Abstract Convex Analysis (Wiley-Interscience and Canadian Mathematics Series of Monographs and Texts)

Front Cover......Page 1
Title Page......Page 4
Copyright Information......Page 5
Contents......Page 6
Foreword......Page 12
Preface......Page 14
0.1 Abstract Convexity of Sets......Page 24
0.1a Inner Approaches......Page 25
0.1b Intersectional and Separational Approaches......Page 27
0.1c Approaches via Convexity Systems and Hull Operators......Page 29
0.2 Abstract Convexity of Functions......Page 32
0.3 Abstract Convexity of Elements of Complete Lattices......Page 35
0.5 Dualities......Page 37
0.6 Abstract Conjugations......Page 39
0.7 Abstract Subdifferentials......Page 42
0.8a Applications to Abstract Lagrangian Duality......Page 43
0.8b Applications to Abstract Surrogate Duality......Page 51
1.1 The Main (Supremal) Approach: M-Convexity of Elements of a Complete Lattice E, Where M \\subset E......Page 57
1.2 Infimal and Supremal Generators and M-Convexity......Page 63
1.3 An Equivalent Approach: Convexity Systems......Page 67
1.4 Another Equivalent Approach: Convexity with Respect to a Hull Operator......Page 70
2.1 M-Convexity of Subsets of a Set X, Where M \\subset 2^{X}......Page 73
2.2a Convex Subsets of a Linear Space X......Page 79
2.2b Closed Convex Subsets of a Locally Convex Space X......Page 81
2.2c Evenly Convex Subsets of a Locally Convex Space X......Page 83
2.2d Closed Affine Subsets of a Locally Convex Space X......Page 84
2.2e Evenly Coaffine Subsets of a Locally Convex Space X......Page 85
2.2f Spherically Convex Subsets of a Metric Space X......Page 86
2.2h Order Ideals and Order Convex Subsets of a Poset X......Page 87
2.2i Parametrizations of Families M \\subet 2^{X}, Where M Is a Set......Page 93
2.3 An Equivalent Approach, via Separation by Functions: W-Convexity of Subsets of a Set X, Where W \\subset cl(R)^{X}......Page 95
2.5 Other Concepts of Convexity of Subsets of a Set X, with Respect to a Set of Functions W \\subset cl(R)^{X}......Page 106
2.6 (W,φ)-Convexity of Subsets of a Set X, Where W Is a Set and φ : X×W→ cl(R) Is a Coupling Function......Page 108
3.1 W-Convexity of Functions on a Set X, Where W \\subset cl(R)^{X}......Page 115
3.2a C(X*+R), Where X Is a Locally Convex Space......Page 127
3.2b C(X*), Where X Is a Locally Convex Space......Page 131
3.2c The Case Where X = {0,1}^n and W \\subset (R^n)*|_X......Page 134
3.2d The Case Where X = {0,1}^n and W = (R^n)*|_X + R......Page 137
3.2e α-Holder Continuous Functions with Constant N, Where 0<α≤1 and 03.2f Suprema of α-Holder Continuous Functions, Where 0<α≤1......Page 144
3.2g The Case Where α > 1......Page 148
3.3 (W,φ)-Convexity of Functions on a Set X, Where W Is a Set and φ : X×W → cl(R) Is a Coupling Function......Page 150
4.1 M-Quasi-Convexity of Functions on a Set X, Where M \\subset 2^{X}......Page 152
4.2a Quasi-Convex Functions on a Linear Space X......Page 163
4.2b Lower Semicontinuous Quasi-Convex Functions on a Locally Convex Space X......Page 165
4.2c Evenly Quasi-Convex Functions on a Locally Convex Space X......Page 166
4.2d Evenly Quasi-Coaffine Functions on a Locally Convex Space X......Page 167
4.2f Nondecreasing Functions on a Poset X......Page 168
4.3 An Equivalent Approach: W-Quasi-Convexity of Functions on a Set X, Where W \\subset cl(R)^{X}......Page 169
4.4 Relations Between W-Convexity and W-Quasi-Convexity of Functions on a Set X, Where W \\subset cl(R)^{X}......Page 174
4.5a Lower Semicontinuous Quasi-Convex Functions Revisited......Page 177
4.5b Evenly Quasi-Convex Functions Revisited......Page 181
4.5c Evenly Quasi-Coaffine Functions Revisited......Page 184
4.6 (W,φ)-Quasi-Convexity of Functions on a Set X, Where W Is a Set and φ : X×W → cl(R) Is a Coupling Function......Page 185
4.7 Other Equivalent Approaches: Quasi-Convexity of Functions on a Set X, with Respect to Convexity Systems B \\subset 2^{X} and Hull Operators u : 2^{X} → 2^{X}......Page 188
4.8 Some Characterizations of Quasi-Convex Hull Operators among Hull Operators on cl(R)^{X}......Page 189
5.1 Dualities and Infimal Generators......Page 195
