Convex Analysis and Minimization Algorithms I

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1 Jean-Baptiste Hiriart-Urruty Claude Lemarechal Convex Analysis and Minimization Algorithms I Fundamentals With 113 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

2 Table of Contents Part I Introduction XV I. Convex Functions of One Real Variable 1 1 Basic Definitions and Examples First Definitions of a Convex Function Inequalities with More Than Two Points Modern Definition of Convexity 8 2 First Properties Stability Under Functional Operations Limits of Convex Functions Behaviour at Infinity 14 3 Continuity Properties Continuity on the Interior of the Domain Lower Semi-Continuity: Closed Convex Functions Properties of Closed Convex Functions 19 4 First-Order Differentiation One-Sided Differentiability of Convex Functions Basic Properties of Subderivatives Calculus Rules 27 5 Second-Order Differentiation The Second Derivative ofa Convex Function One-Sided Second Derivatives How to Recognize a Convex Function 33 6 First Steps into the Theory of Conjugate Functions Basic Properties of the Conjugate Differentiation of the Conjugate Calculus Rules with Conjugacy 43 II. Introduction to Optimization Algorithms 47 1 Generalities The Problem General Structure of Optimization Schemes General Structure of Optimization Algorithms 52 2 Defining the Direction 54

3 VI Table of Contents Parti 2.1 Descent and Steepest-Descent Directions First-Order Methods Newtonian Methods Conjugate-Gradient Methods 65 3 Line-Searches General Structure of a Line-Search Designing the Test (0), (R), (L) The Wolfe Line-Search Updating the Trial Stepsize 81 III. Convex Sets 87 1 Generalities Definition and First Examples Convexity-Preserving Operations on Sets Convex Combinations and Convex Hulls Closed Convex Sets and Hulls 99 2 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 The Lemma of Minkowski-Farkas Conical Approximations of Convex Sets Convenient Definitions of Tangent Cones The Tangent and Normal Cones to a Convex Set Some Properties of Tangent and Normal Cones 139 IV. Convex Functions of Several Variables Basic Definitions and Examples The Definitions ofa Convex Function Special Convex Functions: Affinity and Closedness First Examples Functional Operations Preserving Convexity Operations Preserving Closedness Dilations and Perspectives ofa Function Infimal Convolution Image ofa Function Undera Linear Mapping Convex Hüll and Closed Convex Hüll of a Function 169

4 Table of Contents Part I VII 3 Local and Global Behaviour ofa Convex Function Continuity Properties Behaviour at Infinity First- and Second-Order Differentiation Differentiable Convex Functions Nondifferentiable Convex Functions Second-Order Differentiation 190 V. 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 The Isomorphism Between Closed Convex Sets and Closed Sublinear Functions The Fundamental Correspondence Example: Norms and Their Duals, Polarity Calculus with Support Functions Example: Support Functions of Closed Convex Polyhedra VI. Subdifferentials offinite Convex Functions The Subdifferential: Definitions and Interpretations First Definition: Directional Derivatives Second Definition: Minorization by Affine Functions Geometrie Constructions and Interpretations A Constructive Approach to the Existence of a Subgradient 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 LargestEigenvalueofa Symmetrie Matrix 275

5 VIII Table of Contents Part I 5.2 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 Gradients 284 VII. Constrained Convex Minimization Problems: Minimality Conditions, Elements of Duality Theory Abstract Minimality Conditions A Geometrie Characterization Conceptual Exact Penalty Minimality Conditions Involving Constraints Explicitly Expressing the Normal and Tangent Cones in Terms of the Constraint-Functions Constraint Qualification Conditions The Strang Slater Assumption Tackling the Minimization Problem with Its Data Directly Properties and Interpretations of the Multipliers Multipliers as a Means to Eliminate Constraints: the Lagrange Function Multipliers and Exact Penalty Multipliers as Sensitivity Parameters with Respect to Perrurbations Minimality Conditions and Saddle-Points Saddle-Points: Definitions and First Properties Mini-Maximization Problems An Existence Result Saddle-Points of Lagrange Functions A First Step into Duality Theory 338 VIII. Descent Theory for Convex Minimization: The Case of Complete Information Descent Directions and Steepest-Descent Schemes Basic Definitions Solving the Direction-Finding Problem Some Particular Cases Conclusion Illustration. The Finite Minimax Problem The Steepest-Descent Method for Finite Minimax Problems Non-Convergence of the Steepest-Descent Method Connection with Nonlinear Programming 366

6 Table of Contents Part I IX 3 The Practical Value of Descent Schemes Large Minimax Problems Infinite Minimax Problems Smooth but Stiff Functions The Steepest-Descent Trajectory Conclusion 383 Appendix: Notations Some Facts About Optimization The Set ofextended Real Numbers Linear and Bilinear Algebra Differentiation in a Euclidean Space Set-Valued Analysis A Bird's Eye View of Measure Theory and Integration 399 Bibliographical Comments 401 References 407 Index 415

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