Preface ........................................................ xv
1 Introduction ................................................. 1
1.1 Set-valued analysis ..................................... 1
1.2 Examples of point-to-set mappings ....................... 2
1.3 Description of the contents ............................. 3
2 Set Convergence and Point-to-Set Mappings .................... 5
2.1 Convergence of sets ..................................... 5
2.1.1 Nets and subnets ................................. 6
2.2 Nets of sets ............................................ 9
Exercises .................................................... 1
2.3 Point-to-set mappings .................................. 22
2.4 Operating with point-to-set mappings ................... 23
2.5 Semicontinuity of point-to-set mappings ................ 24
2.5.1 Weak and weak topologies ........................ 30
Exercises ................................................... 44
2.6 Semilimits of point-to-set mappings .................... 46
Exercises ................................................... 48
2.7 Generic continuity ..................................... 48
2.8 The closed graph theorem for point-to-set mappings ..... 52
Exercises ................................................... 55
2.9 Historical notes ....................................... 56
3 Convex Analysis and Fixed Point Theorems .................... 57
3.1 Lower-semicontinuous functions ......................... 57
Exercises ................................................... 64
3.2 Ekeland's variational principle ........................ 64
3.3 Caristi's fixed point theorem .......................... 66
3.4 Convex functions and conjugacy ......................... 68
3.5 The subdifferential of a convex function ............... 76
3.5.1 Subdifferential of a sum ........................ 78
3.6 Tangent and normal cones ............................... 79
Exercises ................................................... 86
3.7 Differentiation of point-to-set mappings ............... 87
Exercises ................................................... 94
3.8 Marginal functions ..................................... 94
3.9 Paracontinuous mappings ................................ 97
3.10 Ky Fan's inequality ................................... 100
3.11 Kakutani's fixed point theorem ........................ 103
3.12 Fixed point theorems with coercivity .................. 106
3.13 Duality for variational problems ...................... 109
3.13.1 Application to convex optimization ............. 111
3.13.2 Application to the Clarke-Ekeland least
action principle ............................... 112
3.13.3 Singer-Toland duality .......................... 114
3.13.4 Application to normal mappings ................. 116
3.13.5 Composition duality ............................ 117
Exercises .................................................. 118
3.14 Historical notes ...................................... 119
4 Maximal Monotone Operators ................................. 121
4.1 Definition and examples ............................... 121
4.2 Outer semicontinuity .................................. 124
4.2.1 Local boundedness .............................. 127
4.3 The extension theorem for monotone sets ............... 131
4.4 Domains and ranges in the reflexive case .............. 133
4.5 Domains and ranges without reflexivity ................ 141
4.6 Inner semicontinuity .................................. 145
4.7 Maximality of subdifferentials ........................ 150
4.8 Sum of maximal monotone operators ..................... 153
Exercises .................................................. 158
4.9 Historical notes ...................................... 159
5 Enlargements of Monotone Operators ......................... 161
5.1 Motivation ............................................ 161
5.2 The Te-enlargement .................................... 162
5.3 Theoretical properties of Te .......................... 164
5.3.1 Affine local boundedness ....................... 165
5.3.2 Transportation formula ......................... 168
5.3.3 The Brøndsted-Rockafellar property ............. 170
5.4 The family  (T) ....................................... 176
5.4.1 Linear skew-symmetric operators are
nonenlargeable ................................. 180
5.4.2 Continuity properties .......................... 182
5.4.3 Nonenlargeable operators are linear skew-
symmetric ...................................... 187
5.5 Algorithmic applications of Te ........................ 192
5.5.1 An extragradient algorithm for point-to-set
operators ...................................... 196
5.5.2 A bundle-like algorithm for point-to-set
operators ...................................... 200
5.5.3 Convergence analysis ........................... 207
5.6 Theoretical applications of Te ........................ 210
5.6.1 An alternative concept of sum of maximal
monotone operators ............................. 210
5.6.2 Properties of the extended sum ................. 212
5.6.3 Preservation of well-posedness using
enlargements ................................... 215
5.6.4 Well-posedness with respect to the family of
perturbations .................................. 216
Exercises .................................................. 217
5.7 Historical notes ...................................... 219
6 Recent Topics in Proximal Theory ........................... 221
6.1 The proximal point method ............................. 221
6.2 Existence of regularizing functions ................... 225
6.3 An inexact proximal point method in Banach spaces ..... 234
6.4 Convergence analysis of Algorithm IPPM ................ 236
6.5 Finite-dimensional augmented Lagrangians .............. 241
6.6 Augmented Lagrangians for Јp-constrained problems ..... 245
6.7 Convergence analysis of Algorithm IDAL ................ 251
6.8 Augmented Lagrangians for cone constrained problems ... 254
6.9 Nonmonotone proximal point methods .................... 257
6.10 Convergence analysis of IPPH1 and IPPH2 ............... 261
Exercises .................................................. 268
6.11 Historical notes ...................................... 268
Bibliography ............................................... 271
Notation ................................................... 287
Index ......................................................... 289
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