Discrete convex analysis /
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| Author / Creator: | Murota, Kazuo, 1955- |
|---|---|
| Imprint: | Philadelphia : Society for Industrial and Applied Mathematics, c2003. |
| Description: | xxii, 389 p. : ill. ; 26 cm. |
| Language: | English |
| Series: | SIAM monographs on discrete mathematics and applications |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/4927073 |
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
