Discrete convex analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Philadelphia
SIAM
2003
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| Series: | SIAM monographs on discrete mathematics and applications
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXII, 389 S. Diagramme |
| ISBN: | 9781611972559 0898715407 |
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MARC
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| adam_text | DISCRETE CONVEX ANALYSIS *O KAZUO MUROTA UNIVERSITY OF TOKYO; PRESTO,
JST TOKYO, JAPAN SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS
PHILADELPHIA CONTENTS LIST OF FIGURES XI NOTATION XIII PREFACE XXI 1
INTRODUCTION TO THE CENTRAL CONCEPTS 1 1.1 AIM AND HISTORY OF DISCRETE
CONVEX ANALYSIS 1 1.1.1 AIM 1 1.1.2 HISTORY 5 1.2 USEFUL PROPERTIES OF
CONVEX FUNCTIONS 9 1.3 SUBMODULAR FUNCTIONS AND BASE POLYHEDRA 15 1.3.1
SUBMODULAR FUNCTIONS 16 1.3.2 BASE POLYHEDRA 18 1.4 DISCRETE CONVEX
FUNCTIONS 21 1.4.1 L-CONVEX FUNCTIONS 21 1.4.2 M-CONVEX FUNCTIONS 25
1.4.3 CONJUGACY 29 1.4.4 DUALITY 32 1.4.5 CLASSES OF DISCRETE CONVEX
FUNCTIONS 36 BIBLIOGRAPHICAL NOTES 36 2 CONVEX FUNCTIONS WITH
COMBINATORIAL STRUCTURES 39 2.1 QUADRATIC FUNCTIONS 39 2.1.1 CONVEX
QUADRATIC FUNCTIONS 39 2.1.2 SYMMETRIC M-MATRICES . 41 2.1.3
COMBINATORIAL PROPERTY OF CONJUGATE FUNCTIONS . . 47 2.1.4 GENERAL
QUADRATIC L-/M-CONVEX FUNCTIONS 51 2.2 NONLINEAR NETWORKS 52 2.2.1
REAL-VALUED FLOWS . . 52 2.2.2 INTEGER-VALUED FLOWS 56 2.2.3 TECHNICAL
SUPPLEMENTS 58 2.3 SUBSTITUTES AND COMPLEMENTS IN NETWORK FLOWS 61 2.3.1
CONVEXITY AND SUBMODULARITY 61 VI CONTENTS 2.3.2 TECHNICAL SUPPLEMENTS
63 2.4 MATROIDS 68 2.4.1 FROM MATRICES TO MATROIDS 68 2.4.2 FROM
POLYNOMIAL MATRICES TO VALUATED MATROIDS . . 71 BIBLIOGRAPHICAL NOTES 74
3 CONVEX ANALYSIS, LINEAR PROGRAMMING, AND INTEGRALITY 77 3.1 CONVEX
ANALYSIS 77 3.2 LINEAR PROGRAMMING 86 3.3 INTEGRALITY FOR A PAIR OF
INTEGRAL POLYHEDRA 90 3.4 INTEGRALLY CONVEX FUNCTIONS 92 BIBLIOGRAPHICAL
NOTES 99 4 M-CONVEX SETS AND SUBMODULAR SET FUNCTIONS 101 4.1 DEFINITION
101 4.2 EXCHANGE AXIOMS 102 4.3 SUBMODULAR FUNCTIONS AND BASE POLYHEDRA
103 4.4 POLYHEDRAL DESCRIPTION OF M-CONVEX SETS 108 4.5 SUBMODULAR
FUNCTIONS AS DISCRETE CONVEX FUNCTIONS ILL 4.6 M-CONVEX SETS AS DISCRETE
CONVEX SETS 114 4.7 M-CONVEX SETS 116 4.8 M-CONVEX POLYHEDRA 118
BIBLIOGRAPHICAL NOTES 119 5 L-CONVEX SETS AND DISTANCE FUNCTIONS 121 5.1
DEFINITION 121 5.2 DISTANCE FUNCTIONS AND ASSOCIATED POLYHEDRA 122 5.3
POLYHEDRAL DESCRIPTION OF L-CONVEX SETS 123 5.4 L-CONVEX SETS AS
DISCRETE CONVEX SETS 125 5.5 L^-CONVEX SETS 128 5.6 L-CONVEX POLYHEDRA
131 BIBLIOGRAPHICAL NOTES 131 6 M-CONVEX FUNCTIONS 133 6.1 M-CONVEX
FUNCTIONS AND M^-CONVEX FUNCTIONS 133 6.2 LOCAL EXCHANGE AXIOM 135 6.3
EXAMPLES 138 6.4 BASIC OPERATIONS 142 6.5 SUPERMODULARITY 145 6.6
