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eBook ISBN:  9781470443429 
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AMS Member Price:  $60.00 
Hardcover ISBN:  9781470441142 
eBook: ISBN:  9781470443429 
Product Code:  AMSTEXT/30.B 
List Price:  $160.00 $122.50 
MAA Member Price:  $144.00 $110.25 
AMS Member Price:  $128.00 $98.00 
Hardcover ISBN:  9781470441142 
Product Code:  AMSTEXT/30 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
eBook ISBN:  9781470443429 
Product Code:  AMSTEXT/30.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $60.00 
Hardcover ISBN:  9781470441142 
eBook ISBN:  9781470443429 
Product Code:  AMSTEXT/30.B 
List Price:  $160.00 $122.50 
MAA Member Price:  $144.00 $110.25 
AMS Member Price:  $128.00 $98.00 

Book DetailsPure and Applied Undergraduate TextsVolume: 30; 2017; 327 ppMSC: Primary 46; 65; 90; 97; 58; 11; 68
Optimization Theory is an active area of research with numerous applications; many of the books are designed for engineering classes, and thus have an emphasis on problems from such fields. Covering much of the same material, there is less emphasis on coding and detailed applications as the intended audience is more mathematical. There are still several important problems discussed (especially scheduling problems), but there is more emphasis on theory and less on the nuts and bolts of coding. A constant theme of the text is the “why” and the “how” in the subject. Why are we able to do a calculation efficiently? How should we look at a problem? Extensive effort is made to motivate the mathematics and isolate how one can apply ideas/perspectives to a variety of problems. As many of the key algorithms in the subject require too much time or detail to analyze in a first course (such as the runtime of the Simplex Algorithm), there are numerous comparisons to simpler algorithms which students have either seen or can quickly learn (such as the Euclidean algorithm) to motivate the type of results on runtime savings.
ReadershipUndergraduate and graduate students interested in learning and teaching optimization and operation research.

