Abstract convex analysis:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Wiley
1997
|
| Schriftenreihe: | Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts
[20] |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIX, 491 S. |
| ISBN: | 0471160156 |
Internformat
MARC
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| 100 | 1 | |a Singer, Ivan |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Abstract convex analysis |c Ivan Singer |
| 264 | 1 | |a New York [u.a.] |b Wiley |c 1997 | |
| 300 | |a XIX, 491 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts |v [20] | |
| 490 | 0 | |a A Wiley-Interscience publication | |
| 650 | 7 | |a Convexidade (análise funcional) |2 larpcal | |
| 650 | 4 | |a Convex functions | |
| 650 | 4 | |a Convex sets | |
| 650 | 4 | |a Mathematical optimization | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
Foreword xi
Preface xiii
Introduction: From Convex Analysis to Abstract Convex Analysis 1
0.1 Abstract Convexity of Sets 1
0.1a Inner Approaches 2
0.1b Intersectional and Separational Approaches 4
0. lc Approaches via Convexity Systems and Hull Operators 6
0.2 Abstract Convexity of Functions 9
0.3 Abstract Convexity of Elements of Complete Lattices 12
0.4 Abstract Quasi Convexity of Functions 14
0.5 Dualities 14
0.6 Abstract Conjugations 16
0.7 Abstract Subdifferentials 19
0.8 Some Applications of Abstract Convex Analysis to Optimization
Theory 20
0.8a Applications to Abstract Lagrangian Duality 20
0.8b Applications to Abstract Surrogate Duality 28
Chapter One Abstract Convexity of Elements of a Complete Lattice 34
1.1 The Main (Supremal) Approach: .M Convexity of Elements of a
Complete Lattice E, Where M c E 34
1.2 Infimal and Supremal Generators and M Convexity 40
1.3 An Equivalent Approach: Convexity Systems 44
1.4 Another Equivalent Approach: Convexity with Respect to a Hull
Operator 47
Chapter Two Abstract Convexity of Subsets of a Set 50
2.1 X Convexity of Subsets of a Set X, Where M C. 2X 50
2.2 Some Particular Cases 56
2.2a Convex Subsets of a Linear Space X 56
2.2b Closed Convex Subsets of a Locally Convex Space X 58
V
vi Contents
2.2c Evenly Convex Subsets of a Locally Convex Space X 60
2.2d Closed Affine Subsets of a Locally Convex Space X 61
2.2e Evenly Coaffine Subsets of a Locally Convex Space X 62
2.2f Spherically Convex Subsets of a Metric Space X 63
2.2g Closed Subsets of a Topological Space X 64
2.2h Order Ideals and Order Convex Subsets of a Poset X 64
2.2i Parametrizations of Families M c 2X, Where M
Is a Set 70
2.3 An Equivalent Approach, via Separation by Functions:
W Convexity of Subsets of a Set X, Where W c ~RX 72
2.4 A Particular Case: Closed Convex Sets Revisited 83
2.5 Other Concepts of Convexity of Subsets of a Set X, with Respect
to a Set of Functions W c ~RX 83
2.6 (W, p) Convexity of Subsets of a Set X, Where W Is a Set and
p:XxW »/?Isa Coupling Function 85
Chapter Three Abstract Convexity of Functions on a Set 92
3.1 W Convexity of Functions on a Set X, Where W c fl* 92
3.2 Some Particular Cases 104
3.2a C(X* + R), Where X Is a Locally Convex Space 104
3.2b C{X*), Where X Is a Locally Convex Space 108
3.2c The Case Where X = {0, 1) and W c (R )* x 111
3.2d The Case Where X = {0, 1} and W = (R )* x + R 114
3.2e a Holder Continuous Functions with Constant N,
Where 0 a 1 andO TV +oo 119
3.2f Suprema of « H61der Continuous Functions, Where
0 a ^ 1 121
3.2g The Case Where a 1 125
3.3 (W, (^) Convexity of Functions on a Set X, Where W Is a Set and
p:XxW— /?Isa Coupling Function 127
Chapter Four Abstract Quasi Convexity of Functions on a Set 129
4.1 M Quasi Convexity of Functions on a Set X, Where M Q 2X 129
4.2 Some Particular Cases 140
4.2a Quasi Convex Functions on a Linear Space X 140