DESCENT DIRECTIONS 146 6.7 MINIMIZERS 148 6.8 GROSS SUBSTITUTES PROPERTY
152 6.9 PROXIMITY THEOREM 156 6.10 CONVEX EXTENSION 158 6.11 POLYHEDRAL
M-CONVEX FUNCTIONS 160 6.12 POSITIVELY HOMOGENEOUS M-CONVEX FUNCTIONS
164 CONTENTS VII 6.13 DIRECTIONAL DERIVATIVES AND SUBGRADIENTS 166 6.14
QUASI M-CONVEX FUNCTIONS 168 BIBLIOGRAPHICAL NOTES 175 7 L-CONVEX
FUNCTIONS 177 7.1 L-CONVEX FUNCTIONS AND L^-CONVEX FUNCTIONS 177 7.2
DISCRETE MIDPOINT CONVEXITY 180 7.3 EXAMPLES 181 7.4 BASIC OPERATIONS
183 7.5 MINIMIZERS 185 7.6 PROXIMITY THEOREM 186 7.7 CONVEX EXTENSION
187 7.8 POLYHEDRAL L-CONVEX FUNCTIONS 189 7.9 POSITIVELY HOMOGENEOUS
L-CONVEX FUNCTIONS 193 7.10 DIRECTIONAL DERIVATIVES AND SUBGRADIENTS 196
7.11 QUASI L-CONVEX FUNCTIONS 198 BIBLIOGRAPHICAL NOTES 202 8 CONJUGACY
AND DUALITY 205 8.1 CONJUGACY 205 8.1.1 SUBMODULARITY UNDER CONJUGACY
206 8.1.2 POLYHEDRAL M-/L-CONVEX FUNCTIONS 208 8.1.3 INTEGRAL
M-/L-CONVEX FUNCTIONS 212 8.2 DUALITY 216 8.2.1 SEPARATION THEOREMS 216
8.2.2 FENCHEL-TYPE DUALITY THEOREM 221 8.2.3 IMPLICATIONS 224 8.3 M 2
-CONVEX FUNCTIONS AND L 2 -CONVEX FUNCTIONS 226 8.3.1 M 2 -CONVEX
FUNCTIONS 226 8.3.2 L 2 -CONVEX FUNCTIONS 229 8.3.3 RELATIONSHIP 234 8.4
LAGRANGE DUALITY FOR OPTIMIZATION 234 8.4.1 OUTLINE 234 8.4.2 GENERAL
DUALITY FRAMEWORK 235 . 8.4.3 LAGRANGIAN FUNCTION BASED ON M-CONVEXITY .
. . . 238 8.4.4 SYMMETRY IN DUALITY 241 BIBLIOGRAPHICAL NOTES 244 9
NETWORK FLOWS 245 9.1 MINIMUM COST FLOW AND FENCHEL DUALITY 245 9.1.1
MINIMUM COST FLOW PROBLEM 245 9.1.2 FEASIBILITY 247 9.1.3 OPTIMALITY
CRITERIA 248 9.1.4 RELATIONSHIP TO FENCHEL DUALITY 253 9.2 M-CONVEX
SUBMODULAR FLOW PROBLEM 255 9.3 FEASIBILITY OF SUBMODULAR FLOW PROBLEM
258 VIII CONTENTS 9.4 OPTIMALITY CRITERION BY POTENTIALS 260 9.5
OPTIMALITY CRITERION BY NEGATIVE CYCLES 263 9.5.1 NEGATIVE-CYCLE
CRITERION 263 9.5.2 CYCLE CANCELLATION 265 9.6 NETWORK DUALITY 268 9.6.1
TRANSFORMATION BY NETWORKS 269 9.6.2 TECHNICAL SUPPLEMENTS 273
BIBLIOGRAPHICAL NOTES 278 10 ALGORITHMS 281 10.1 MINIMIZATION OF
M-CONVEX FUNCTIONS 281 10.1.1 STEEPEST DESCENT ALGORITHM 281 10.1.2
STEEPEST DESCENT SCALING ALGORITHM 283 10.1.3 DOMAIN REDUCTION ALGORITHM
284 10.1.4 DOMAIN REDUCTION SCALING ALGORITHM 286 10.2 MINIMIZATION OF
SUBMODULAR SET FUNCTIONS 288 10.2.1 BASIC FRAMEWORK 288 10.2.2
SCHRIJVER S ALGORITHM 293 10.2.3 IWATA-FLEISCHER-FUJISHIGE S ALGORITHM
296 10.3 MINIMIZATION OF L-CONVEX FUNCTIONS 305 10.3.1 STEEPEST DESCENT
ALGORITHM 305 10.3.2 STEEPEST DESCENT SCALING ALGORITHM 308 10.3.3
REDUCTION TO SUBMODULAR FUNCTION MINIMIZATION . . 308 10.4 ALGORITHMS
FOR M-CONVEX SUBMODULAR FLOWS 308 10.4.1 TWO-STAGE ALGORITHM 309 10.4.2
SUCCESSIVE SHORTEST PATH ALGORITHM 311 10.4.3 CYCLE-CANCELING ALGORITHM
312 10.4.4 PRIMAL-DUAL ALGORITHM 313 10.4.5 CONJUGATE SCALING ALGORITHM
318 BIBLIOGRAPHICAL NOTES 321 11 APPLICATION TO MATHEMATICAL ECONOMICS
