DISCRETE CONVEX ANALYSIS

Transcription

1 DISCRETE CONVEX ANALYSIS o KAZUO MUROTA University of Tokyo; PRESTO, JST Tokyo, Japan Society for Industrial and Applied Mathematics Philadelphia

2 List of Figures Notation Preface xi xiii xxi 1 Introduction to the Central Concepts Aim and History of Discrete Convex Analysis Aim History Useful Properties of Convex Functions Submodular Functions and Base Polyhedra Submodular Functions Base Polyhedra Discrete Convex Functions L-Convex Functions M-Convex Functions Conjugacy Duality Classes of Discrete Convex Functions 36 Bibliographical Notes 36 2 Convex Functions with Combinatorial Structures Quadratic Functions Convex Quadratic Functions Symmetric M-Matrices Combinatorial Property of Conjugate Functions General Quadratic L-/M-Convex Functions Nonlinear Networks Real-Valued Flows Integer-Valued Flows Technical Supplements Substitutes and Complements in Network Flows Convexity and Submodularity 61

3 vi Technical Supplements Matroids From Matrices to Matroids From Polynomial Matrices to Valuated Matroids.. 71 Bibliographical Notes 74 3 Convex Analysis, Linear Programming, and Integrality Convex Analysis Linear Programming Integrality for a Pair of Integral Polyhedra Integrally Convex Functions 92 Bibliographical Notes 99 4 M-Convex Sets and Submodular Set Functions Definition Exchange Axioms Submodular Functions and Base Polyhedra Polyhedral Description of M-Convex Sets Submodular Functions as Discrete Convex Functions Ill 4.6 M-Convex Sets as Discrete Convex Sets M-Convex Sets M-Convex Polyhedra 118 Bibliographical Notes L-Convex Sets and Distance Functions Definition Distance Functions and Associated Polyhedra Polyhedral Description of L-Convex Sets L-Convex Sets as Discrete Convex Sets L^-Convex Sets L-Convex Polyhedra 131 Bibliographical Notes M-Convex Functions M-Convex Functions and M^-Convex Functions Local Exchange Axiom Examples Basic Operations Supermodularity Descent Directions Minimizers Gross Substitutes Property Proximity Theorem Convex Extension Polyhedral M-Convex Functions Positively Homogeneous M-Convex Functions 164

4 vii 6.13 Directional Derivatives and Subgradients Quasi M-Convex Functions 168 Bibliographical Notes L-Convex Functions L-Convex Functions and L^-Convex Functions Discrete Midpoint Convexity Examples Basic Operations Minimizers Proximity Theorem Convex Extension Polyhedral L-Convex Functions Positively Homogeneous L-Convex Functions Directional Derivatives and Subgradients Quasi L-Convex Functions 198 Bibliographical Notes Conjugacy and Duality Conjugacy Submodularity under Conjugacy Polyhedral M-/L-Convex Functions Integral M-/L-Convex Functions Duality Separation Theorems Fenchel-Type Duality Theorem Implications M 2 -Convex Functions and L 2 -Convex Functions M 2 -Convex Functions L 2 -Convex Functions Relationship Lagrange Duality for Optimization Outline General Duality Framework Lagrangian Function Based on M-Convexity Symmetry in Duality 241 Bibliographical Notes Network Flows Minimum Cost Flow and Fenchel Duality Minimum Cost Flow Problem Feasibility Optimality Criteria Relationship to Fenchel Duality M-Convex Submodular Flow Problem Feasibility of Submodular Flow Problem 258

5 viii 9.4 Optimality Criterion by Potentials Optimality Criterion by Negative Cycles Negative-Cycle Criterion Cycle Cancellation Network Duality Transformation by Networks Technical Supplements 273 Bibliographical Notes Algorithms Minimization of M-Convex Functions Steepest Descent Algorithm Steepest Descent Scaling Algorithm Domain Reduction Algorithm Domain Reduction Scaling Algorithm Minimization of Submodular Set Functions Basic Framework Schrijver's Algorithm Iwata-Fleischer-Fujishige's Algorithm Minimization of L-Convex Functions Steepest Descent Algorithm Steepest Descent Scaling Algorithm Reduction to Submodular Function Minimization Algorithms for M-Convex Submodular Flows Two-Stage Algorithm Successive Shortest Path Algorithm Cycle-Canceling Algorithm Primal-Dual Algorithm Conjugate Scaling Algorithm 318 Bibliographical Notes Application to Mathematical Economics Economic Model with Indivisible Commodities Difficulty with Indivisibility M^-Concave Utility Functions Existence of Equilibria General Case M^-Convex Case Computation of Equilibria 340 Bibliographical Notes Application to Systems Analysis by Mixed Matrices Two Kinds of Numbers Mixed Matrices and Mixed Polynomial Matrices Rank of Mixed Matrices Degree of Determinant of Mixed Polynomial Matrices 359

6 jx Bibliographical Notes 361 Bibliography 363 Index 379