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1.

図書

図書
Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal
出版情報: Berlin : Springer, c2001  x, 259 p. ; 24 cm
シリーズ名: Grundlehren text editions / editors, A. Chenciner ... [et al.] ; managing editors, M. Berger, J. Coates, S.R.S. Varadhan
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目次情報: 続きを見る
Preface
Introduction: Notation, Elementary Results / 0:
Some Facts About Lower and Upper Bounds / 1:
The Set of ExtendedReal Numbers / 2:
Linear and Bilinear Algebra / 3:
Differentiationin a Euclidean Space / 4:
Set-Valued Analysis / 5:
Recalls on Convex Functions of the Real Variable / 6:
Exercises
Convex Sets / A:
Generalities
Definition and First Examples / 1.1:
Convexity-PreservingOperationsonSets / 1.2:
ConvexCombinationsandConvexHulls / 1.3:
ClosedConvexSetsandHulls / 1.4:
ConvexSetsAttachedtoaConvexSet
TheRelativeInterior / 2.1:
TheAsymptoticCone / 2.2:
ExtremePoints / 2.3:
Exposed Faces / 2.4:
ProjectionontoClosedConvexSets
TheProjectionOperator / 3.1:
ProjectionontoaClosedConvexCone / 3.2:
Separation and Applications
SeparationBetweenConvexSets / 4.1:
First Consequences of the Separation Properties / 4.2:
Existence of Supporting Hyperplanes
Outer Description of Closed ConvexSets
Proof of Minkowski's Theorem
Bipolar of a ConvexCone
The Lemma of Minkowski-Farkas / 4.3:
ConicalApproximationsofConvexSets
ConvenientDefinitions of Tangent Cones / 5.1:
TheTangentandNormalConestoaConvexSet / 5.2:
SomePropertiesofTangentandNormalCones / 5.3:
Convex Functions / B:
Basic Definitions and Examples
The Definitions of a ConvexFunction
Special Convex Functions: Affinity and Closedness
Linear and Affine Functions
ClosedConvexFunctions
OuterConstructionofClosedConvexFunctions
FirstExamples
FunctionalOperationsPreservingConvexity
OperationsPreservingClosedness
Dilations and Perspectives of a Function
Infimal Convolution
Image of a Function Under a Linear Mapping
Convex Hull and Closed Convex Hull of a Function / 2.5:
Local and Global Behaviour of a Convex Function
Continuity Properties
Behaviour at Infinity
First- and Second-Order Differentiation
Differentiable ConvexFunctions
Nondifferentiable Convex Functions
Second-Order Differentiation
Sublinearity and Support Functions / C:
SublinearFunctions
Definitions and First Propertie
SomeExamples
TheConvexConeofAllClosedSublinearFunctions
The Support Function of a Nonempty Set
Definitions, Interpretations
BasicProperties
Examples
Correspondence Between Convex Sets and Sublinear Functions
The Fundamental Correspondence
Example: Norms and Their Duals, Polarity
Calculus with Support Functions / 3.3:
Example: Support Functions of Closed Convex Polyhedra / 3.4:
Subdifferentials of Finite Convex Functions / D:
The Subdifferential: Definitions and Interpretations
First Definition: Directional Derivatives
Second Definition: Minorizationby Affine Functions
GeometricConstructionsandInterpretations
Local Properties of the Subdifferential
First-OrderDevelopments
Minimality Conditions
Mean-ValueTheorems
Calculus Rules with Subdifferentials
Positive Combinations of Functions
Pre-Composition with an Affine Mapping
Post-Composition with an Increasing Convex Function of Several Variables
Supremum of Convex Functions / 4.4:
FurtherExamples / 4.5:
Largest Eigenvalue of a Symmetric Matrix
NestedOptimization
Best Approximation of a Continuous Function on a Compact Interval
The Subdifferential as a Multifunction
Monotonicity Properties of the Subdifferential / 6.1:
Continuity Properties of the Subdifferential / 6.2:
Subdifferentials and Limits of Subgradients / 6.3:
Conjugacy in Convex Analysis / E:
The Convex Conjugate of a Function
Interpretations
FirstProperties
-Elementary Calculus Rules
-The Biconjugate of a Function
-ConjugacyandCoercivity
Subdifferentials of Extended-Valued Functions
Calculus Rules on the Conjugacy Operation
Sum of Two Functions
Infima and Suprema
Post-Composition with an Increasing Convex Function
Various Examples
The Cramer Transformation
The Conjugate of Convex Partially Quadratic Functions
PolyhedralFunctions
Differentiability of a Conjugate Function
First-Order Differentiability
Lipschitz Continuity of the Gradient Mapping
Bibliographical Comments
The Founding Fathers of the Discipline
References
Index
Preface
Introduction: Notation, Elementary Results / 0:
Some Facts About Lower and Upper Bounds / 1:
2.

