Discrete Convex Analysis

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. Discrete Convex Analysis provides the information that professionals in optimization will need to “catch up” with this new theoretical development. It also presents an unexpected connection between matroid theory and mathematical economics and expounds a deeper connection between matrices and matroids than most standard textbooks.

Spis treści:
List of Figures
Notation
Preface
1. Introduction to the Central Concepts
2. Convex Functions with Combinatorial Structures
3. Convex Analysis, Linear Programming, and Integrality
4. M-Convex Sets and Submodular Set Functions
5. L-Convex Sets and Distance Functions
6. M-Convex Functions
7. L-Convex Functions
8. Conjugacy and Duality
9. Network Flows
10. Algorithms
11. Application to Mathematical Economics
12. Application to Systems Analysis by Mixed Matrices
Bibliography
Index..