5.2 Duals of Dualities......Page 199
5.3 Relations Between Dualities and M-Convex Hulls......Page 205
5.4 Partial Order and Lattice Operations for Dualities......Page 209
6.1 Dualities Δ : 2^{X} → 2^{W}, Where X and W Are Two Sets......Page 213
6.2 Some Particular Cases......Page 223
6.2b Some Dualities Obtained from the Minkowski-Type Dualities Δ_M, by Parametrizing the Family M......Page 224
6.3 Representations of Dualities Δ : 2^{X} → 2^{W} with the Aid of Subsets Ω of X×W and Coupling Functions φ : X×W → cl(R)......Page 231
6.4a Representations with the Aid of Subsets Ω of X×W......Page 239
6.4b Representations with the Aid of Coupling Functions φ : X×W → cl(R)......Page 240
7.1 Dualities Δ : cl(R)^{X} → cl(R)^{W}, Where X and W Are Two Sets......Page 242
7.2 Representations of Dualities Δ : A^{X} → F, Where X Is a Set and (A,≤) \\subset (cl(R),≤) and F Are Complete Lattices......Page 247
7.3 Dualities Δ : A^{X} → B^{W}, Where X Is a Set and (A,≤),{B,≤) \\subset (cl(R),≤) Are Complete Lattices......Page 253
7.4a The Case Where A = {0,+∞}......Page 260
7.4b The Case Where A = B = [0,1]......Page 262
7.5 Strict Dualities Δ : A^{X} → B^{W}......Page 263
7.6 Dualitylike Mappings Δ : A^{X} → B^{W}......Page 264
8.1 Conjugations c : cl(R)^{X} → cl(R)^{W}, Where X and W Are Two Sets......Page 265
8.2 Representations of Conjugations c : cl(R)^{X} → cl(R)^{W} with the Aid of Coupling Functions φ : X×W → cl(R)......Page 269
8.3 Biconjugates and Abstract Convex Hulls......Page 279
8.4a The Case Where X = {0,1}^n, W = (R^n)*|_X and φ = φ_nat......Page 284
8.4b The Case Where X Is a Metric Space, W = X, and φ = φ_α,N......Page 286
8.4c The Case Where X Is a Metric Space, W = X×(R_+ \\ {0}), and φ = φ_α......Page 289
8.5 The Conjugate of f + -h, Where f,h \\in cl(R)^{X}......Page 290
8.6 Conjugations of Type Lau......Page 297
8.7a Conjugations of Type Lau Associated to a Family M of Subsets of a Set X......Page 307
8.7b Quasi-Conjugation......Page 310
8.7c Semiconjugation......Page 313
8.7d Pseudoconjugation......Page 314
8.7e Some Extensions of the Preceding Conjugations......Page 315
8.8 Relations Between Conjugations c : cl(R)^{X} → cl(R)^{W} and Dualities Δ : 2^{X} → 2^{W}, Where X and W Are Two Sets......Page 319
8.9b Conjugations of Type Lau Associated to Parametrized Minkowski-Type Dualities......Page 327
8.10 The Conjugate of Type Lau of max{f,-h}, Where f,-h \\in cl(R)^{X}......Page 331
8.11 Conjugate Functions and Level Sets......Page 341
9.1 The Binary Operations \\perp and T......Page 358
9.2 \\vee-Dualities......Page 361
9.3 \\perp-Dualities......Page 365
9.4 The Duals of \\vee-Dualities......Page 370
9.5 The Duals of \\perp-Dualities......Page 374
9.6 Characterizations of Conjugations of Type Lau with the Aid of \\vee-Dualities and \\perp-Dualities......Page 378
10.1 Subdifferentials with Respect to a Duality Δ : cl(R)^{X} → cl(R)^{W}, Where X and W Are Two Sets......Page 382
10.2 Subdifferentials with Respect to a Conjugation c : cl(R)^{X} → cl(R)^{W}, Where X and W Are Two Sets......Page 387
10.3a The Case Where X = {0,1}^n, W = (R^n)*|_X, and φ = φ_nat......Page 390
10.3b The Case Where X Is a Metric Space, W = X, and φ = φ_α,N......Page 391
10.3c The Case Where X Is a Metric Space, W = X×(R_+ \\ {0}), and φ = φ_α......Page 392
10.4 The Subdifferential of f + -h at x_0, Where f,h \\in cl(R)^{X} and x_0 \\in X......Page 393
10.5 Subdifferentials with Respect to Conjugations of Type Lau......Page 394
10.6a L(Δ)-Subdifferentials, for Minkowski-Type and Parametrized Minkowski-Type Set-Dualities A......Page 398
10.6b Subdifferentials with Respect to Quasi-Conjugations; Quasi-Subdifferentials......Page 399
10.6c Subdifferentials with Respect to Semiconjugations; Semisubdifferentials......Page 402
10.6d Subdifferentials with Respect to Pseudoconjugations; Pseudosubdifferentials......Page 406
10.7 Subdifferentials with Respect to v-Dualities and _L-Dualities......Page 407
Notes and Remarks......Page 410
References......Page 484
Notation Index......Page 498
Author Index......Page 504
Subject Index......Page 508