Approximation Methods for Polynomial Optimization: Models, Algorithms, and Applications

129 Pages·2012·0.72 MB·English

by  Zhening Li

SpringerBriefs in Optimization
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Ja´nosD.Pinte´r
StephenM.Robinson
Tama´sTerlaky
MyT.Thai
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Zhening Li (cid:129) Simai He (cid:129) Shuzhong Zhang
Approximation Methods
for Polynomial Optimization
Models, Algorithms, and Applications
123
ZheningLi SimaiHe
DepartmentofMathematics DepartmentofManagementSciences
ShanghaiUniversity CityUniversityofHongKong
Shanghai KowloonTong
China HongKong
ShuzhongZhang
IndustrialandSystemsEngineering
UniversityofMinnesota
Minneapolis,MN
USA
ISSN2190-8354 ISSN2191-575X(electronic)
ISBN978-1-4614-3983-7 ISBN978-1-4614-3984-4(eBook)
DOI10.1007/978-1-4614-3984-4
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2012936832
Mathematics Subject Classification (2010): 90C59, 68W25, 65Y20, 15A69, 15A72, 90C26, 68W20,
90C10,90C11
©ZheningLi,SimaiHe,ShuzhongZhang2012
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Preface
Polynomial optimization, as its name suggests, is used to optimize a generic
multivariate polynomial function, subject to some suitable polynomial equality
and/orinequalityconstraints.Suchproblemformulationdatesbacktothenineteenth
centurywhentherelationshipbetweennonnegativepolynomialsandsumofsquares
(SOS)wasdiscussedbyHilbert.Polynomialoptimizationisoneofthefundamental
problems in Operations Research and has applications in a wide range of areas,
includingbiomedicalengineering,controltheory,graphtheory,investmentscience,
materialscience, numericallinearalgebra,quantummechanics,signalprocessing,
speech recognition, among many others. This brief discusses some important
subclasses of polynomial optimization models arising from various applications.
The focus is on optimizing a high degree polynomial function over some fre-
quently encountered constraint sets, such as the Euclidean ball, the Euclidean
sphere, intersection of co-centered ellipsoids, binary hypercube, general convex
compactset, and possibly a combinationof the above constraints. All the models
under consideration are NP-hard in general. In particular, this brief presents a
study on the design and analysis of polynomial-time approximation algorithms,
with guaranteed worst-case performance ratios. We aim at deriving the worst-
case performance/approximation ratios that are solely dependent on the problem
dimensions, meaning that they are independentof any other types of the problem
parametersorinputdata.Thenewtechniquescanbeappliedtosolveevenbroader
classes of polynomial/tensor optimization models. Given the wide applicability
of the polynomial optimization models, the ability to solve such models—albeit
approximately—is clearly beneficial. To illustrate how such benefits might be,
we present a variety of examples in this brief so as to showcase the potential
applicationsofpolynomialoptimization.
Shanghai,China ZheningLi
KowloonTong,HongKong SimaiHe
Minnesota,MN,USA ShuzhongZhang
v

