Math 408 Section A WINTER 2004
NONLINEAR OPTIMIZATION
Instructor: E-Mail: Phone: Jim Burke burke@math.washington.edu 543-6183 Mon. 11:30-12:20 and Fri. 11:30- 12:20 C-443 Padelford & by appointment Math 327 and 308 (or 318)

#### URL for the course website:

• http://www.math.washington.edu/~burke/crs/408f/

### Text:

David Luenberger, Investment Science (1998)

### Course Content:

A mathematical optimization problem is one in which a given function is either minimized or maximized relative to some set or range of choices available in a given situation. Optimization problems arise in a multitude of ways as a means of solving problems in engineering design, portfolio design, system management, parameter estimation, statistics, and in the modeling of physical and behavioral phenomena. Math 408 is an introductory course in numerical methods for continuous optimization in finite dimensions. This term the course is being taught with an emphasis on applications in financial optimization. We will learn how the techniques of optimization modeling and numerical solution methodology can be applied to a range of important problems in finance. The financial models we consider include linear portfolio optimization, the mean-variance portfolio theory of Markowitz, the capital asset pricing model, value-at-risk, expected value at risk, and the construction of indexed funds. The optimization tools that we consider cover much of what is known as mathematical programming. We begin with linear programming and then progress through quadratic programming to nonlinear programming and integer programming. Special emphasis is given to the duality theory and numerical methods.

### Background and Prerequisites:

This course requires a background in multi-variable calculus. In order to succeed you will need to be conversant with the differential properties of smooth vector valued mappings In particular, you will need to know properties of the gradient and Hessian. Moreover, some background in linear algebra is also required. In particular, you will need to know some results concerning the eigenvalue decomposition of a symmetric matrix, Gaussian elimination (LU factorization), and Gram--Schmidt orthogonalization (QR factorization). However, I do not expect everyone to have the the same level of preparation. Consequently, all of the material discussed above will be reviewed with most proofs omitted.

Quizzes: There are 9 fifteen minute quizzes each worth 50 points. The quizzes are given each Friday except February 13. The quizzes cover the homework of the previous week. The potential content of the quiz will be announced the Wednesday before the quiz. The top 7 of your quiz scores count toward your grade.

Assignments: Three take-home assignments will be given each worth 50 points. These are extended homeworks that will require the use of a range of skills. You will be asked to model a financial optimization problem, solve the problem using a computer package, and then write a report on the model and its solution.

Midterms: There is one midterm: Friday, February 13. The content of the midterm will be discussed in advance and a sample midterm will be distributed before the exam. The midterm is worth 200 points.

Final Exam: The final exam is to be given on Monday, March 15 from 8:30 to 10:20 am. The final exam is comprehensive. A sample final exam will be distributed. The final exam is worth 300 points.

Final Grade: The total number of possible points is 1000:

 350 quiz pts $+$ 150 assignment pts $+$ 200 midterm pts $+$ 300 final exam pts $=$ 1000 points.

### Time Conflicts with an Exam:

There will be no make-up exams except in the case of a documented emergency. In the event of an unavoidable conflict with a midterm (an athletic meet, wedding, funeral, etc...), you must notify me as soon as you are aware of the conflict (a minimum of 1 week prior to the exam date) so that we can arrange for you to take the exam BEFORE the actual exam date. In the event of an unavoidable conflict with the final exam, you will need to submit a written petition for this purpose to me by Friday, March 5, or as soon as you are aware of the conflict. No make-up quizzes will be given since your lowest two quiz scores will be dropped.

### Incomplete:

A grade of Incomplete will be given only if a student is doing satisfactory work up until the end of the quarter, and then misses the final exam due to a documented medical or family emergency.

### OFFICE HOUR RULES

If no student arrives within the first 15 minutes of office hours, then I will assume that no student will be coming for office hours that day unless other arrnagements have been made. In this case, I will attend to my other University duties, and so I may leave my office at that time. However, it is important to remember that you are always free to make an appointment to see me at a time other than the scheduled office hours. You can arrange such an appointment by either speaking directly with me in person, by phone, or by by email.

### Important Dates:

Holidays: January 19, Martin Luther King Day: February 16, Presidents Day.

Quiz Dates: Jan. 9,16,23,30: Feb. 6,20,27: Mar. 5,12.

Midterm Date: Friday, February 13.

Final Exam: Monday, March 15, 8:30-10:20 am.