図書

図書
Alexander Koldobsky
出版情報: Providence, R.I. : American Mathematical Society, c2005  vi, 170 p. ; 26 cm
シリーズ名: Mathematical surveys and monographs ; v. 116
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3.

図書

図書
Ivan Singer
出版情報: New York : John Wiley & Sons, c1997  xix, 491 p. ; 25 cm
シリーズ名: Canadian Mathematical Society series of monographs and advanced texts
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目次情報: 続きを見る
Abstract Convexity of Elements of a Complete Lattice
Abstract Convexity of Subsets of a Set
Abstract Convexity of Functions on a Set
Abstract Quasi-Convexity of Functions on a Set
Dualities Between Complete Lattices
Dualities Between Families of Subsets
Dualities Between Sets of Functions
Conjugations. (-Dualities and <$$$>-Dualities
Abstract Subdifferentials
Notes and Remarks
References
Indexes
Abstract Convexity of Elements of a Complete Lattice
Abstract Convexity of Subsets of a Set
Abstract Convexity of Functions on a Set
4.

図書

図書
Heinz Bauer ; edited by Herbert Heyer, Niels Jacob, Ivan Netuka
出版情報: Berlin : Walter de Gruyter, 2003  xiv, 597 p. ; 25 cm
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5.

図書

図書
Steven R. Lay
出版情報: New York : Wiley, c1982  xvi, 244 p. ; 25 cm
シリーズ名: Pure and applied mathematics : a Wiley-Interscience series of texts, monographs, and tracts
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目次情報: 続きを見る
Fundamentals / 1:
Linear Algebra and Topology
Convex Sets / 2:
Hyperplanes
Hyperplanes and Linear Functionals / 3:
Separating Hyperplanes / 4:
Supporting Hyperplanes / 5:
Helly-Type Theorems
Helly's Theorem / 6:
Kirchberger's Theorem / 7:
Kirchberger-type Theorems
Separation by a Spherical Surface / 8:
Separation by a Cylinder / 9:
Separation by a Parallelotope / 10:
Special Topics in E[superscript 2]
Sets of Constant Width / 11:
Universal Covers / 12:
The Isoperimetric Problem / 13:
Families of Convex Sets
Parallel Bodies / 14:
The Blaschke Selection Theorem / 15:
The Existence of Extremal Sets / 16:
Characterizations of Convex Sets
Local Convexity / 17:
Local Support Properties / 18:
Nearest-Point Properties / 19:
Polytopes
The Faces of a Polytope / 20:
Special Types of Polytopes and Euler's Formula / 21:
Approximation by Polytopes / 22:
Duality
Polarity and Polytopes / 23:
Dual Cones / 24:
Optimization
Finite Matrix Games / 25:
Linear Programming / 26:
The Simplex Method / 27:
Convex Functions
Basic Properties / 28:
Support and Distance Functions / 29:
Continuity and Differentiability / 30:
Solutions, Hints, and References for Exercises
Bibliography
Index
Fundamentals / 1:
Linear Algebra and Topology
Convex Sets / 2:
6.

図書

図書
Jacques Bair, René Fourneau
出版情報: Berlin ; New York : Springer-Verlag, 1975-1980  2 v. ; 25 cm
シリーズ名: Lecture notes in mathematics ; 489, 802
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目次情報:
[1] Une introduction
2. Polyèdres et polytopes convexes
[1] Une introduction
2. Polyèdres et polytopes convexes
7.

図書

図書
Richard D. Bourgin
出版情報: Berlin ; New York : Springer-Verlag, 1983  xii, 474 p. ; 24 cm
シリーズ名: Lecture notes in mathematics ; 993
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8.

図書

図書
L. Asimow, A.J. Ellis
出版情報: London ; New York : Academic Press, 1980  x, 266 p. ; 24 cm
シリーズ名: L.M.S. monographs ; no. 16
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9.

図書

図書
[by] Erik M. Alfsen
出版情報: Berlin ; New York : Springer-Verlag, 1971  ix, 210 p. ; 24 cm
シリーズ名: Ergebnisse der Mathematik und ihrer Grenzgebiete ; Bd. 57
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10.

図書

図書
by Diethard Pallaschke and Stefan Rolewicz
出版情報: Dordrecht ; Boston : Kluwer Academic, c1997  xii, 582 p. ; 25 cm
シリーズ名: Mathematics and its applications ; v. 388
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