Contents
1 Introduction .................................................................. 1
1.1 History................................................................... 1
1.1.1 Applications..................................................... 2
1.1.2 Algorithms...................................................... 6
1.2 Contributions............................................................ 9
1.3 NotationsandModels................................................... 10
1.3.1 ObjectiveFunctions............................................. 10
1.3.2 ConstraintSets.................................................. 12
1.3.3 ModelsandOrganization....................................... 13
1.4 Preliminary.............................................................. 14
1.4.1 TensorOperations .............................................. 15
1.4.2 ApproximationAlgorithms..................................... 17
1.4.3 RandomizedAlgorithms ....................................... 18
1.4.4 SemidefiniteProgrammingRelaxationandRandomization.. 19
2 PolynomialOptimizationOvertheEuclideanBall....................... 23
2.1 MultilinearForm........................................................ 24
2.1.1 ComputationalComplexity..................................... 25
2.1.2 CubicCase...................................................... 26
2.1.3 GeneralFixedDegree .......................................... 29
2.2 HomogeneousForm .................................................... 30
2.2.1 LinkBetweenMultilinearFormandHomogeneous
Form............................................................. 31
2.2.2 TheOddDegreeCase .......................................... 32
2.2.3 TheEvenDegreeCase ......................................... 33
2.3 MixedForm ............................................................. 35
2.3.1 ComplexityandaStep-by-StepAdjustment .................. 36
2.3.2 ExtendedLinkBetweenMultilinearFormandMixed
Form............................................................. 39
2.4 InhomogeneousPolynomial............................................ 40
2.4.1 Homogenization ................................................ 43
2.4.2 MultilinearFormRelaxation................................... 44
vii
viii Contents
2.4.3 AdjustingtheHomogenizingComponents.................... 46
2.4.4 FeasibleSolutionAssembling.................................. 49
3 ExtensionsoftheConstraintSets .......................................... 53
3.1 HypercubeandBinaryHypercube..................................... 53
3.1.1 MultilinearForm................................................ 55
3.1.2 HomogeneousForm............................................ 59
3.1.3 MixedForm..................................................... 62
3.1.4 InhomogeneousPolynomial.................................... 64
3.1.5 Hypercube....................................................... 70
3.2 TheEuclideanSphere................................................... 70
3.3 IntersectionofCo-centeredEllipsoids................................. 74
3.3.1 MultilinearForm................................................ 76
3.3.2 HomogeneousForm............................................ 81
3.3.3 MixedForm..................................................... 83
3.3.4 InhomogeneousPolynomial.................................... 84
3.4 ConvexCompactSet.................................................... 87
3.5 MixtureofBinaryHypercubeandtheEuclideanSphere............. 90
3.5.1 MultilinearForm................................................ 91
3.5.2 HomogeneousForm............................................ 94
3.5.3 MixedForm..................................................... 96
4 Applications .................................................................. 99
4.1 Homogeneous Polynomial Optimization Over
theEuclideanSphere.................................................... 99
4.1.1 SingularValuesofTrilinearForms ............................ 99
4.1.2 Rank-OneApproximationofTensors.......................... 100
4.1.3 EigenvaluesandApproximationofTensors................... 101
4.1.4 DensityApproximationinQuantumPhysics.................. 103
4.2 InhomogeneousPolynomialOptimizationOveraGeneralSet....... 104
4.2.1 PortfolioSelectionwithHigherMoments..................... 104
4.2.2 SensorNetworkLocalization .................................. 105
4.3 DiscretePolynomialOptimization..................................... 106
4.3.1 TheCut-NormofTensors...................................... 106
4.3.2 MaximumCompleteSatisfiability............................. 107
4.3.3 Box-ConstrainedDiophantineEquation....................... 108
4.4 MixedIntegerProgramming............................................ 109
4.4.1 MatrixCombinatorialProblem ................................ 109
4.4.2 Vector-ValuedMaximumCut.................................. 110
5 ConcludingRemarks........................................................ 113
References......................................................................... 119
Chapter 1
Introduction
Polynomialoptimizationistooptimizeapolynomialfunctionsubjecttopolynomial
equalityand/orinequalityconstraints,specifically,thefollowinggenericoptimiza-
tionmodel:
(PO)min p(xxx)
s.t. f(xxx)≤0,i=1,2,...,m ,
i 1
g (xxx)=0, j=1,2,...,m ,
j 2
xxx=(x ,x ,...,x )T∈Rn,
1 2 n
wherep(xxx), f(xxx)(i=1,2,...,m )andg (xxx)(j=1,2,...,m )aresomemultivariate
i 1 j 2
polynomial functions. This problem is a fundamental model in the field of op-
timization, and has applications in a wide range of areas. Many algorithms have
beenproposedforsubclassesof(PO),andspecializedsoftwarepackageshavebeen
developed.
1.1 History
The modern history of polynomial optimization may date back to the nineteenth
century when the relationship between nonnegative polynomial function and the
sumofsquares(SOS)ofpolynomialswasstudied.Givenamultivariatepolynomial
function that takes only nonnegative values over the real domain, can it be
representedasanSOSofpolynomialfunctions?Hilbert[51]gaveaconcreteanswer
in 1888, which asserted that the only cases for a nonnegative polynomial to be a
SOSare:univariatepolynomials;multivariatequadraticpolynomials;andbivariate
quarticpolynomials.Later,theissueaboutnonnegativepolynomialswasformulated
in Hilbert’s17th problem—oneof the famous23 problemsthat Hilbertaddressed
in his celebrated speech in 1900 at the Paris conference of the International
Congress of Mathematicians. Hilbert conjectured that a nonnegative polynomial
entailsexpressionofdefiniterationalfunctionsasquotientsoftwosumsofsquares.
Z.Lietal.,ApproximationMethodsforPolynomialOptimization:Models,Algorithms, 1
andApplications,SpringerBriefsinOptimization,DOI10.1007/978-1-4614-3984-4 1,
©ZheningLi,SimaiHe,ShuzhongZhang2012