Table of Contents

Cover

Title page

Contents

Acknowledgements

Preface

Course Outlines

Part 1 . Classical Algorithms

Chapter 1. Efficient Multiplication, I

1.1. Introduction

1.2. Babylonian Multiplication

1.3. Horner’s Algorithm

1.4. Fast Multiplication

1.5. Strassen’s Algorithm

1.6. Eigenvalues, Eigenvectors and the Fibonacci Numbers

1.7. Exercises

Chapter 2. Efficient Multiplication, II

2.1. Binomial Coefficients

2.2. Pascal’s Triangle

2.3. Dimension

2.4. From the Pascal to the Sierpinski Triangle

2.5. The Euclidean Algorithm

2.6. Exercises

Part 2 . Introduction to Linear Programming

Chapter 3. Introduction to Linear Programming

3.1. Linear Algebra

3.2. Finding Solutions

3.3. Calculus Review: Local versus Global

3.4. An Introduction to the Diet Problem

3.5. Solving the Diet Problem

3.6. Applications of the Diet Problem

3.7. Exercises

Chapter 4. The Canonical Linear Programming Problem

4.1. Real Analysis Review

4.2. Canonical Forms and Quadratic Equations

4.3. Canonical Forms in Linear Programming: Statement

4.4. Canonical Forms in Linear Programming: Conversion

4.5. The Diet Problem: Round 2

4.6. A Short Theoretical Aside: Strict Inequalities

4.7. Canonical is Not Always Best

4.8. The Oil Problem

4.9. Exercises

Chapter 5. Symmetries and Dualities

5.1. TicTacToe and a Chess Problem

5.2. Duality and Linear Programming

5.3. Appendix: Fun Versions of TicTacToe

5.4. Exercises

Chapter 6. Basic Feasible and Basic Optimal Solutions

6.1. Review of Linear Independence

6.2. Basic Feasible and Basic Optimal Solutions

6.3. Properties of Basic Feasible Solutions

6.4. Optimal and Basic Optimal Solutions

6.5. Efficiency and Euclid’s Prime Theorem

6.6. Exercises

Chapter 7. The Simplex Method

7.1. The Simplex Method: Preliminary Assumptions

7.2. The Simplex Method: Statement

7.3. Phase II implies Phase I

7.4. Phase II of the Simplex Method

7.5. Runtime of the Simplex Method

7.6. Efficient Sorting

7.7. Exercises

Part 3 . Advanced Linear Programming

Chapter 8. Integer Programming

8.1. The Movie Theater Problem

8.2. Binary Indicator Variables

8.3. Logical Statements

8.4. Truncation, Extrema and Absolute Values

8.5. Linearizing Quadratic Expressions

8.6. The Law of the Hammer and Sudoku

8.7. Bus Route Example

8.8. Exercises

Chapter 9. Integer Optimization

9.1. Maximizing a Product

9.2. The Knapsack Problem

9.3. Solving Integer Programs: Branch and Bound

9.4. Exercises

Chapter 10. MultiObjective and Quadratic Programming

10.1. MultiObjective Linear Programming

10.2. Quadratic Programming

10.3. Example: Quadratic Objective Function

10.4. Removing Quadratic (and Higher Order) Terms in Constraints

10.5. Summary

10.6. Exercises

Chapter 11. The Traveling Salesman Problem

11.1. Integer Linear Programming Version of the TSP

11.2. Greedy Algorithm to the TSP

11.3. The Insertion Algorithm

11.4. Subproblems Method

11.5. Exercises

Chapter 12. Introduction to Stochastic Linear Programming

12.1. Deterministic and Stochastic Oil Problems

12.2. Expected Value approach

12.3. Recourse Approach

12.4. Probabilistic Constraints

12.5. Exercises

Part 4 . Fixed Point Theorems

Chapter 13. Introduction to Fixed Point Theorems

13.1. Definitions and Uses

13.2. Examples

13.3. Real Analysis Preliminaries

13.4. OneDimensional Fixed Point Theorem

13.5. Newton’s Method versus Divide and Conquer

13.6. Equivalent Regions and Fixed Points

13.7. Exercises

Chapter 14. Contraction Maps

14.1. Definitions

14.2. Fixed Points of Contraction Maps

14.3. Introduction to Differential Equations

14.4. Real Analysis Review

14.5. First Order Differential Equations Theorem

14.6. Examples of Picard’s Iteration Method

14.7. Exercises

Chapter 15. Sperner’s Lemma

15.1. Statement of Sperner’s Lemma

15.2. Proof Strategies for Sperner’s Lemma

15.3. Proof of Sperner’s Lemma

15.4. Rental Harmony

15.5. Exercises

Chapter 16. Brouwer’s Fixed Point Theorem

16.1. BolzanoWeierstrass Theorem

16.2. Barycentric Coordinates

16.3. Preliminaries for Brouwer’s Fixed Point Theorem

16.4. Proof of Brouwer’s Fixed Point Theorem

16.5. Nash Equilibrium

16.6. Exercises

Part 5 . Advanced Topics

Chapter 17. GaleShapley Algorithm

17.1. Introduction

17.2. Three Parties

17.3. GaleShapley Algorithm

17.4. Generalization

17.5. Applications

17.6. Exercises

Chapter 18. Interpolating Functions

18.1. Lagrange Interpolation

18.2. Interpolation Error

18.3. Chebyshev Polynomials and Interpolation

18.4. Splines

18.5. Exercises

Chapter 19. The Four Color Problem

19.1. A Brief History

19.2. Preliminaries

19.3. Birkhoff and the Modern Proof

19.4. AppelHaken Proof

19.5. Computational Improvements

19.6. Exercises

Chapter 20. The Kepler Conjecture

20.1. Introduction

20.2. Sphere Packing

20.3. Challenges in Proving the Kepler Conjecture

20.4. Local Density Inequalities

20.5. ComputerAided Proof

20.6. Exercises

Bibliography

Index

Back Cover


Additional Material

Reviews

I think that this book offers a vast and useful outline of many mathematical problems arising from the common ground of optimization theory and operations research. Surely it can be useful and of interest to advanced undergraduates and beginning graduate students concerned with applications of mathematics to optimization problems and related fields.
Giorgio Giorgi, Mathematical Reviews 
I started reading "Mathematics of Optimization: How to do Things Faster" without a significant background in optimization, linear programming, or operations research. Hence, I really did not know what to expect from the book. I was pleasantly surprised to find the book to be so much fun to work through. The writing is upbeat, entertaining and enlightening and the mathematics is varied, interesting, and inspiring...I am really impressed by "Mathematics of Optimization," and I would love to teach a course based on this book just in order to spend more time going through it myself...I think that the book is unique and should be relevant and of interest to advanced undergraduate and beginning graduate students in pure and applied mathematics and some closely related areas.
Jason M. Graham, MAA Reviews


RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a coursePermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
Optimization Theory is an active area of research with numerous applications; many of the books are designed for engineering classes, and thus have an emphasis on problems from such fields. Covering much of the same material, there is less emphasis on coding and detailed applications as the intended audience is more mathematical. There are still several important problems discussed (especially scheduling problems), but there is more emphasis on theory and less on the nuts and bolts of coding. A constant theme of the text is the “why” and the “how” in the subject. Why are we able to do a calculation efficiently? How should we look at a problem? Extensive effort is made to motivate the mathematics and isolate how one can apply ideas/perspectives to a variety of problems. As many of the key algorithms in the subject require too much time or detail to analyze in a first course (such as the runtime of the Simplex Algorithm), there are numerous comparisons to simpler algorithms which students have either seen or can quickly learn (such as the Euclidean algorithm) to motivate the type of results on runtime savings.
Undergraduate and graduate students interested in learning and teaching optimization and operation research.

Cover

Title page

Contents

Acknowledgements

Preface

Course Outlines

Part 1 . Classical Algorithms

Chapter 1. Efficient Multiplication, I

1.1. Introduction

1.2. Babylonian Multiplication

1.3. Horner’s Algorithm

1.4. Fast Multiplication

1.5. Strassen’s Algorithm

1.6. Eigenvalues, Eigenvectors and the Fibonacci Numbers

1.7. Exercises

Chapter 2. Efficient Multiplication, II

2.1. Binomial Coefficients

2.2. Pascal’s Triangle

2.3. Dimension

2.4. From the Pascal to the Sierpinski Triangle

2.5. The Euclidean Algorithm

2.6. Exercises

Part 2 . Introduction to Linear Programming

Chapter 3. Introduction to Linear Programming

3.1. Linear Algebra

3.2. Finding Solutions

3.3. Calculus Review: Local versus Global

3.4. An Introduction to the Diet Problem

3.5. Solving the Diet Problem

3.6. Applications of the Diet Problem

3.7. Exercises

Chapter 4. The Canonical Linear Programming Problem

4.1. Real Analysis Review

4.2. Canonical Forms and Quadratic Equations

4.3. Canonical Forms in Linear Programming: Statement

4.4. Canonical Forms in Linear Programming: Conversion

4.5. The Diet Problem: Round 2

4.6. A Short Theoretical Aside: Strict Inequalities

4.7. Canonical is Not Always Best

4.8. The Oil Problem

4.9. Exercises

Chapter 5. Symmetries and Dualities

5.1. TicTacToe and a Chess Problem

5.2. Duality and Linear Programming

5.3. Appendix: Fun Versions of TicTacToe

5.4. Exercises

Chapter 6. Basic Feasible and Basic Optimal Solutions

6.1. Review of Linear Independence

6.2. Basic Feasible and Basic Optimal Solutions

6.3. Properties of Basic Feasible Solutions

6.4. Optimal and Basic Optimal Solutions

6.5. Efficiency and Euclid’s Prime Theorem

6.6. Exercises

Chapter 7. The Simplex Method

7.1. The Simplex Method: Preliminary Assumptions

7.2. The Simplex Method: Statement

7.3. Phase II implies Phase I

7.4. Phase II of the Simplex Method

7.5. Runtime of the Simplex Method

7.6. Efficient Sorting

7.7. Exercises

Part 3 . Advanced Linear Programming

Chapter 8. Integer Programming

8.1. The Movie Theater Problem

8.2. Binary Indicator Variables

8.3. Logical Statements

8.4. Truncation, Extrema and Absolute Values

8.5. Linearizing Quadratic Expressions

8.6. The Law of the Hammer and Sudoku

8.7. Bus Route Example

8.8. Exercises

Chapter 9. Integer Optimization

9.1. Maximizing a Product

9.2. The Knapsack Problem

