Large Scale Linear and Integer Optimization: A Unified Approach

1999
ISBN: 978-0-7923-8202-7
Verlag: Springer US

Buch, Englisch, 740 Seiten, HC runder Rücken kaschiert, Format (B × H): 160 mm x 241 mm, Gewicht: 1294 g

This is a textbook about linear and integer linear optimization. There is a growing need in industries such as airline, trucking, and financial engineering to solve very large linear and integer linear optimization problems. Building these models requires uniquely trained individuals. Not only must they have a thorough understanding of the theory behind mathematical programming, they must have substantial knowledge of how to solve very large models in today's computing environment. The major goal of the book is to develop the theory of linear and integer linear optimization in a unified manner and then demonstrate how to use this theory in a modern computing environment to solve very large real world problems. After presenting introductory material in Part I, Part II of this book is de­ voted to the theory of linear and integer linear optimization. This theory is developed using two simple, but unifying ideas: projection and inverse projec­ tion. Through projection we take a system of linear inequalities and replace some of the variables with additional linear inequalities. Inverse projection, the dual of this process, involves replacing linear inequalities with additional variables. Fundamental results such as weak and strong duality, theorems of the alternative, complementary slackness, sensitivity analysis, finite basis the­ orems, etc. are all explained using projection or inverse projection. Indeed, a unique feature of this book is that these fundamental results are developed and explained before the simplex and interior point algorithms are presented.

Zielgruppe

Research

Autoren/Hrsg.

Martin, Richard Kipp

Fachgebiete

• Technische Wissenschaften Elektronik | Nachrichtentechnik Elektronik Robotik
• Wirtschaftswissenschaften Volkswirtschaftslehre Volkswirtschaftslehre Allgemein Ökonometrie
• Mathematik | Informatik Mathematik Mathematische Analysis Variationsrechnung
• Mathematik | Informatik Mathematik Operations Research Spieltheorie
• Wirtschaftswissenschaften Betriebswirtschaft Unternehmensforschung
• Mathematik | Informatik EDV | Informatik Informatik Künstliche Intelligenz Wissensbasierte Systeme, Expertensysteme
• Wirtschaftswissenschaften Betriebswirtschaft Wirtschaftsmathematik und -statistik

Inhaltsverzeichnis

1 Linear and Integer Linear Optimization.- 2 Linear Systems and Projection.- 3 Linear Systems and Inverse Projection.- 4 Integer Linear Systems: Projection and Inverse Projection.- 5 The Simplex Algorithm.- 6 More on Simplex.- 7 Interior Point Algorithms: Polyhedral Transformations.- 8 Interior Point Algorithms: Barrier Methods.- 9 Integer Programming.- 10 Projection: Benders’ Decomposition.- 11 Inverse Projection: Dantzig-Wolfe Decomposition.- 12 Lagrangian Methods.- 13 Sparse Methods.- 14 Network Flow Linear Programs.- 15 Large Integer Programs: Preprocessing and Cutting Planes.- 16 Large Integer Programs: Projection and Inverse Projection.- VI Appendix.- A Polyhedral Theory.- A.1 Introduction.- A.2 Concepts and Definitions.- A.3 Faces of Polyhedra.- A.4 Finite Basis Theorems.- A.5 Inner Products, Subspaces and Orthogonal Subspaces.- A. 6 Exercises.- B Complexity Theory.- B.1 Introduction.- B.2 Solution Sizes.- B.3 The Turing Machine.- B.4 Complexity Classes.- B.5 Satisfiability.- B.7 Complexity of Gaussian Elimination.- B.8 Exercises.- C Basic Graph Theory.- D Software And Test Problems.- E NOTATION.- References.- Author Index.- Topic Index.