323 11.1 ECONOMIC MODEL WITH INDIVISIBLE COMMODITIES 323 11.2 DIFFICULTY
WITH INDIVISIBILITY 327 11.3 M^-CONCAVE UTILITY FUNCTIONS 330 11.4
EXISTENCE OF EQUILIBRIA 334 11.4.1 GENERAL CASE 334 11.4.2 M^-CONVEX
CASE 337 11.5 COMPUTATION OF EQUILIBRIA 340 BIBLIOGRAPHICAL NOTES 344 12
APPLICATION TO SYSTEMS ANALYSIS BY MIXED MATRICES 347 12.1 TWO KINDS OF
NUMBERS 347 12.2 MIXED MATRICES AND MIXED POLYNOMIAL MATRICES 353 12.3
RANK OF MIXED MATRICES 356 12.4 DEGREE OF DETERMINANT OF MIXED
POLYNOMIAL MATRICES 359 CONTENTS JX BIBLIOGRAPHICAL NOTES 361
BIBLIOGRAPHY 363 INDEX 379
|
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| indexdate | 2024-07-09T20:10:52Z |
| institution | BVB |
| isbn | 9781611972559 0898715407 |
| language | English |
| lccn | 2003042468 |
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| physical | XXII, 389 S. Diagramme |
| publishDate | 2003 |
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| publisher | SIAM |
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| series2 | SIAM monographs on discrete mathematics and applications |
| spelling | Murota, Kazuo 1955- Verfasser (DE-588)121432424 aut Discrete convex analysis Kazuo Murota Philadelphia SIAM 2003 XXII, 389 S. Diagramme txt rdacontent n rdamedia nc rdacarrier SIAM monographs on discrete mathematics and applications aConvex functions aConvex sets aMathematical analysis Konvexe Funktion (DE-588)4139679-0 gnd rswk-swf Konvexe Menge (DE-588)4165212-5 gnd rswk-swf Konvexe Analysis (DE-588)4138566-4 gnd rswk-swf Konvexe Analysis (DE-588)4138566-4 s DE-604 Konvexe Menge (DE-588)4165212-5 s Konvexe Funktion (DE-588)4139679-0 s GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013336463&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Murota, Kazuo 1955- Discrete convex analysis aConvex functions aConvex sets aMathematical analysis Konvexe Funktion (DE-588)4139679-0 gnd Konvexe Menge (DE-588)4165212-5 gnd Konvexe Analysis (DE-588)4138566-4 gnd |
| subject_GND | (DE-588)4139679-0 (DE-588)4165212-5 (DE-588)4138566-4 |
| title | Discrete convex analysis |
| title_auth | Discrete convex analysis |
| title_exact_search | Discrete convex analysis |
| title_full | Discrete convex analysis Kazuo Murota |
| title_fullStr | Discrete convex analysis Kazuo Murota |
| title_full_unstemmed | Discrete convex analysis Kazuo Murota |
| title_short | Discrete convex analysis |
| title_sort | discrete convex analysis |
| topic | aConvex functions aConvex sets aMathematical analysis Konvexe Funktion (DE-588)4139679-0 gnd Konvexe Menge (DE-588)4165212-5 gnd Konvexe Analysis (DE-588)4138566-4 gnd |
| topic_facet | aConvex functions aConvex sets aMathematical analysis Konvexe Funktion Konvexe Menge Konvexe Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013336463&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT murotakazuo discreteconvexanalysis |