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CONTENTS
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 87
D. Coercivity Properties 91
E∗. Cones and Orderings 96
F∗. Cosmic Convexity 97
G∗. Positive Hulls 100
Commentary 105
Chapter 4. Set Convergence 108
A. Inner and Outer Limits 109
B. Painlev′e-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
v
C. Local Boundedness 157
D. Total Continuity 164
E. Pointwise and Graphical Convergence 166
F. Equicontinuity of Sequences 173
G. Continuous 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 270
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 Subdifferential 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
vi
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 and Consequences 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 499
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 558
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
vii
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