4.2b Lower Semicontinuous Quasi Convex Functions on a
Locally Convex Space X 142
4.2c Evenly Quasi Convex Functions on a Locally Convex
Space X 143
4.2d Evenly Quasi Coaffine Functions on a Locally Convex
Space X 144
4.2e Lower Semicontinuous Functions on a Topological
Space X 145
4.2f Nondecreasing Functions on a Poset X 145
4.3 An Equivalent Approach: W Quasi Convexity of Functions on a
Set X, Where W c R~x 146
Contents vii
4.4 Relations Between IV Convexity and W Quasi Convexity of
Functions on a Set X, Where W C ~RX 151
4.5 Some Particular Cases 154
4.5a Lower Semicontinuous Quasi Convex Functions
Revisited 154
4.5b Evenly Quasi Convex Functions Revisited 158
4.5c Evenly Quasi Coaffine Functions Revisited 161
4.6 (W, p) Quasi Convexity of Functions on a Set X, Where W Is a
Set and cp : X x W — R Is a Coupling Function 162
4.7 Other Equivalent Approaches: Quasi Convexity of Functions on
a Set X, with Respect to Convexity Systems BU* and Hull
Operators u : 2X ¦ 2X 165
4.8 Some Characterizations of Quasi Convex Hull Operators
among Hull Operators on Rx 166
Chapter Five Dualities Between Complete Lattices 172
5.1 Dualities and Infimal Generators 172
5.2 Duals of Dualities 176
5.3 Relations Between Dualities and M Convex Hulls 182
5.4 Partial Order and Lattice Operations for Dualities 186
Chapter Six Dualities Between Families of Subsets 190
6.1 Dualities A : 2X » 2W, Where X and W Are Two Sets 190
6.2 Some Particular Cases 200
6.2a Some Minkowski Type Dualities 201
6.2b Some Dualities Obtained from the Minkowski Type
Dualities A vf, by Parametrizing the Family M. 201
6.3 Representations of Dualities A : 2X 2W with the Aid of
Subsets Q of X x W and Coupling Functions p : X x W ¦ ^ 208
6.4 Some Particular Cases 216
6.4a Representations with the Aid of Subsets Q of X x W 216
6.4b Representations with the Aid of Coupling Functions
p : X x W ¦ ~R 217
Chapter Seven Dualities Between Sets of Functions 219
7.1 Dualities A : ~RX + ~RW, Where X and W Are Two Sets 219
7.2 Representations of Dualities A : Ax — ¦ F, Where X Is a Set
and (A, ) c (^, ^) and F Are Complete Lattices 224
7.3 Dualities A^AX — Bw, Where X Is a Set and (/I, sC),
(fi, ) C (R, ) Are Complete Lattices 230
7.4 Some Particular Cases 237
7.4a The Case Where A = {0, +oo) 237
7.4b The Case Where A = B = [0, 1] 239
7.5 Strict Dualities A : Ax + Bw 240
7.6 Duality like Mappings A : Ax »• Bw 241
viii Contents
Chapter Eight Conjugations 242
8.1 Conjugations c : ~RX Rw, Where X and W Are Two Sets 242
8.2 Representations of Conjugations c : Rx /?w with the Aid of
Coupling Functions p : X x W — ¦ /? 246
8.3 Biconjugates and Abstract Convex Hulls 256
8.4 Some Particular Cases 261
8.4a The Case Where X = {0, 1} , W = (/? )*|x and
P = Pnat 261
8.4b The Case Where X Is a Metric Space, IV = X,
and ^ = ipa jv 263
8.4c The Case Where X Is a Metric Space, W = X x
(/?+ {0}), and p = pa _ 266
8.5 The Conjugate of / j h, Where f,heRx 267
8.6 Conjugations of Type Lau 274
8.7 Some Particular Cases 284
8.7a Conjugations of Type Lau Associated to a Family M of
Subsets of a Set X 283
8.7b Quasi Conjugation 287
8.7c Semiconjugation 290
8.7d Pseudoconjugation 291
8.7e Some Extensions of the Preceding Conjugations 292
8.8 Relations Between Conjugations c : Rx ¦ Rw and Dualities
A : 2X » 2W, Where X and W Are Two Sets 296
8.9 Some Particular Cases 304
8.9a The Conjugation of Type Lau Associated to a
Minkowski Type Duality 304
8.9b Conjugations of Type Lau Associated to Parametrized
Minkowski Type Dualities _ 304
8.10 The Conjugate of Type Lau of max{/, h], Where /, h e R~x 308
8.11 Conjugate Functions and Level Sets 318
Chapter Nine v Dualities and L Dualities 335
9.1 The Binary Operations _L and T 335
9.2 v Dualities 338
9.3 _L Dualities 342
