Calculus Without Derivatives

Calculus Without Derivatives expounds the foundations and recent advances in nonsmooth analysis, a powerful compound of mathematical tools that obviates the usual smoothness assumptions. This textbook also provides significant tools and methods towards applications, in particular optimization problems. Whereas most books on this subject focus on a particular theory, this text takes a general approach including all main theories. In order to be self-contained, the book includes three chapters of preliminary material, each of which can be used as an independent course if needed. The first chapter deals with metric properties, variational principles, decrease principles, methods of error bounds, calmness and metric regularity. The second one presents the classical tools of differential calculus and includes a section about the calculus of variations. The third contains a clear exposition of convex analysis. Table of Contents Cover Calculus Without Derivatives Preface Contents Notation Chapter 1 Metric and Topological Tools 1.1 Convergences and Topologies 1.1.1 Sets and Orders 1.1.2 A Short Refresher About Topologies and Convergences 1.1.3 Weak Topologies 1.1.4 Semicontinuity of Functions and Existence Results 1.1.5 Baire Spaces and the Uniform Boundedness Theorem 1.2 Set-Valued Mappings 1.2.1 Generalities About Sets and Correspondences 1.2.2 Continuity Properties of Multimaps 1.3 Limits of Sets and Functions 1.3.1 Convergence of Sets 1.3.2 Supplement: Variational Convergences 1.4 Convexity and Separation Properties 1.4.1 Convex Sets and Convex Functions 1.4.2 Separation and Extension Theorems 1.5 Variational Principles 1.5.1 The Ekeland Variational Principle 1.5.2 Supplement: Some Consequences of the Ekeland Principle 1.5.3 Supplement: Fixed-Point Theorems via Variational Principles 1.5.4 Supplement: Metric Convexity 1.5.5 Supplement: Geometric Principles 1.5.6 Supplement: The Banach-Schauder Open Mapping Theorem 1.6 Decrease Principle, Error Bounds, and Metric Estimates 1.6.1 Decrease Principle and Error Bounds 1.6.2 Supplement: A Palais-Smale Condition 1.6.3 Penalization Methods 1.6.4 Robust and Stabilized Infima 1.6.5 Links Between Penalization and Robust Infima 1.6.6 Metric Regularity, Lipschitz Behavior, and Openness 1.6.7 Characterizations of the Pseudo-Lipschitz Property 1.6.8 Supplement: Convex-Valued Pseudo-LipschitzianMultimaps 1.6.9 Calmness and Metric Regularity Criteria 1.7 Well-Posedness and Variational Principles 1.7.1 Supplement: Stegall's Principle 1.8 Notes and Remarks Chapter 2 Elements of Differential Calculus 2.1 Derivatives of One-Variable Functions 2.1.1 Differentiation of One-Variable Functions 2.1.2 The Mean Value Theorem 2.2 Primitives and Integrals 2.3 Directional Differential Calculus 2.4 Fr�chet Differential Calculus 2.5 Inversion of Differentiable Maps 2.5.1 Newton's Method 2.5.2 The Inverse Mapping Theorem 2.5.3 The Implicit Function Theorem 2.5.4 The Legendre Transform 2.5.5 Geometric Applications 2.5.6 The Method of Characteristics 2.6 Applications to Optimization 2.6.1 Normal Cones, Tangent Cones, and Constraints 2.6.2 Calculus of Tangent and Normal Cones 2.6.3 Lagrange Multiplier Rule 2.7 Introduction to the Calculus of Variations 2.8 Notes and Remarks Chapter 3 Elements of Convex Analysis 3.1 Continuity Properties of Convex Functions 3.1.1 Supplement: Another Proof of the Robinson-UrsescuTheorem 3.2 Differentiability Properties of Convex Functions 3.2.1 Derivatives and Subdifferentials of Convex Functions 3.2.2 Differentiability of Convex Functions 3.3 Calculus Rules for Subdifferentials 3.3.1 Supplement: Subdifferentials of Marginal ConvexFunctions 3.4 The Legendre-Fenchel Transform and Its Uses 3.4.1 