of Convex Analysis Fundamentals Jean-Baptiste Hiriart-Urruty Claude Lemarechal Springer With 66 Figures

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1 2008 AGI-Information Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network. Jean-Baptiste Hiriart-Urruty Claude Lemarechal Fundamentals of Convex Analysis With 66 Figures Springer

2 Contents Preface... V 0. Introduction: Notation. Elementary Results Some Facts About Lower and Upper Bounds The Set of Extended Real Numbers Linear and Bilinear Algebra Differentiation in a Euclidean Space Set-Valued Analysis Recalls on Convex Functions of the Real Variable Exercises A. Convex Sets Generalities Definition and First Examples Convexity-Preserving Operations on Sets Convex Combinations and Convex Hulls Closed Convex Sets and Hulls Convex Sets Attached to a Convex Set The Relative Interior The Asymptotic Cone Extreme Points Exposed Faces Projection onto Closed Convex Sets The Projection Operator Projection onto a Closed Convex Cone Separation and Applications Separation Between Convex Sets First Consequences of the Separation Properties Existence of Supporting Hyperplanes... - Outer Description of Closed Convex Sets... - Proof of Minkowski s Theorem Bipolar of a Convex Cone The Lemma of Minkowski-Farkas Conical Approximations of Convex Sets

3 VI11 Contents 5.1 Convenient Definitions of Tangent Cones The Tangent and Normal Cones to a Convex Set Some Properties of Tangent and Normal Cones Exercises B. Convex Functions Basic Definitions and Examples The Definitions of a Convex Function Special Convex Functions: Affinity and Closedness Linear and Affine Functions Closed Convex Functions Outer Construction of Closed Convex Functions First Examples Functional Operations Preserving Convexity Operations Preserving Closedness 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 Local and Global Behaviour of a Convex Function Continuity Properties Behaviour at Infinity First- and Second-Order Differentiation Differentiable Convex Functions Nondifferentiable Convex Functions Second-Order Differentiation Exercises C. Sublinearity and Support Functions Sublinear Functions Definitions and First Properties Some Examples The Convex Cone of All Closed Sublinear Functions The Support Function of a Nonempty Set Definitions, Interpretations Basic Properties Examples Correspondence Between Convex Sets and Sublinear Functions The Fundamental Correspondence Example: Norms and Their Duals, Polarity Calculus with Support Functions Example: Support Functions of Closed Convex Polyhedra Exercises

4 Contents IX D. Subdifferentials of Finite Convex Functions The Subdifferential: Definitions and Interpretations First Definition: Directional Derivatives I. 2 Second Definition: Minorization by Affine Functions Geometric Constructions and Interpretations Local Properties of the Subdifferential First-Order Developments Minimality Conditions Mean-Value Theorems First Examples 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 Image of a Function Under a Linear Mapping Further Examples Largest Eigenvalue of a Symmetric Matrix Nested Optimization Best Approximation of a Continuous Function on a Compact Interval The Subdifferential as a Multifunction Monotonicity Properties of the Subdifferential Continuity Properties of the Subdifferential Subdifferentials and Limits of Subgradients Exercises E. Conjugacy in Convex Analysis The Convex Conjugate of a Function Definition and First Examples Interpretations First Properties Elementary Calculus Rules The Biconjugate of a Function Conjugacy and Coercivity Subdifferentials of Extended-Valued Functions Calculus Rules on the Conjugacy Operation Image of a Function Under a Linear Mapping Pre-Composition with an Affine Mapping Sum of Two Functions Infima and Suprema Post-Composition with an Increasing Convex Function Various Examples The Cramer Transformation

5 X Contents 3.2 The Conjugate of Convex Partially Quadratic Functions Polyhedral Functions Differentiability of a Conjugate Function First-Order Differentiability Lipschitz Continuity of the Gradient Mapping Exercises Bibliographical Comments The Founding Fathers of the Discipline References Index