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Record Nr. |
UNISA996472037403316 |
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Autore |
Mordukhovich Boris S. |
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Titolo |
Convex analysis and beyond . Volume I : basic theory / / Boris S. Mordukhovich and Nguyen Mau Nam |
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Pubbl/distr/stampa |
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Cham, Switzerland : , : Springer International Publishing, , [2022] |
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©2022 |
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ISBN |
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Descrizione fisica |
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1 online resource (597 pages) |
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Collana |
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Springer Series in Operations Research and Financial Engineering |
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Disciplina |
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Soggetti |
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Convex geometry |
Convex functions |
Geometria convexa |
Funcions convexes |
Llibres electrònics |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Nota di contenuto |
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Intro -- Preface -- Contents -- 1 FUNDAMENTALS -- 1.1 Topological Spaces -- 1.1.1 Definitions and Examples -- 1.1.2 Topological Interior and Closure of Sets -- 1.1.3 Continuity of Mappings -- 1.1.4 Bases for Topologies -- 1.1.5 Topologies Generated by Families of Mappings -- 1.1.6 Product Topology and Quotient Topology -- 1.1.7 Subspace Topology -- 1.1.8 Separation Axioms -- 1.1.9 Compactness -- 1.1.10 Connectedness and Disconnectedness -- 1.1.11 Net Convergence in Topological Spaces -- 1.2 Topological Vector Spaces -- 1.2.1 Basic Concepts in Topological Vector Spaces -- 1.2.2 Weak Topology and Weak* Topology -- 1.2.3 Quotient Spaces -- 1.3 Some Fundamental Theorems of Functional Analysis -- 1.3.1 Hahn-Banach Extension Theorem -- 1.3.2 Baire Category Theorem -- 1.3.3 Open Mapping Theorem -- 1.3.4 Closed Graph Theorem and Uniform Boundedness Principle -- 1.4 Exercises for Chapter 1 -- 1.5 Commentaries to Chapter 1 -- 2 BASIC THEORY OF CONVEXITY -- 2.1 Convexity of Sets -- 2.1.1 Basic Definitions and Elementary Properties -- 2.1.2 Operations on Convex Sets and Convex Hulls -- 2.2 Cores, Minkowski Functions, and Seminorms -- 2.2.1 Algebraic Interior and Linear |
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Closure -- 2.2.2 Minkowski Gauges -- 2.2.3 Seminorms and Locally Convex Topologies -- 2.3 Convex Separation Theorems -- 2.3.1 Convex Separation in Vector Spaces -- 2.3.2 Convex Separation in Topological Vector Spaces -- 2.3.3 Convex Separation in Finite Dimensions -- 2.3.4 Extreme Points of Convex Sets -- 2.4 Convexity of Functions -- 2.4.1 Descriptions and Properties of Convex Functions -- 2.4.2 Convexity under Differentiability -- 2.4.3 Operations Preserving Convexity of Functions -- 2.4.4 Continuity of Convex Functions -- 2.4.5 Lower Semicontinuity and Convexity -- 2.5 Extended Relative Interiors in Infinite Dimensions -- 2.5.1 Intrinsic Relative and Quasi-Relative Interiors. |
2.5.2 Convex Separation via Extended Relative Interiors -- 2.5.3 Extended Relative Interiors of Graphs and Epigraphs -- 2.6 Exercises for Chapter 2 -- 2.7 Commentaries to Chapter 2 -- 3 CONVEX GENERALIZED DIFFERENTIATION -- 3.1 The Normal Cone and Set Extremality -- 3.1.1 Basic Definition and Normal Cone Properties -- 3.1.2 Set Extremality and Convex Extremal Principle -- 3.1.3 Normal Cone Intersection Rule in Topological Vector Spaces -- 3.1.4 Normal Cone Intersection Rule in Finite Dimensions -- 3.2 Coderivatives of Convex-Graph Mappings -- 3.2.1 Coderivative Definition and Elementary Properties -- 3.2.2 Coderivative Calculus in Topological Vector Spaces -- 3.2.3 Coderivative Calculus in Finite Dimensions -- 3.3 Subgradients of Convex Functions -- 3.3.1 Basic Definitions and Examples -- 3.3.2 Subdifferential Sum Rules -- 3.3.3 Subdifferential Chain Rules -- 3.3.4 Subdifferentiation of Maximum Functions -- 3.3.5 Distance Functions and Their Subgradients -- 3.4 Generalized Differentiation under Polyhedrality -- 3.4.1 Polyhedral Convex Separation -- 3.4.2 Polyhedral Normal Cone Intersection Rule -- 3.4.3 Polyhedral Calculus for Coderivatives and Subdifferentials -- 3.5 Exercises for Chapter 3 -- 3.6 Commentaries to Chapter 3 -- 4 ENHANCED CALCULUS AND FENCHEL DUALITY -- 4.1 Fenchel Conjugates -- 4.1.1 Definitions, Examples, and Basic Properties -- 4.1.2 Support Functions -- 4.1.3 Conjugate Calculus -- 4.2 Enhanced Calculus in Banach Spaces -- 4.2.1 Support Functions of Set Intersections -- 4.2.2 Refined Calculus Rules -- 4.3 Directional Derivatives -- 4.3.1 Definitions and Elementary Properties -- 4.3.2 Relationships with Subgradients -- 4.4 Subgradients of Supremum Functions -- 4.4.1 Supremum of Convex Functions -- 4.4.2 Subdifferential Formula for Supremum Functions -- 4.5 Subgradients and Conjugates of Marginal Functions. |
