Linear Programming Duality

Linear Programming Duality is one of the cornerstones in combinatorial optimization. The book is written by two authors who have been working in the field of combinatorial optimization for many years. They give an elementary introduction to the theory of oriented matroids. Their approach clarifies the theoretical basis of Linear Programming and simplifies the proofs of standard results. The book contains numerous figures and the authors have included suggestions for further reading after each chapter.

1 Prerequisites. - 7. 1 Sets and Relations. - 10. 2 Linear Algebra. - 14. 3 Topology. - 15. 4 Polyhedra. - 2 Linear Duality in Graphs. - 2. 1 Some Definitions. - 2. 2 FARKAS Lemma for Graphs. - 2. 3 Subspaces Associated with Graphs. - 2. 4 Planar Graphs. - 2. 5 Further Reading. - 3 Linear Duality and Optimization. - 3. 1 Optimization Problems. - 3. 2 Recognizing Optimal Solutions. - 3. 3 Further Reading. - 4 The FARKAS Lemma. - 4. 1 A first version. - 4. 2 Homogenization. - 4. 3 Linearization. - 4. 4 Delinearization. - 4. 5 Dehomogenization. - 4. 6 Further Reading. - 5 Oriented Matroids. - 5. 1 Sign Vectors. - 5. 2 Minors. - 5. 3 Oriented Matroids. - 5. 4 Abstract Orthogonality. - 5. 5 Abstract Elimination Property. - 5. 6 Elementary vectors. - 5. 7 The Composition Theorem. - 5. 8 Elimination Axioms. - 5. 9 Approximation Axioms. - 5. 10 Proof of FARKAS Lemma in OMs. - 5. 11 Duality. - 5. 12 Further Reading. - 6 Linear Programming Duality. - 6. 1 The Dual Program. - 6. 2 The Combinatorial Problem. - 6. 3 Network Programming. - 6. 4 Further Reading. - 7 Basic Facts in Polyhedral Theory. - 7. 1 MINKOWSKI S Theorem. - 7. 2 Polarity. - 7. 3 Faces of Polyhedral Cones. - 7. 4 Faces and Interior Points. - 7. 5 The Canonical Map. - 7. 6 Lattices. - 7. 7 Face Lattices of Polars. - 7. 8 General Polyhedra. - 7. 9 Further Reading. - 8 The Poset (O, ?). - 8. 1 Simplifications. - 8. 2 Basic Results. - 8. 3 Shellability of Topes. - 8. 4 Constructibility of O. - 8. 5 Further Reading. - 9 Topological Realizations. - 9. 1 Linear Sphere Systems. - 9. 2 A Nonlinear OM. - 9. 3 Sphere Systems. - 9. 4 PL Ball Complexes. - 9. 5 Further Reading.