变分分析 英文PDF电子书下载

Chapter 1．Max and Min 1

A．Penalties and Constraints 2

B．Epigraphs and Semicontinuity 7

C．Attainment of a Minimum 11

D．Continuity,Closure and Growth 13

E．Extended Arithmetic 15

F．Parametric Dependence 16

G．Moreau Envelopes 20

H．Epi-Addition and Epi-Multiplication 23

I．Auxiliary Facts and Principles 28

Commentary 34

Chapter 2．Convexity 38

A．Convex Sets and Functions 38

B．Level Sets and Intersections 42

C．Derivative Tests 45

D．Convexity in Operations 49

E．Convex Hulls 53

F．Closures and Continuity 57

G．Separation 62

H．Relative Interiors 64

I．Piecewise Linear Functions 67

J．Other Examples 71

Commentary 74

Chapter 3．Cones and Cosmic Closure 77

A．Direction Points 77

B．Horizon Cones 80

C．Horizon Functions 86

D．Coercivity Properties 90

E．Cones and Orderings 95

F．Cosmic Convexity 97

G．Positive Hulls 99

Commentary 105

Chapter 4．Set Convergence 108

A．Inner and Outer Limits 109

B．Painlevé-Kuratowski Convergence 111

C．Pompeiu-Hausdorff Distance 117

D．Cones and Convex Sets 118

E．Compactness Properties 120

F．Horizon Limits 122

G．Continuity of Operations 125

H．Quantification of Convergence 131

I．Hyperspace Metrics 138

Commentary 144

Chapter 5．Set-Valued Mappings 148

A．Domains,Ranges and Inverses 149

B．Continuity and Semicontinuity 152

C．Local Boundedness 157

D．Total Continuity 164

E．Pointwise and Graphical Convergence 166

F．Equicontinuity of Sequences 173

G．Continuons and Uniform Convergence 175

H．Metric Descriptions of Convergence 181

I．Operations on Mappings 183

J．Generic Continuity and Selections 187

Commentary 192

Chapter 6．Variational Geometry 196

A．Tangent Cones 196

B．Normal Cones and Clarke Regularity 199

C．Smooth Manifolds and Convex Sets 202

D．Optimality and Lagrange Multipliers 205

E．Proximal Normals and Polarity 212

F．Tangent-Normal Relations 217

G．Recession Properties 222

H．Irregularity and Convexification 225

I．Other Formulas 227

Commentary 232

Chapter 7．Epigraphical Limits 238

A．Pointwise Convergence 239

B．Epi-Convergence 240

C．Continuous and Uniform Convergence 250

D．Generalized Differentiability 255

E．Convergence in Minimization 262

F．Epi-Continuity of Function-Valued Mappings 269

G．Continuity of Operations 275

H．Total Epi-Convergence 278

I．Epi-Distances 282

J．Solution Estimates 286

Commentary 292

Chapter 8．Subderivatives and Subgradients 298

A．Subderivatives of Functions 299

B．Subgradients of Functions 300

C．Convexity and Optimality 308

D．Regular Subderivatives 311

E．Support Functions and Subdifierential Duality 317

F．Calmness 322

G．Graphical Differentiation of Mappings 324

H．Proto-Differentiability and Graphical Regularity 329

I．Proximal Subgradients 333

J．Other Results 336

Commentary 343

Chapter 9．Lipschitzian Properties 349

A．Single-Valued Mappings 349

B．Estimates of the Lipschitz Modulus 354

C．Subdifferential Characterizations 358

D．Derivative Mappings and Their Norms 364

E．Lipschitzian Concepts for Set-Valued Mappings 368

F．Aubin Property and Mordukhovich Criterion 376

G．Metric Regularity and Openness 386

H．Semiderivatives and Strict Graphical Derivatives 390

I．Other Properties 399

J．Rademacher's Theorem and Consequences 403

K．Mollifiers and Extremals 408

Commentary 415

Chapter 10．Subdifferential Calculus 421

A．Optimality and Normals to Level Sets 421

B．Basic Chain Rule 426

C．Parametric Optimality 432

D．Rescaling 438

E．Piecewise Linear-Quadratic Functions 440

F．Amenable Sets and Functions 442

G．Semiderivatives and Subsmoothness 446

H．Coderivative Calculus 452

I．Extensions 458

Commentary 469

Chapter 11．Dualization 473

A．Legendre-Fenchel Transform 473

B．Special Cases of Conjugacy 476

C．The Role of Differentiability 480

D．Piecewise Linear-Quadratic Functions 484

E．Polar Sets and Gauges 490

F．Dual Operations 493

G．Duality in Convergence 500

H．Dual Problems of Optimization 502

I．Lagrangian Functions 508

J．Minimax Problems 514

K．Augmented Lagrangians and Nonconvex Duality 518

L．Generalized Conjugacy 525

Commentary 529

Chapter 12．Monotone Mappings 533

A．Monotonicity Tests and Maximality 533

B．Minty Parameterization 537

C．Connections with Convex Functions 542

D．Graphical Convergence 551

E．Domains and Ranges 553

F．Preservation of Maximality 556

G．Monotone Variational Inequalities 559

H．Strong Monotonicity and Strong Convexity 562

I．Continuity and Differentiability 567

Commentary 575

Chapter 13．Second-Order Theory 579

A．Second-Order Differentiability 579

B．Second Subderivatives 582

C．Calculus Rules 591

D．Convex Functions and Duality 603

E．Second-Order Optimality 606

F．Prox-Regularity 609

G．Subgradient Proto-Differentiability 618

H．Subgradient Coderivatives and Perturbation 622

I．Further Derivative Properties 625

J．Parabolic Subderivatives 633

Commentary 638

Chapter 14．Measurability 642

A．Measurable Mappings and Selections 643

B．Preservation of Measurability 651

C．Limit Operations 655

D．Normal Integrands 660

E．Operations on Integrands 669

F．Integral Functionals 675

Commentary 679

References 684

Index of Statements 710

Index of Notation 725

Index of Topics 726