Discrete convex analysis /

Discrete convex analysis /

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Bibliographic Details
Author / Creator:Murota, Kazuo, 1955-
Imprint:Philadelphia : Society for Industrial and Applied Mathematics, ©2003.
Description:1 online resource (xxii, 389 pages) : illustrations
Language:English
Series:SIAM monographs on discrete mathematics and applications
SIAM monographs on discrete mathematics and applications.
Subject:
Convex functions.
Convex sets.
Mathematical analysis.
Convex functions
Convex sets
Mathematical analysis
Format: E-Resource Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/12577265
Hidden Bibliographic Details
ISBN:0898715407
9780898715408
9780898718508
0898718503
Notes:Includes bibliographical references (pages 363-377) and index.
Also available in print version.
Summary:Discrete Convex Analysis is a novel paradigm for discrete optimization that combines the ideas in continuous optimization (convex analysis) and combinatorial optimization (matroid/submodular function theory) to establish a unified theoretical framework for nonlinear discrete optimization. The study of this theory is expanding with the development of efficient algorithms and applications to a number of diverse disciplines like matrix theory, operations research, and economics. This self-contained book is designed to provide a novel insight into optimization on discrete structures and should reveal unexpected links among different disciplines. It is the first and only English-language monograph on the theory and applications of discrete convex analysis.
Other form:Print version: Murota, Kazuo, 1955- Discrete convex analysis. Philadelphia : Society for Industrial and Applied Mathematics, ©2003 0898715407
Publisher's no.:DT10 siam
Table of Contents:
  • List of Figures
  • Notation
  • Preface
  • 1. Introduction to the Central Concepts
  • 1.1. Aim and History of Discrete Convex Analysis
  • 1.1.1. Aim
  • 1.1.2. History
  • 1.2. Useful Properties of Convex Functions
  • 1.3. Submodular Functions and Base Polyhedra
  • 1.3.1. Submodular Functions
  • 1.3.2. Base Polyhedra
  • 1.4. Discrete Convex Functions
  • 1.4.1. L-Convex Functions
  • 1.4.2. M-Convex Functions
  • 1.4.3. Conjugacy
  • 1.4.4. Duality
  • 1.4.5. Classes of Discrete Convex Functions
  • Bibliographical Notes
  • 2. Convex Functions with Combinatorial Structures
  • 2.1. Quadratic Functions
  • 2.1.1. Convex Quadratic Functions
  • 2.1.2. Symmetric M-Matrices
  • 2.1.3. Combinatorial Property of Conjugate Functions
  • 2.1.4. General Quadratic L-/M-Convex Functions
  • 2.2. Nonlinear Networks
  • 2.2.1. Real-Valued Flows
  • 2.2.2. Integer-Valued Flows
  • 2.2.3. Technical Supplements
  • 2.3. Substitutes and Complements in Network Flows
  • 2.3.1. Convexity and Submodularity
  • 2.3.2. Technical Supplements
  • 2.4. Matroids
  • 2.4.1. From Matrices to Matroids
  • 2.4.2. From Polynomial Matrices to Valuated Matroids
  • Bibliographical Notes
  • 3. Convex Analysis, Linear Programming, and Integrality
  • 3.1. Convex Analysis
  • 3.2. Linear Programming
  • 3.3. Integrality for a Pair of Integral Polyhedra
  • 3.4. Integrally Convex Functions
  • Bibliographical Notes
  • 4. M-Convex Sets and Submodular Set Functions
  • 4.1. Definition
  • 4.2. Exchange Axioms
  • 4.3. Submodular Functions and Base Polyhedra
  • 4.4. Polyhedral Description of M-Convex Sets
  • 4.5. Submodular Functions as Discrete Convex Functions
  • 4.6. M-Convex Sets as Discrete Convex Sets
  • 4.7. M-Convex Sets
  • 4.8. M-Convex Polyhedra
  • Bibliographical Notes
  • 5. L-Convex Sets and Distance Functions
  • 5.1. Definition
  • 5.2. Distance Functions and Associated Polyhedra
  • 5.3. Polyhedral Description of L-Convex Sets
  • 5.4. L-Convex Sets as Discrete Convex Sets
  • 5.5. L-Convex Sets