9.4 The Duals of v Dualities 347
9.5 The Duals of ± Dualities 351
9.6 Characterizations of Conjugations of Type Lau with the Aid of
v Dualities and ± Dualities 355
Chapter Ten Abstract Subdifferentials 359
10.1 Subdifferentials with Respect to a Duality A : R~x + ~RW,
Where X and W Are Two Sets _ _ 359
10.2 Subdifferentials with Respect to a Conjugation c : Rx Rw,
Where X and W Are Two Sets 364
Contents ix
10.3 Some Particular Cases 367
10.3a The Case Where X = {0, }n,W = (R )* x, and
p = (pmt 367
10.3b The Case Where X Is a Metric Space, W = X, and
P = Pa,N 368
10.3c The Case Where X Is a Metric Space, W = X x
(R+ {0}), and p pa _ 369
10.4 The Subdifferential of / i h at x0, Where f,h e ~RX and
x0 e X 370
10.5 Subdifferentials with Respect to Conjugations of Type Lau 371
10.6 Some Particular Cases 375
10.6a L(A) Subdifferentials, for Minkowski Type and
Parametrized Minkowski Type Set Dualities A 375
10.6b Subdifferentials with Respect to Quasi Conjugations;
Quasi Subdifferentials 376
10.6c Subdifferentials with Respect to Semiconjugations;
Semisubdifferentials 379
10.6d Subdifferentials with Respect to Pseudoconjugations;
Pseudosubdifferentials 383
10.7 Subdifferentials with Respect to v Dualities and _L Dualities 384
Notes and Remarks 387
References 461
Notation Index 475
Author Index 481
Subject Index 485
|
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| id | DE-604.BV011622131 |
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| isbn | 0471160156 |
| language | English |
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| physical | XIX, 491 S. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Wiley |
| record_format | marc |
| series | Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts |
| series2 | Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts A Wiley-Interscience publication |
| spelling | Singer, Ivan Verfasser aut Abstract convex analysis Ivan Singer New York [u.a.] Wiley 1997 XIX, 491 S. txt rdacontent n rdamedia nc rdacarrier Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts [20] A Wiley-Interscience publication Convexidade (análise funcional) larpcal Convex functions Convex sets Mathematical optimization Konvexe Analysis (DE-588)4138566-4 gnd rswk-swf Erweiterung (DE-588)4128080-5 gnd rswk-swf Konvexe Analysis (DE-588)4138566-4 s Erweiterung (DE-588)4128080-5 s DE-604 Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts [20] (DE-604)BV012067942 20 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007830510&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Singer, Ivan Abstract convex analysis Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts Convexidade (análise funcional) larpcal Convex functions Convex sets Mathematical optimization Konvexe Analysis (DE-588)4138566-4 gnd Erweiterung (DE-588)4128080-5 gnd |
| subject_GND | (DE-588)4138566-4 (DE-588)4128080-5 |
| title | Abstract convex analysis |
| title_auth | Abstract convex analysis |
| title_exact_search | Abstract convex analysis |
| title_full | Abstract convex analysis Ivan Singer |
| title_fullStr | Abstract convex analysis Ivan Singer |
| title_full_unstemmed | Abstract convex analysis Ivan Singer |
| title_short | Abstract convex analysis |
| title_sort | abstract convex analysis |
| topic | Convexidade (análise funcional) larpcal Convex functions Convex sets Mathematical optimization Konvexe Analysis (DE-588)4138566-4 gnd Erweiterung (DE-588)4128080-5 gnd |
| topic_facet | Convexidade (análise funcional) Convex functions Convex sets Mathematical optimization Konvexe Analysis Erweiterung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007830510&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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