Table Of ContentEURO Advanced Tutorials on Operational Research
Series Editors: M. Grazia Speranza · José Fernando Oliveira
Giancarlo Bigi
Marco Castellani
Massimo Pappalardo
Mauro Passacantando
Nonlinear
Programming
Techniques
for Equilibria
EURO Advanced Tutorials on Operational
Research
Serieseditors
M.GraziaSperanza,Brescia,Italy
JoséFernandoOliveira,Porto,Portugal
Moreinformationaboutthisseriesathttp://www.springer.com/series/13840
Giancarlo Bigi (cid:129) Marco Castellani (cid:129)
Massimo Pappalardo (cid:129) Mauro Passacantando
Nonlinear Programming
Techniques for Equilibria
123
GiancarloBigi MarcoCastellani
DepartmentofComputerScience DepartmentofInformationEngineering,
UniversityofPisa ComputerScienceandMathematics
Pisa,Italy UniversityofL’Aquila
L’Aquila,Italy
MassimoPappalardo MauroPassacantando
DepartmentofComputerScience DepartmentofComputerScience
UniversityofPisa UniversityofPisa
Pisa,Italy Pisa,Italy
ISSN2364-687X ISSN2364-6888 (electronic)
EUROAdvancedTutorialsonOperationalResearch
ISBN978-3-030-00204-6 ISBN978-3-030-00205-3 (eBook)
https://doi.org/10.1007/978-3-030-00205-3
LibraryofCongressControlNumber:2018956154
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Preface
In science, the term “equilibrium” has been widely used in physics, chemistry,
biology,engineering,andeconomics,amongothers,withindifferentframeworks.It
generallyreferstoconditionsorstatesofasysteminwhichallcompetinginfluences
are balanced. For instance, in physics the mechanical equilibrium is the state in
whichthe sumofallthe forcesandtorquesoneachparticleofthesystem iszero,
while a fluid is said to be in hydrostatic equilibrium when it is at rest, or when
the flow velocity at each point is constant over time. In chemistry, the dynamic
equilibrium is the state of a reversible reaction where the forward reaction rate is
equal to the reverse one. In biology, the genetic equilibrium denotes a situation
in whicha genotypedoesnotevolveanymore in a populationfromgenerationto
generation.Inengineering,thetrafficequilibriumistheexpectedsteadydistribution
oftrafficoverpublicroadsorovercomputerandtelecommunicationnetworks.Even
more, the well-known equilibrium theory is a fundamental branch of economics
studyingthe dynamicsofsupply,demand,andpricesin an economywithin either
one(partialequilibrium)orseveral(generalequilibrium)markets:thebasicmodel
of supply and demand is an example of the former, while the Arrow–Debreuand
Radnermodelsareexamplesofthelatter.
Actually,theterm“equilibrium”hasalsoalwaysbeenveryrelevantinmathemat-
ics, particularly in dynamical systems, partial differential equations, and calculus
of variations. After the breakthrough of game theory and the concept of Nash
equilibrium, the term has been used in mathematics in much larger contexts
involvingrelevantaspects of operationsresearch and mathematicalprogramming.
Indeed, many “equilibrium problems”, including some of them mentioned above,
can be modeled in this framework through different mathematical models such
asoptimization,complementarity,variationalinequalities,multiobjectiveoptimiza-
tion, noncooperative games, and inverse optimization, among others. All these
mathematical models share an underlying common structure that allows us to
convenientlyformulatetheminauniqueformat.
Thisbookfocusesontheanalysisofthisunifyingformatforequilibriumprob-
lems. Since it allows describinga large numberof applications,many researchers
devoted their efforts to study it, and nowadays, many results and algorithms are
v
vi Preface
available: as optimization fits in this format, nonlinear programming techniques
haveoftenbeenthekeytooloftheirwork.Thebookaimsataddressinginparticular
twocoreissuessuchastheexistenceandcomputationofequilibria.Thefirstchapter
illustratesasampleofapplications,thesecondaddressesthemaintheoreticalissues,
andthethirdintroducesthemainalgorithmsavailableforcomputingequilibria.The
final chapter is devoted to quasi-equilibria, a more general format that is needed
to cover more complex applications having additional features such as shared
resources in noncooperativegames. Finally, basic material on sets, functions, and
multivalued maps that are exploited throughout the book are summarized in the
appendix. To make the book as readable as possible, examples and applications
havebeenincluded.Wehopethatthisbookmayserveasabasisforasecond-level
academic course or a specialized course in a Ph.D. programand stimulate further
interestinequilibriumproblems.