  • 5.6. L-Convex Polyhedra
  • Bibliographical Notes
  • 6. M-Convex Functions
  • 6.1. M-Convex Functions and M-Convex Functions
  • 6.2. Local Exchange Axiom
  • 6.3. Examples
  • 6.4. Basic Operations
  • 6.5. Supermodularity
  • 6.6. Descent Directions
  • 6.7. Minimizers
  • 6.8. Gross Substitutes Property
  • 6.9. Proximity Theorem
  • 6.10. Convex Extension
  • 6.11. Polyhedral M-Convex Functions
  • 6.12. Positively Homogeneous M-Convex Functions
  • 6.13. Directional Derivatives and Subgradients
  • 6.14. Quasi M-Convex Functions
  • Bibliographical Notes
  • 7. L-Convex Functions
  • 7.1. L-Convex Functions and L[sharp]-Convex Functions
  • 7.2. Discrete Midpoint Convexity
  • 7.3. Examples
  • 7.4. Basic Operations
  • 7.5. Minimizers
  • 7.6. Proximity Theorem
  • 7.7. Convex Extension
  • 7.8. Polyhedral L-Convex Functions
  • 7.9. Positively Homogeneous L-Convex Functions
  • 7.10. Directional Derivatives and Subgradients
  • 7.11. Quasi L-Convex Functions
  • Bibliographical Notes
  • 8. Conjugacy and Duality
  • 8.1. Conjugacy
  • 8.1.1. Submodularity under Conjugacy
  • 8.1.2. Polyhedral M-/L-Convex Functions
  • 8.1.3. Integral M-/L-Convex Functions
  • 8.2. Duality
  • 8.2.1. Separation Theorems
  • 8.2.2. Fenchel-Type Duality Theorem
  • 8.2.3. Implications
  • 8.3. M[subscript 2]-Convex Functions and L[subscript 2]-Convex Functions
  • 8.3.1. M[subscript 2]-Convex Functions
  • 8.3.2. L[subscript 2]-Convex Functions
  • 8.3.3. Relationship
  • 8.4. Lagrange Duality for Optimization
  • 8.4.1. Outline
  • 8.4.2. General Duality Framework
  • 8.4.3. Lagrangian Function Based on M-Convexity
  • 8.4.4. Symmetry in Duality
  • Bibliographical Notes
  • 9. Network Flows
  • 9.1. Minimum Cost Flow and Fenchel Duality
  • 9.1.1. Minimum Cost Flow Problem
  • 9.1.2. Feasibility
  • 9.1.3. Optimality Criteria
  • 9.1.4. Relationship to Fenchel Duality
  • 9.2. M-Convex Submodular Flow Problem
  • 9.3. Feasibility of Submodular Flow Problem
  • 9.4. Optimality Criterion by Potentials
  • 9.5. Optimality Criterion by Negative Cycles
  • 9.5.1. Negative-Cycle Criterion
  • 9.5.2. Cycle Cancellation
  • 9.6. Network Duality
  • 9.6.1. Transformation by Networks
  • 9.6.2. Technical Supplements
  • Bibliographical Notes
  • 10. Algorithms
  • 10.1. Minimization of M-Convex Functions
  • 10.1.1. Steepest Descent Algorithm
  • 10.1.2. Steepest Descent Scaling Algorithm
  • 10.1.3. Domain Reduction Algorithm
  • 10.1.4. Domain Reduction Scaling Algorithm
  • 10.2. Minimization of Submodular Set Functions
  • 10.2.1. Basic Framework
  • 10.2.2. Schrijver's Algorithm
  • 10.2.3. Iwata-Fleischer-Fujishige's Algorithm
  • 10.3. Minimization of L-Convex Functions
  • 10.3.1. Steepest Descent Algorithm
  • 10.3.2. Steepest Descent Scaling Algorithm
  • 10.3.3. Reduction to Submodular Function Minimization
  • 10.4. Algorithms for M-Convex Submodular Flows
  • 10.4.1. Two-Stage Algorithm
  • 10.4.2. Successive Shortest Path Algorithm
  • 10.4.3. Cycle-Canceling Algorithm
  • 10.4.4. Primal-Dual Algorithm
  • 10.4.5. Conjugate Scaling Algorithm
  • Bibliographical Notes
  • 11. Application to Mathematical Economics
  • 11.1. Economic Model with Indivisible Commodities
  • 11.2. Difficulty with Indivisibility
  • 11.3. M[sharp]-Concave Utility Functions
  • 11.4. Existence of Equilibria
  • 11.4.1. General Case
  • 11.4.2. M[sharp]-Convex Case
  • 11.5. Computation of Equilibria
  • Bibliographical Notes
  • 12. Application to Systems Analysis by Mixed Matrices
  • 12.1. Two Kinds of Numbers
  • 12.2. Mixed Matrices and Mixed Polynomial Matrices
  • 12.3. Rank of Mixed Matrices
  • 12.4. Degree of Determinant of Mixed Polynomial Matrices
  • Bibliographical Notes
  • Bibliography
  • Index

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