We discuss modeling in Section There is a fair number of exercises that are given at the end of each chapter. Most of them are intended to deepen the understanding of the subject , or to explore extensions of the theory in the text , as opposed to routine drills. However, several numerical exercises are also included. Starred exercises are supposed to be fairly hard. A solutions manual for qualified instructors can be obtained from the authors.

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We discuss modeling in Section There is a fair number of exercises that are given at the end of each chapter. Most of them are intended to deepen the understanding of the subject , or to explore extensions of the theory in the text , as opposed to routine drills. However, several numerical exercises are also included. Starred exercises are supposed to be fairly hard. A solutions manual for qualified instructors can be obtained from the authors.

We have made a special effort to keep the text as modular as possible , allowing the reader to omit certain topics without loss of continuity. For example, much of the material in Chapters 5 and 6 is rarely used in the rest of the book. Furthermore, in Chapter 7 on network flow problems , a reader who has gone through the problem formulation Sections 7.

Also, the interior point algorithms of Chapter 9 are not used later, with the exception of some of the applications in Chapter Even within the core chapters Chapters , there are many sections that can be skipped during a first reading. Some sections have been marked with a star indicating that they contain somewhat more advanced material that is not usually covered in an introductory course.

The book was developed while we took turns teaching a first-year graduate course at M. The only prerequisite is a working knowledge of linear algebra. In fact , it is only a small subset of linear algebra that is needed e. However, these elementary tools are sometimes used in subtle ways, and some mathematical maturity on the part of the reader can lead to a better appreciation of the subject.

The book can be used to teach several different types of courses. The first two suggestions below are one-semester variants that we have tried at M. The core of such a course could consist of Chapter 1, Sections 2. There is a truly large literature on linear optimization, and we make no attempt to provide a comprehensive bibliography.

To a great extent , the sources that we cite are either original references of historical interest , or recent texts where additional information can be found. For those topics , however, that touch upon current research, we also provide pointers to recent journal articles. We would like to express our thanks to a number of individuals. We are grateful to our colleagues Dimitri Bertsekas and Rob Freund, for many discussions on the subjects in this book, as well as for reading parts of the manuscript.

But mostly, we are grateful to our families for their patience, love, and support in the course of this long project. Dimitris Bertsimas John N.

Variants of the linear programming problem 1. Examples of linear programming problems 1. Piecewise linear convex objective functions 1. Graphical representation and solution 1. Linear algebra background and notation 1. Algorithms and operation counts 1. Exercises 1. History, notes , and sources 1 2 Chap. We consider a few equivalent forms and then present a number of examples to illustrate the applicability of linear programming to a wide variety of contexts.

We also solve a few simple examples and obtain some basic geometric intuition on the nature of the problem. The chapter ends with a review of linear algebra and of the conventions used in describing the computational requirements operation count of algorithms. Rather than starting abstractly, we first state a concrete example, which is meant to facilitate understanding of the formal definition that will follow. The example we give is devoid of any interpretation. Later on, in Section 1.

Example 1. Some of these constraints, such as Xl 2 0 and X3 0 , amount to simple restrictions on the sign of certain variables. We now generalize. Row vectors are indicated as transposes of column vectors. The reader who is unfamiliar with our notation may wish to consult Section 1. In particular , let MI, M2, M3 be some finite index sets, and suppose that for every i in any one of these sets, we are given an n-dimensional vector ai and a scalar bi , that will be used to form the ith constraint.

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## Solutions Of Introduction To Linear Optimization

Anticycling: lexicography and Blands rule 1C8 7. Network flow prablems ,,.. Finding an initial basic feasible solution 3. Column geometry and the simplex method 7. Grphs 3. Computational eficiency o the simplex method 7.

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## Introduction to Linear Optimization

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## MS&E 310: Linear Optimization, Fall 2018

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