Basic Polyhedral Theory

TL;DR

This chapter provides an overview of key concepts in convex polyhedral theory relevant to optimization, including cones, the Weyl-Minkowski Theorem, faces, projections, and integrality properties.

Contribution

It summarizes fundamental polyhedral concepts and theorems essential for understanding optimization problems involving convex polyhedra.

Findings

Clarifies the structure of convex polyhedra and cones

Explains the significance of total dual integrality and unimodularity

Provides foundational knowledge for optimization applications

Abstract

This is a chapter (planned to appear in Wiley's upcoming Encyclopedia of Operations Research and Management Science) describing parts of the theory of convex polyhedra that are particularly important for optimization. The topics include polyhedral and finitely generated cones, the Weyl-Minkowski Theorem, faces of polyhedra, projections of polyhedra, integral polyhedra, total dual integrality, and total unimodularity.