Fundamentals of Convex Analysis

Foreword 0. Introduction: Notation, Elementary Results 1 Come Facts About Lower and Upper Bounds 2 The Set of Extended Real Numbers 3 Linear and Bilinear Algebra 4 Differentiation in a Euclidean Space 5 Set-Valued Analysis 6 Recalls on Convex Functions of the Real Variable Exercises A. Convex Sets 1. Generalities 1.1 Definitions and First Examples 1.2 Convexity-Preserving Operations on Sets 1.3 Convex Combinations and Convex Hulls 1.4 Closed Convex Sets and Hulls 2. Convex Sets Attached to a Convex Set 1.1 The Relative Interior 2.2 The Asymptotic Cone 2.3 Extreme Points 2.4 Exposed Faces 3. Projection onto Closed Convex Sets 3.1 The Projection Operator 3.2 Projection onto a Closed Convex Cone 4. Separation and Applications 4.1 Separation Between Convex Sets 4.2 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 4.3 The Lemma of Minkowski-Farkas 5. Conical Approximations of Convex Sets 5.1 Convenient Definitions of Tangent Cones 5.2 The Tangent and Normal Cones to a Convex Set 5.3 Some Properties of Tangent and Normal Cones Exercises B. Convex Functions 1. Basic Definitions and Examples 1.1 The Definitions of a Convex Function 1.2 Special Convex Functions: Affinity and Closedness - Linear and Affine Functions - Closed Convex Functions - Outer Construction of Closed Convex Functions 1.3 First Examples 2. Functional Operations Preserving Convexity 2.1 Operations Preserving Closedness 2.2 Dilations and Perspectives of a Function 2.3 Infimal Convolution 2.4 Image of a Functions Under a Linear Mapping 2.5 Convex Hull and Closed Convex Hull of a Function 3. Local and Global Behaviour of a Convex Function 3.1 Continuity Properties 3.2 Behaviour at Infinity 4. Fist- and Second-Order Differentiation 4.1 Differentiable Convex Functions 4.2 Nondifferentiable Convex Functions 4.3 Second-Order Differentiation Exercises C. Sublinearity and Support Functions 1. Sublinear Functions 1.1 Definitions and First Properties 1.2 Some Examples 1.3 The Convex Cone of All closed Sublinear Functions 2. The Support Function of a Nonempty Set 2.1 Definitions, Interpretations 2.2 Basic Properties 2.3 Examples 3. Correspondence Between Convex Sets and Sublinear Functions 3.1 The Fundamental Correspondence 3.2 Example: Norms and Their Duals, Polarity 3.3 Calculus with Support Functions 3.4 Example: Support Functions of Closed Convex Polyhedra Exercises D. Subdifferentials of Finite Convex Functions 1. The Subdifferential: Definitions and Interpretations 1.1 First Definition: Directional Derivatives 1.2 Second Definition: Minorization by Affine Functions 1.3 Geometric Constructions and Interpretations 2. Local Properties of the Subdifferential 2.1 First-Order Developments 2.2 Minimality conditions 2.3 Mean-Value Theorems 3. First Examples 4. Calculus Rules with Subdifferentials 4.1 Positive combinations of Functions 4.2 Pre-Compositions with an Affine Mapping 4.3 Post-composition with an Increasing Convex Functions of Several Variables 4.4 Supremum of Convex Functions 4.5 Image of a Functions Under a Linear Mapping 5. Further Examples 5.1 Largest Eigenvaule of a Symmetric Matrix 5.2 Nested Optimization 5.3 Best Approximation of a Continuous Function on a Compact Interval 6. The Subdifferential as a Multifunction 6.1 Monotonicity Properties of Subdifferential 6.2 Continuity Properties of the Subdifferential 6.3 Subdifferentials and Limits of Subgradients Exercises E. Conjugacy in Convex Analysis 1. The Convex Conjugate of a Function 1.1 Definition and First Examples 1.2 Interpretations 1.3 First Properties - Elementary Calculus Rules - The Biconjugate of a Function - Conjugacy and Coercivity 1.4 Subdifferntials of Extended-Valued Functions 2. Calculus Rules on the Conjugacy Operation 2.1 Image of a Function Under a Linear Mapping 2.2 Pre-Composition with an Affine Mapping 2.3 Sum of Two Functions 2.4 Infima and Suprema 2.5 Post-