4.5.1 Computing Subgradients and Another Chain Rule -- 4.5.2 Conjugate Calculations for Marginal Functions -- 4.6 Fenchel Duality -- 4.6.1 Fenchel Duality for Convex Composite Problems -- 4.6.2 Duality Theorems via Generalized Relative Interiors -- 4.7 Exercises for Chapter 4 -- 4.8 Commentaries to Chapter 4 -- 5 VARIATIONAL TECHNIQUES AND FURTHER SUBGRADIENT STUDY -- 5.1 Variational Principles and Convex Geometry -- 5.1.1 Ekeland's Variational Principle and Related Results -- 5.1.2 Convex Extremal Principles in Banach Spaces -- 5.1.3 Density of ε-Subgradients and Some Consequences -- 5.2 Calculus Rules for ε-Subgradients -- 5.2.1 Exact Sum and Chain Rules for ε-Subgradients -- 5.2.2 Asymptotic ε-Subdifferential Calculus -- 5.3 Mean Value Theorems for Convex Functions -- 5.3.1 Mean Value Theorem for Continuous Functions -- 5.3.2 Approximate Mean Value Theorem -- 5.4 Maximal Monotonicity of Subgradient Mappings -- 5.5 Subdifferential Characterizations of Differentiability -- 5.5.1 Gâteaux and Fréchet Differentiability -- 5.5.2 Characterizations of Gâteaux Differentiability -- 5.5.3 Characterizations of Fréchet Differentiability -- 5.6 Generic Differentiability of Convex Functions -- |
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5.6.1 Generic Gâteaux Differentiability -- 5.6.2 Generic Fréchet Differentiability -- 5.7 Spectral and Singular Functions in Convex Analysis -- 5.7.1 Von Neumann Trace Inequality -- 5.7.2 Spectral and Symmetric Functions -- 5.7.3 Singular Functions and Their Subgradients -- 5.8 Exercises for Chapter 5 -- 5.9 Commentaries to Chapter 5 -- 6 MISCELLANEOUS TOPICS ON CONVEXITY -- 6.1 Strong Convexity and Strong Smoothness -- 6.1.1 Basic Definitions and Relationships -- 6.1.2 Strong Convexity/Strong Smoothness via Derivatives -- 6.2 Derivatives of Conjugates and Nesterov's Smoothing -- 6.2.1 Differentiability of Conjugate Compositions -- 6.2.2 Nesterov's Smoothing Techniques. |
6.3 Convex Sets and Functions at Infinity -- 6.3.1 Horizon Cones and Unboundedness -- 6.3.2 Perspective and Horizon Functions -- 6.4 Signed Distance Functions -- 6.4.1 Basic Definition and Elementary Properties -- 6.4.2 Lipschitz Continuity and Convexity -- 6.5 Minimal Time Functions -- 6.5.1 Minimal Time Functions with Constant Dynamics -- 6.5.2 Subgradients of Minimal Time Functions -- 6.5.3 Signed Minimal Time Functions -- 6.6 Convex Geometry in Finite Dimensions -- 6.6.1 Carathéodory Theorem on Convex Hulls -- 6.6.2 Geometric Version of Farkas Lemma -- 6.6.3 Radon and Helly Theorems on Set Intersections -- 6.7 Approximations of Sets and Geometric Duality -- 6.7.1 Full Duality between Tangent and Normal Cones -- 6.7.2 Tangents and Normals for Polyhedral Sets -- 6.8 Exercises for Chapter 6 -- 6.9 Commentaries to Chapter 6 -- 7 CONVEXIFIED LIPSCHITZIAN ANALYSIS -- 7.1 Generalized Directional Derivatives -- 7.1.1 Definitions and Relationships -- 7.1.2 Properties of Extended Directional Derivatives -- 7.2 Generalized Derivative and Subderivative Calculus -- 7.2.1 Calculus Rules for Subderivatives -- 7.2.2 Calculus of Generalized Directional Derivatives -- 7.3 Directionally Generated Subdifferentials -- 7.3.1 Basic Definitions and Some Properties -- 7.3.2 Calculus Rules for Generalized Gradients -- 7.3.3 Calculus of Contingent Subgradients -- 7.4 Mean Value Theorems and More Calculus -- 7.4.1 Mean Value Theorems for Lipschitzian Functions -- 7.4.2 Additional Calculus Rules for Generalized Gradients -- 7.5 Strict Differentiability and Generalized Gradients -- 7.5.1 Notions of Strict Differentiability -- 7.5.2 Single-Valuedness of Generalized Gradients -- 7.6 Generalized Gradients in Finite Dimensions -- 7.6.1 Rademacher Differentiability Theorem -- 7.6.2 Gradient Representation of Generalized Gradients -- 7.6.3 Generalized Gradients of Antiderivatives. |
7.7 Subgradient Analysis of Distance Functions -- 7.7.1 Regular and Limiting Subgradients of Lipschitzian Functions -- 7.7.2 Regular and Limiting Subgradients of Distance Functions -- 7.7.3 Subgradients of Convex Signed Distance Functions -- 7.8 Differences of Convex Functions -- 7.8.1 Continuous DC Functions -- 7.8.2 The Mixing Property of DC Functions -- 7.8.3 Locally DC Functions -- 7.8.4 Subgradients and Conjugates of DC Functions -- 7.9 Exercises for Chapter 7 -- 7.10 Commentaries to Chapter 7 -- Glossary of Notation and Acronyms -- Glossary of Notation and Acronyms -- List of Figures -- References -- -- Subject Index -- Index. |
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