The Legendre-Fenchel Transform 3.4.2 The Interplay Between a Function and Its Conjugate 3.4.3 A Short Account of Convex Duality Theory 3.4.4 Duality and Subdifferentiability Results 3.5 General Convex Calculus Rules 3.5.1 Fuzzy Calculus Rules in Convex Analysis 3.5.2 Exact Rules in Convex Analysis 3.5.3 Mean Value Theorems 3.5.4 Application to Optimality Conditions 3.6 Smoothness of Norms 3.7 Favorable Classes of Spaces 3.8 Notes and Remarks Chapter 4 Elementary and Viscosity Subdifferentials 4.1 Elementary Subderivatives and Subdifferentials 4.1.1 Definitions and Characterizations 4.1.2 Some Elementary Properties 4.1.3 Relationships with Geometrical Notions 4.1.4 Coderivatives 4.1.5 Supplement: Incident and Proximal Notions 4.1.6 Supplement: Bornological Subdifferentials 4.2 Elementary Calculus Rules 4.2.1 Elementary Sum Rules 4.2.2 Elementary Composition Rules 4.2.3 Rules Involving Order 4.2.4 Elementary Rules for Marginal and PerformanceFunctions 4.3 Viscosity Subdifferentials 4.4 Approximate Calculus Rules 4.4.1 Approximate Minimization Rules 4.4.2 Approximate Calculus in Smooth Banach Spaces 4.4.3 Metric Estimates and Calculus Rules 4.4.4 Supplement: Weak Fuzzy Rules 4.4.5 Mean Value Theorems and Superdifferentials 4.5 Soft Functions 4.6 Calculus Rules in Asplund Spaces 4.6.1 A Characterization of Fr�chet Subdifferentiability 4.6.2 Separable Reduction 4.6.3 Application to Fuzzy Calculus 4.7 Applications 4.7.1 Subdifferentials of Value Functions o 4.7.1.1 Subdifferentiability of Marginal Functions o 4.7.1.2 Subdifferentiability of Performance Functions o 4.7.1.3 Differentiability Properties 4.7.2 Application to Regularization 4.7.3 Mathematical Programming Problems and Sensitivity 4.7.4 Openness and Metric Regularity Criteria 4.7.5 Stability of Dynamical Systems and Lyapunov Functions 4.8 Notes and Remarks Chapter 5 Circa-Subdifferentials, Clarke Subdifferentials 5.1 The Locally Lipschitzian Case 5.1.1 Definitions and First Properties 5.1.2 Calculus Rules in the Locally Lipschitzian Case 5.1.3 The Clarke Jacobian and the Clarke Subdifferentialin Finite Dimensions 5.2 Circa-Normal and Circa-Tangent Cones 5.3 Subdifferentials of Arbitrary Functions 5.3.1 Definitions and First Properties 5.3.2 Regularity 5.3.3 Calculus Rules 5.4 Limits of Tangent and Normal Cones 5.5 Moderate Subdifferentials 5.5.1 Moderate Tangent Cones 5.5.2 Moderate Subdifferentials 5.5.3 Calculus Rules for Moderate Subdifferentials 5.6 Notes and Remarks Chapter 6 Limiting Subdifferentials 6.1 Limiting Constructions with Firm Subdifferentials 6.1.1 Limiting Subdifferentials and Limiting Normals 6.1.2 Limiting Coderivatives 6.1.3 Some Elementary Properties 6.1.4 Calculus Rules Under Lipschitz Assumptions 6.2 Some Compactness Properties 6.3 Calculus Rules for Coderivatives and Normal Cones 6.3.1 Normal Cone to an Intersection 6.3.2 Coderivative to an Intersection of Multimaps 6.3.3 Normal Cone to a Direct Image 6.3.4 Normal Cone to an Inverse Image 6.3.5 Coderivatives of Compositions 6.3.6 Coderivatives of Sums 6.4 General Subdifferential Calculus 6.5 Error Bounds and Metric Estimates 6.5.1 Upper Limiting Subdifferentials and Conditioning 6.5.2 Application to Regularity and Openness 6.6 Limiting Directional Subdifferentials 6.6.1 Some Elementary Properties 6.6.2 Calculus Rules Under Lipschitz Assumptions 6.7 Notes and Remarks Chapter 7 Graded Subdifferentials, Ioffe Subdifferentials 7.1 The Lipschitzian Case 7.1.1 Some Uses of Separable Subspaces 7.1.2 The Graded Subdifferential and the Graded Normal Cone 7.1.3 Relationships with Other Subdifferentials 7.1.4 Elementary Properties in the Lipschitzian Case 7.2 Subdifferentials of Lower Semicontinuous Functions 7.3 Notes and Remarks References Index