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
L 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
R 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 Painleve-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 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-Vahied 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 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
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
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