Introduction
PartI.NonlinearAnalysis:Theory
1.MinimisationProblems:GeneralTheorems
1.1Introduction
1.2Definitions
1.3Epigraph
1.4LowerSections
1.5LowerSemi-continuousFunctions
1.6LowerSemi-compactFunctions
1.7ApproximateMinimisationofLowerSemi-continuousFunctions
onaCompleteSpace
1.8ApplicationtoFixed-pointTheorems
2.ConvexFunctionsandProximation,Projectionand
SeparationTheorems
2.1Introduction
2.2Definitions
2.3ExamplesofConvexFunctions
2.4ContinuousConvexFunctions
2.5TheProximationTheorem
2.6SeparationTheorems
3.ConjugateFunctionsandConvexMinimisationProblems
3.1Introduction
3.2CharacterisationofConvexLowerSemi-continuousFunctions
3.3Fenchel'sTheorem
3.4PropertiesofConjugateFunctions
3.5SupportFunctions
4.SubdifferentialsofConvexFunctions
4.1Introduction
4.2Definitions
4.3SubdifferentiabilityofConvexContinuousFunctions
4.4SubdifferentiabilityofConvexLowerSemi-continuousFunctions
4.5SubdifferentialCalculus
4.6TangentandNormalCones
5.MarginalPropertiesofSolutionsofConvexMinimisation
Problems
5.1Introduction
5.2Fermat'sRule
5.3MinimisationProblemswithConstraints
5.4PrincipleofPriceDecentralisation
5.5RegularisationandPenalisation
6.GeneralisedGradientsofLocallyLipschitzFunctions
6.1Introduction
6.2Definitions
6.3ElementaryProperties
6.4GeneralisedGradients
6.5NormalandTangentConestoaSubset
6.6Fermat'sRuleforMinimisationProblemswithConstraints
7.Two-personGames.FundamentalConceptsandExamples
7.1Introduction
7.2DecisionRulesandConsistentPairsofStrategies
7.3Brouwer'sFixed-pointTheorem(1910)
7.4TheNeedtoConvexify:MixedStrategies
7.5GamesinNormal(Strategic)Form
7.6ParetoOptima
7.7ConservativeStrategies
7.8SomeFiniteGames
7.9Cournot'sDuopoly
8.Two-personZero-sumGames:TheoremsofVonNeumann
andKyFan
8.1Introduction
8.2ValueandSaddlePointsofaGame
8.3ExistenceofConservativeStrategies
8.4ContinuousPartitionsofUnity
8.5OptimalDecisionRules
9.SolutionofNonlinearEquationsandInclusions
9.1Introduction
9.2UpperHemi-continuousSet-valuedMaps
9.3TheDebreu-Gale-Nikai'doTheorem
9.4TheTangentialCondition
9.5TheFundamentalTheoremfortheExistenceofZerosof
aSet-valuedMap
9.6Fixed-pointTheorems
9.7TheViabilityTheorem
9.8VariationalInequalities
9.9TheLeray-SchauderTheorem
9.10Quasi-variationalInequalities
9.11Shapley'sGeneralisationoftheThree-PolesLemma
10.IntroductiontotheTheoryofEconomicEquilibrium
10.1Introduction
10.2ExchangeEconomies
10.3TheWalrasian~Mechanism
10.4AnotherMechanismforPriceDecentralisation
10.5CollectiveFinancialRule
11.TheVonNeumannGrowthModel
11.1Introduction
11.2TheVonNeumannModel
11.3ThePerron-FrobeniusTheorem
11.4SurjectivityoftheMmatrices
12.n-personGames
12.1Introduction
12.2Non-cooperativeBehaviour
12.3n-personGamesinNormal(Strategic)Form
12.4Non-cooperativeGameswithConstraints(Metagames)
12.5ParctoOptima
12.6BehaviourofPlayersinCoalitions
12.7CooperativeGamesWithoutSidePayments
13.CooperativeGamesandFuzzyGames
13.1Introduction
13.2Coalitions,FuzzyCoalitionsandGeneralisedCoalitions
ofnPlayers
13.3ActionGamesandEquilibriumCoalitions
13.4Share-outGameswithSidePayments
13.5CoreandShapleyValueofStandardGames
PartII.NonlinearAnalysis:Examples
14.Exercises
14.1ExercisesforChapter1
14.2ExercisesforChapter2
14.3ExercisesforChapter3
14.4ExercisesforChapter4
14.5ExercisesforChapter5
14.6ExercisesforChapter6
14.7ExercisesforChapter8
14.8ExercisesforChapter9
14.9ExercisesforChapter10
14.10ExercisesforChapter11
14.11ExercisesforChapter12
14.12ExercisesforChapter13
15.StatementsofProblems
15.1Problem1-Set-valuedMapswithaClosedGraph
15.2Problem2-UpperSemi-continuousSet-valuedMaps
15.3Problem3-ImageofaSet-valuedMap
15.4Problem4-InverseImageofaSet-valuedMap
15.5Problem5-PolarsofaSet-valuedMap