9.3. Solving Integer Programs: Branch and Bound

9.4. Exercises

Chapter 10. MultiObjective and Quadratic Programming

10.1. MultiObjective Linear Programming

10.2. Quadratic Programming

10.3. Example: Quadratic Objective Function

10.4. Removing Quadratic (and Higher Order) Terms in Constraints

10.5. Summary

10.6. Exercises

Chapter 11. The Traveling Salesman Problem

11.1. Integer Linear Programming Version of the TSP

11.2. Greedy Algorithm to the TSP

11.3. The Insertion Algorithm

11.4. Subproblems Method

11.5. Exercises

Chapter 12. Introduction to Stochastic Linear Programming

12.1. Deterministic and Stochastic Oil Problems

12.2. Expected Value approach

12.3. Recourse Approach

12.4. Probabilistic Constraints

12.5. Exercises

Part 4 . Fixed Point Theorems

Chapter 13. Introduction to Fixed Point Theorems

13.1. Definitions and Uses

13.2. Examples

13.3. Real Analysis Preliminaries

13.4. OneDimensional Fixed Point Theorem

13.5. Newton’s Method versus Divide and Conquer

13.6. Equivalent Regions and Fixed Points

13.7. Exercises

Chapter 14. Contraction Maps

14.1. Definitions

14.2. Fixed Points of Contraction Maps

14.3. Introduction to Differential Equations

14.4. Real Analysis Review

14.5. First Order Differential Equations Theorem

14.6. Examples of Picard’s Iteration Method

14.7. Exercises

Chapter 15. Sperner’s Lemma

15.1. Statement of Sperner’s Lemma

15.2. Proof Strategies for Sperner’s Lemma

15.3. Proof of Sperner’s Lemma

15.4. Rental Harmony

15.5. Exercises

Chapter 16. Brouwer’s Fixed Point Theorem

16.1. BolzanoWeierstrass Theorem

16.2. Barycentric Coordinates

16.3. Preliminaries for Brouwer’s Fixed Point Theorem

16.4. Proof of Brouwer’s Fixed Point Theorem

16.5. Nash Equilibrium

16.6. Exercises

Part 5 . Advanced Topics

Chapter 17. GaleShapley Algorithm

17.1. Introduction

17.2. Three Parties

17.3. GaleShapley Algorithm

17.4. Generalization

17.5. Applications

17.6. Exercises

Chapter 18. Interpolating Functions

18.1. Lagrange Interpolation

18.2. Interpolation Error

18.3. Chebyshev Polynomials and Interpolation

18.4. Splines

18.5. Exercises

Chapter 19. The Four Color Problem

19.1. A Brief History

19.2. Preliminaries

19.3. Birkhoff and the Modern Proof

19.4. AppelHaken Proof

19.5. Computational Improvements

19.6. Exercises

Chapter 20. The Kepler Conjecture

20.1. Introduction

20.2. Sphere Packing

20.3. Challenges in Proving the Kepler Conjecture

20.4. Local Density Inequalities

20.5. ComputerAided Proof

20.6. Exercises

Bibliography

Index

Back Cover

I think that this book offers a vast and useful outline of many mathematical problems arising from the common ground of optimization theory and operations research. Surely it can be useful and of interest to advanced undergraduates and beginning graduate students concerned with applications of mathematics to optimization problems and related fields.
Giorgio Giorgi, Mathematical Reviews 
I started reading "Mathematics of Optimization: How to do Things Faster" without a significant background in optimization, linear programming, or operations research. Hence, I really did not know what to expect from the book. I was pleasantly surprised to find the book to be so much fun to work through. The writing is upbeat, entertaining and enlightening and the mathematics is varied, interesting, and inspiring...I am really impressed by "Mathematics of Optimization," and I would love to teach a course based on this book just in order to spend more time going through it myself...I think that the book is unique and should be relevant and of interest to advanced undergraduate and beginning graduate students in pure and applied mathematics and some closely related areas.
Jason M. Graham, MAA Reviews