group of the University of Leth-
bridgefortheirsupport. IamgratefultoNabiandUmairfortheirhelp. Ialsowanttothank
AnamaySarkarwithwhomIworkedonsomeimportantpartsofmythesis.
A very special thanks goes to my husband and best friend Imtiaz, who has been a constant
source of support and encouragement during all the challenges of my life. This accom-
plishment would not have been possible without him. Thanks for making me realize that
thedreamscanactuallycometrue. Thanksforeverything.
I would also like to thank my family for the support they provided me through my entire
life. I must acknowledge my parents, sister, parents-in-law. Without their unconditional
love,sacrifices,andprayers,Iwouldnothavefinishedthisthesis.
I want to thank my friends Fatema, Tasnuba and Jeeshan who made my life easier in
Canada.
v
Contents
Contents vi
ListofTables viii
ListofFigures ix
1 Introduction 1
1.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Organizationofthethesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 PreliminariesandRelatedConcepts 4
2.1 Optimizationproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Linearprogramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Simplexmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Othermethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Integerprogramming(IP) . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Integerprogrammingsolutionmethods . . . . . . . . . . . . . . . . . . . . 11
2.5 Lagrangeanrelaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5.1 BasicFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5.2 Subgradientoptimization . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.3 Semi-Lagrangeanrelaxation(SLR) . . . . . . . . . . . . . . . . . 17
2.6 Localsearchheuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6.1 Generalalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 k-medianproblem 21
3.1 Problemdefinition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Relatedresearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Computingtheupperbound . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 Localsearchmethod . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.2 Asimpleranalysisoflocalsearchmethodforthek-medianproblem 26
3.4 Computingthelowerbound . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.1 Semi-Lagrangeanrelaxationfork-median . . . . . . . . . . . . . . 30
3.4.2 TheAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 ExperimentsandResults . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
vi
CONTENTS
4 Dataplacementproblem 40
4.1 ProblemDefinition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Relatedresearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Computingtheupperbound . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.1 Localsearchmethod . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.2 A simpler analysis of local search method for the uncapacitated
facilitylocationproblem . . . . . . . . . . . . . . . . . . . . . . . 47
4.4 Computingthelowerbound . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4.1 Lagrangeanrelaxation1(LR1-DP) . . . . . . . . . . . . . . . . . 53
4.4.2 Lagrangeanrelaxation2(LR2-DP) . . . . . . . . . . . . . . . . . 56
4.5 ExperimentsandResults . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5.1 Generationoftestinstances . . . . . . . . . . . . . . . . . . . . . 58
4.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 ConclusionandFutureworks 68
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Futurework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Bibliography 70
vii
List of Tables
2.1 Notationsusedinsubgradientprocedure . . . . . . . . . . . . . . . . . . . 16
3.1 Notationsusedintheanalysisoflocalsearchmethodforthek-medianprob-
lem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Experimentalresultsfork-medianproblem . . . . . . . . . . . . . . . . . 35
3.2 Experimentalresultsfork-medianproblem . . . . . . . . . . . . . . . . . 36
3.2 Experimentalresultsfork-medianproblem . . . . . . . . . . . . . . . . . 37
4.1 Notationsusedintheanalysisoflocalsearchmethodfortheuncapacitated
facilitylocationproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Experimentalresultsforthedataplacementproblem . . . . . . . . . . . . 61
viii
List of Figures
3.1 Anexamplemappingη:F∗ →F . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Assignmentofclients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1 Assignmentofclienttofacility . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Assignmentofclienttofacility . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 AcomparisonofdualitygapfortwoLagrangeanrelaxationswhilevarying
thenumberofcaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4 A comparison of time for two Lagrangean relaxations while varying the
numberofcaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5 AcomparisonofdualitygapfortwoLagrangeanrelaxationswhilevarying
thecachecapacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6 A comparison of time for two Lagrangean relaxations while varying the
cachecapacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.7 AcomparisonofdualitygapfortwoLagrangeanrelaxationswhilevarying
thenumberofclients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.8 A comparison of time for two Lagrangean relaxations while varying the
numberofclients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.9 AcomparisonofdualitygapfortwoLagrangeanrelaxationswhilevarying
thenumberofobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.10 A comparison of time for two Lagrangean relaxations while varying the
numberofobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
ix
Chapter 1
Introduction
Facility location problems have occupied a major place in Operations Research since the
early 1960s. They model situations such as the placements of warehouses [35], factories,
firestations,hospitalsandsoon. Everylocationproblemconsistsoffourbasiccomponents
[13].
• Asetoflocationswherethefacilitiesareplaced. Foreachsuchlocation,wearegiven
acostofopening(orstoring)thefacilities.
• A set of demand points or clients who need certain services from the facilities and
havetobeassignedtoafacilitysuchthattheirrequirementsarefulfilled.
• Alistofrequirementstobemetbythefacilitiesandassignmentofclientstofacilities.
• Acostfunctionassociatedwiththeassignmentofthedemandpointstothefacilities.
Atypicalobjectiveistoselectasetoffacilitiestoopeninordertooptimizethegivenfunc-
tion. Varioustypesoffacilitylocationproblemscanbeobtainedusingtheabovementioned
features. The location problem that we study in this thesis is the data placement problem
[3]. Data placement problem have been studied extensively in the areas of database man-
agement [24] and cooperative caching in networks [18, 4, 33]. This problem also arises
naturally in the modeling and operation of Content Distribution Networks [10, 9, 50, 19].
Insuchanetwork,thecomputersystemsneedtooptimizethedistributionofInternetpack-
etstotheusersbyreplicatingandcachingthedataatmultiplelocationsinthenetwork. This
reduces the load on the server and also helps to eliminate network congestion. Due to the
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