# Large Scale Linear and Integer Optimization: A Unified Approach

• Richard Kipp Martin
Book

1. Front Matter
Pages i-xvii
2. ### Motivation

1. Front Matter
Pages 1-1
2. Richard Kipp Martin
Pages 3-32
3. ### Theory

1. Front Matter
Pages 33-33
2. Richard Kipp Martin
Pages 35-80
3. Richard Kipp Martin
Pages 81-101
4. Richard Kipp Martin
Pages 103-139
4. ### Algorithms

1. Front Matter
Pages 141-141
2. Richard Kipp Martin
Pages 143-181
3. Richard Kipp Martin
Pages 183-217
4. Richard Kipp Martin
Pages 219-260
5. Richard Kipp Martin
Pages 261-311
6. Richard Kipp Martin
Pages 313-346
5. ### Solving Large Scale Problems: Decomposition Methods

1. Front Matter
Pages 347-347
2. Richard Kipp Martin
Pages 349-367
3. Richard Kipp Martin
Pages 369-392
4. Richard Kipp Martin
Pages 393-436
6. ### Solving Large Scale Problems: Using Special Structure

1. Front Matter
Pages 437-437
2. Richard Kipp Martin
Pages 439-480
3. Richard Kipp Martin
Pages 481-525

### Introduction

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.

### Keywords

Ant algorithm Operations Research Polyhedral Theory algorithms complexity complexity theory linear algebra linear optimization optimization programming

#### Authors and affiliations

• Richard Kipp Martin
• 1