15.6Problem6-MarginalFunctions
15.7Problem7-GenericContinuityofaSet-valuedMapwith
aClosedGraph
15.8Problem8-ApproximateSelectionofanUpperSemi-continuous
Set-valuedMap
15.9Problem9-ContinuousSelectionofaLowerSemi-continuous
Set-valuedMap
15.10Problem10-InterioroftheImageofaConvexClosedCone
15.11Problem11-DiscreteDynamicalSystems
15.12Problem12-FixedPointsofContractiveSet-valuedMaps
15.13Problem13-ApproximateVariationalPrinciple
15.14Problem14-OpenImageTheorem
15.15Problem15-AsymptoticCentres
15.16Problem16-FixedPointsofNon-expansiveMappings
15.17Problem17-OrthogonalProjectorsontoConvexClosedCones
15.18Problem18-Gamma-convexFunctions
15.19Problem19-ProperMappings
15.20Problem20-Fenchel'sTheoremfortheFunctionsL(x,Ax)
15.21Problem21-ConjugateFunctionsofx→L(x,Ax)
15.22Problem22-HamiltoniansandPartialConjugates
15.23Problem23-LackofConvexityandFenchel'sTheoremfor
ParetoOptima
15.24Problem24-DualityinLinearProgramming
15.25Problem25-LagrangianofaConvexMinimisationProblem
15.26Problem26-VariationalPrinciplesforConvexLagrangians
15.27Problem27-VariationalPrinciplesforConvexHamiltonians
15.28Problem28-ApproximationtoFermat'sRule
15.29Problem29-TransposesofConvexProcesses
15.30Problem30-ConeswithaCompactBase
15.31Problem31-RegularityofTangentCones
15.32Problem32-TangentConestoanIntersection
15.33Problem33-DerivativesofSet-valuedMapswithConvex
Graphs
i5.34Problem34-EpiderivativesofConvexFunctions
15.35Problem35-SubdifferentialsofMarginalFunctions
15.36Problem36-ValuesofaGameAssociatedwithaCovering
15.37Problem37-MinimaxTheoremswithWeak
CompactnessAssumptions
15.38Problem38-MinimaxTheoremsforFiniteTopologies
15.39Problem39-KyFan'sInequality
15.40Problem40-KyFan'sInequalityforMonotoneFunctions
15.41Problem41-GeneralisationoftheGale-Nikaido-Debreu
Theorem
15.42Problem42-EquilibriumofCoerciveSet-valuedMaps
15.43Problem43-EigenveetorsofSet-valuedMaps
15.44Problem44-PositiveEigenvectorsofPositiveSet-valuedMaps
15.45Problem45-SomeVariationalPrinciples
15.46Problem46-GeneralisedVariationalInequalities
15.47Problem47-MonotoneSet-valuedMaps
15.48Problem48-WalrasianEquilibriumforSet-valued
DemandMaps
16.SolutionstoProblems
16.1Problem1-Solution
16.2Problem2-Solution
16.3Problem3-Solution
16.4Problem4-Solution
16.5Problem5-Solution
16.6Problem6-Solution
16.7Problem7-Solution
16.8Problem8-Solution
16.9Problem9-Solution
16.10Problem10-Solution
16.11Problem11-Solution
16.12Problem12-Solution
16.13Problem13-Solution
16.14Problem14-Solution
16.15Problem15-Solution
16.16Problem16-Solution
16.17Problem17-Solution
16.18Problem18-Solution
16.19Problem19-Solution
16.20Problem20-Solution
16.21Problem21-Solution
16.22Problem22-Solution
16.23Problem23-Solution
16.24Problem24-Solution
16.25Problem25-Solution
16.26Problem26-Solution
16.27Problem27-Solution
16.28Problem28-Solution
16.29Problem29-Solution
16.30Problem30-Solution
16.31Problem31-Solution
16.32Problem32-Solution
16.33Problem33-Solution
16.34Problem34-Solution
16.35Problem35-Solution
16.36Problem36-Solution
16.37Problem37-Solution
16.38Problem38-Solution
16.39Problem39-Solution
16.40Problem40-Solution
16.41Problem41-Solution
16.42Problem42-Sol'ation
16.43Problem43-Solution
16.44Problem44-Solution
16.45Problem45-Solution
16.46Problem46-Solution
16.47Problem47-Solution
16.48Problem48-Solution
Appendix
17.CompendiumofResults
17.1Strict,Convex,LowerSemi-continuousFunctions
17.2ConvexFunctions
17.3ConjugateFunctions
17.4SeparationTheoremsandSupportFunctions
17.5Subdifferentiability
17.6TangentandNormalCones
17.7Optimisation
17.8Two-personGames
17.9Set-valuedMapsandtheExistenceofZerosandFixedPoints
References
SubjectIndex
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