An Introduction to Optimization with Applications in Machine Learning and Data Analytics

The text introduces students to numerous methods in solving a variety of Optimization problems. Also, the narrow focus of most math textbooks is completely dedicated to nonlinear programming, linear programming, combinatorial or convex optimization.

The primary goal of this text is a practical one. Equipping students with enough knowledge and creating an independent research platform, the author strives to prepare students for professional careers. Providing students with a marketable skill set requires topics from many areas of optimization. The initial goal of this text is to develop a marketable skill set for mathematics majors as well as for students of engineering, computer science, economics, statistics, and business. Optimization reaches into many different fields.

This text provides a balance where one is needed. Mathematics optimization books are often too heavy on theory without enough applications; texts aimed at business students are often strong on applications, but weak on math. The book represents an attempt at overcoming this imbalance for all students taking such a course.

The book contains many practical applications but also explains the mathematics behind the techniques, including stating definitions and proving theorems. Optimization techniques are at the heart of the first spam filters, are used in self-driving cars, play a great role in machine learning, and can be used in such places as determining a batting order in a Major League Baseball game. Additionally, optimization has seemingly limitless other applications in business and industry. In short, knowledge of this subject offers an individual both a very marketable skill set for a wealth of jobs as well as useful tools for research in many academic disciplines.

Many of the problems rely on using a computer. Microsoft's Excel is most often used, as this is common in business, but Python and other languages are considered. The consideration of other programming languages permits experienced mathematics and engineering students to use MATLAB® or Mathematica, and the computer science students to write their own programs in Java or Python.

Autorentext

Jeffrey Paul Wheeler earned his PhD in Combinatorial Number Theory from the University of Memphis by extending what had been a conjecture of Erds on the integers to finite groups. He has published, given talks at numerous schools, and twice been a guest of Trinity College at the University of Cambridge. He has taught mathematics at Miami University (Ohio), the University of Tennessee-Knoxville, the University of Memphis, Rhodes College, the University of Pittsburgh, Carnegie Mellon University, and Duquesne University. He has received numerous teaching awards and is currently in the Department of Mathematics at the University of Pittsburgh. He also occasionally teaches for Pitt's Computer Science Department and the College of Business Administration. Dr. Wheeler's Optimization course was one of the original thirty to participate in the Mathematical Association of America's NSF-funded PIC Math program.

Inhalt

1. 1. Preamble. 2. The Language of Optimization. 3. Computational Complexity. 4. Algebra Review. 5. Matrix Factorization. 6. Linear Programming. 7. Sensitivity Analysis. 8. Integer Linear Programing. 9. Calculus Review. 10. A Calculus Approach to Nonlinear Programming. 11. Constrained Nonlinear Programming: Lagrange Multipliers and the KKT Conditions. 12. Optimization involving Quadratic Forms. 13. Iterative Methods. 14. Derivative-Free Methods. 15. Search Algorithms. 16. Important Sets for Optimization. 17. The Fundamental Theorem of Linear Programming. 18. Convex Functions. 19. Convex Optimization. 20. An Introduction to Combinatorics. 21. An Introduction to Graph Theory. 22. Network Flows. 23. Minimum-Weight Spanning Trees and Shortest Paths. 24. Network Modeling and the Transshipment Problem. 25. The Traveling Salesperson Problem. Probability. 27. Regression Analysis via Least Squares. 28. Forecasting. 29. Introduction to Machine Learning.