Pisa,Italy GiancarloBigi
L’Aquila,Italy MarcoCastellani
Pisa,Italy MassimoPappalardo
Pisa,Italy MauroPassacantando
July2018
Contents
1 EquilibriumModelsandApplications..................................... 1
1.1 ObstacleProblem....................................................... 1
1.2 PowerControlinWirelessCommunications.......................... 3
1.3 TrafficNetwork......................................................... 4
1.4 PortfolioSelection...................................................... 6
1.5 OptimalProductionUnderRestrictedResources..................... 7
1.6 Input-OutputAnalysisinanEconomy................................ 8
1.7 QualityControlinProductionSystems ............................... 10
1.8 KyFanInequalities:AUnifyingEquilibriumModel................. 11
1.9 NotesandReferences .................................................. 16
2 TheoryforEquilibria........................................................ 17
2.1 ConvexityandMonotonicity........................................... 17
2.2 EquivalentReformulations............................................. 24
2.3 Existence................................................................ 31
2.4 Stability................................................................. 39
2.5 ErrorBounds............................................................ 43
2.6 NotesandReferences .................................................. 47
3 AlgorithmsforEquilibria................................................... 51
3.1 Fixed-PointandExtragradientMethods............................... 51
3.2 DescentMethods....................................................... 59
3.3 RegularizationMethods................................................ 65
3.4 ComputationalIssues................................................... 68
3.5 NotesandReferences .................................................. 69
4 Quasi-Equilibria ............................................................. 73
4.1 Applications ............................................................ 73
4.2 Theory................................................................... 75
4.3 Algorithms.............................................................. 80
4.4 NotesandReferences .................................................. 96
vii
viii Contents
A MathematicalBackground.................................................. 99
A.1 TopologicalConcepts .................................................. 99
A.2 Functions................................................................ 103
A.3 MultivaluedMaps ...................................................... 108
References......................................................................... 113
Index............................................................................... 119
Chapter 1
Equilibrium Models and Applications
As already mentioned in the preface, the term “equilibrium” is widespread in
science in the study of different phenomena. In this chapter a small selection of
equilibriumproblemsfromdifferentareasisgiven,eachleadingtoadifferentkind
of mathematical model. The equilibrium position of an elastic string in presence
of an obstacle, which is depicted in Sect.1.1, coincides with the solution of a
complementarity problem, the Nash equilibrium fits in well to model a power
controlmulti-agentsystem described in Sect.1.2, the steady distribution of traffic
overanetworkisrepresentedbyavariationalinequalityinSect.1.3,theMarkowitz
portfolio theory is viewed as a multiobjective problem in Sect.1.4, the shadow
pricetheoryisviewedasasaddlepointproblemforthenonlinearcaseinSect.1.5,
the solution of the input-output model given in Sect.1.6 is a fixed point and the
qualitycontrolprobleminaproductionsystemillustratedinSect.1.7isaninverse
optimization problem. Finally, the last section is devoted to show that all these
mathematicalmodels,whichareapparentlydifferent,haveacommonstructurethat
leadstoaunifiedformat:theKyFaninequalityorthe“equilibriumproblem”using
the “abstract” name introduced by Blum, Muu and Oettli to stress this unifying
feature.
1.1 ObstacleProblem
Consideranelasticone-dimensionalstringboundedonaplane.Theendpointsofthe
stringarekeptfixed,anditsnaturalpositionisastraightlinebetweentheendpoints.
What shape does the string take at the equilibrium if an obstacle is inserted in
between the two endpoints?The string stretches over the obstacle: it sticks to the
obstacle somewhere while it remains stretched as a straight line elsewhere (see
Fig.1.1).
©SpringerNatureSwitzerlandAG2019 1
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