Preface......Page 5
Contents......Page 8
1.1 Subgradient Projection Method......Page 11
1.2 The Mirror Descent Method......Page 20
1.3 Gradient Algorithm with a Smooth Objective Function......Page 28
1.4 Examples......Page 32
2.1 Preliminaries......Page 35
2.2 A Convex Minimization Problem......Page 37
2.3 The Main Lemma......Page 41
2.4 Proof of Theorem 2.4......Page 43
2.5 Subgradient Algorithm on Unbounded Sets......Page 44
2.6 Proof of Theorem 2.8......Page 52
2.7 Proof of Theorem 2.9......Page 55
2.8 Proof of Theorem 2.12......Page 59
2.9 Zero-Sum Games with Two Players......Page 63
2.10 Proof of Proposition 2.13......Page 68
2.11 Zero-Sum Games on Bounded Sets......Page 76
2.12 Zero-Sum Games on Unbounded Sets......Page 81
2.13 Proof of Theorem 2.16......Page 87
2.14 An Example for Theorem 2.16......Page 90
3.1 Optimization on Bounded Sets......Page 92
3.2 The Main Lemma......Page 96
3.3 Proof of Theorem 3.1......Page 99
3.4 Optimization on Unbounded Sets......Page 100
3.5 Proof of Theorem 3.3......Page 105
3.6 Proof of Theorem 3.4......Page 109
3.7 Proof of Theorem 3.5......Page 114
3.8 Zero-Sum Games on Bounded Sets......Page 119
3.9 Zero-Sum Games on Unbounded Sets......Page 125
4.1 Optimization on Bounded Sets......Page 135
4.2 Auxiliary Results......Page 137
4.3 The Main Lemma......Page 138
4.4 Proof of Theorem 4.1......Page 144
4.5 Optimization on Unbounded Sets......Page 145
5.1 Preliminaries and the Main Result......Page 159
5.2 Auxiliary Results......Page 162
5.3 Proof of Theorem 5.1......Page 169
5.4 The First Extension of Theorem 5.1......Page 171
5.5 The Second Extension of Theorem 5.1......Page 175
6.1 Bochner Integrable Functions......Page 180
6.2 Convergence Analysis for Continuous Subgradient Method......Page 181
6.3 An Auxiliary Result......Page 184
6.4 Proof of Theorem 6.1......Page 185
6.5 Continuous Subgradient Method for Zero-Sum Games......Page 188
6.6 An Auxiliary Result......Page 191
6.7 Proof of Theorem 6.5......Page 195
6.8 Continuous Subgradient Projection Method......Page 200
6.9 An Auxiliary Result......Page 202
6.10 Proof of Theorem 6.7......Page 203
6.11 Continuous Subgradient Projection Methodon Unbounded Sets......Page 208
6.12 An Auxiliary Result......Page 209
6.13 The Convergence Result......Page 214
6.14 Subgradient Projection Algorithm for Zero-Sum Games......Page 222
6.15 An Auxiliary Result......Page 223
6.16 A Convergence Result for Games on Bounded Sets......Page 232
6.17 A Convergence Result for Games on Unbounded Sets......Page 241
7.1 Preliminaries......Page 249
7.2 The Algorithm and Main Results......Page 251
7.3 Auxiliary Results......Page 254
7.4 Proof of Theorem 7.4......Page 260
7.5 Proof of Theorem 7.5......Page 263
8.1 The Algorithm and the Main Result......Page 265
8.2 Auxiliary Results......Page 269
8.3 Proof of Theorem 8.1......Page 277
9.1 Preliminaries and the Main Result......Page 282
9.2 Auxiliary Results......Page 286
9.3 Proof of Theorem 9.2 and Examples......Page 290
10.1 Preliminaries......Page 292
10.2 An Auxiliary Result......Page 293
10.3 The Main Result......Page 295
11.1 Preliminaries......Page 299
11.2 The Subdifferential of Weakly Convex Functions......Page 301
11.3 An Auxiliary Result......Page 302
11.4 The First Main Result......Page 305
11.5 An Algorithm with Constant Step Sizes......Page 312
11.6 An Auxiliary Result......Page 313
11.7 The Second Main Result......Page 317
11.8 Convex Problems......Page 319
11.9 An Auxiliary Result......Page 320
11.10 Proof of Theorem 11.7......Page 322
12.1 Preliminaries and Main Results......Page 325
12.2 Auxiliary Results......Page 328
12.3 An Auxiliary Result with Assumption A2......Page 333
12.4 An Auxiliary Result with Assumption A3......Page 339
12.5 Proof of Theorem 12.1......Page 342
12.6 Proof of Theorem 12.2......Page 350
References......Page 359
Index......Page 363