Large Scale Linear and Integer Optimization: A Unified Approach

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.

Klappentext

There is a growing need in major industries such as airline, trucking, financial engineering, etc. to solve very large linear and integer linear optimization problems. Because of the dramatic increase in computing power, it is now possible to solve these problems. Along with the increase in computer power, the mathematical programming community has developed better and more powerful algorithms to solve very large problems. These algorithms are of interest to many researchers in the areas of operations research/management science, computer science, and engineering. In this book, Kipp Martin has systematically provided users with a unified treatment of the algorithms and the implementation of the algorithms that are important in solving large problems. Parts I and II of Large Scale Linear and Integer Programming provide an introduction to linear optimization using two simple but unifying ideas-projection and inverse projection. The ideas of projection and inverse projection are also extended to integer linear optimization. With the projection-inverse projection approach, theoretical results in integer linear optimization become much more analogous to their linear optimization counterparts. Hence, with an understanding of these two concepts, the reader is equipped to understand fundamental theorems in an intuitive way. Part III presents the most important algorithms that are used in commercial software for solving real-world problems. Part IV shows how to take advantage of the special structure in very large scale applications through decomposition. Part V describes how to take advantage of special structureby modifying and enhancing the algorithms developed in Part III. This section contains a discussion of the current research in linear and integer linear programming. The author also shows in Part V how to take different problem formulations and appropriately `modify' them so that the algorithms from Part III are more efficient. Again, the projection and inverse projection concepts are used in Part V to present the current research in linear and integer linear optimization in a very unified way. While the book is written for a mathematically mature audience, no prior knowledge of linear or integer linear optimization is assumed. The audience is upper-level undergraduate students and graduate students in computer science, applied mathematics, industrial engineering and operations research/management science. Course work in linear algebra and analysis is sufficient background.

Zusammenfassung

From the reviews of the first edition:

"This book is a very comprehensive textbook of linear and integer optimization. It presents a unified approach to the subject and is one of the few books treating linear and integer aspects together. ... This is an interesting book with special features. ... The text is written in a clear and readily comprehensible way, and it contains computer programs as well as many examples being helpful for students. This book is an important textbook of mathematical programming." (Johannes Jahn, Zentralblatt MATH, Vol. 1053 (4), 2005)

Inhalt
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.