MATH UN2500 Analysis and Optimization

Section 001 - TTh 8:40--9:55, in 203 Mathematics

Section 002 - TTh 10:10-11:25, in 203 Mathematics

Instructor: Daniel Halpern-Leistner

Office: 716A Mathematics

Office hours: Wednesdays 9:30 - 11:30 in 716A Mathematics, or by appointment if that time does not work

Email: danhl@math.columbia.edu

TAs:

Donghan Kim - dk2571@columbia.edu

Yihao Huang - yh2771@columbia.edu

Peiran Fang - pf2317@columbia.edu

Piazza:

Most of our class communication will take place on our Piazza page:

https://piazza.com/columbia/spring2017/mathv2500/home

Textbook:

The official textbook is:

-Further Mathematics for Economic Analysis, Second Edition by K. Sydsaeter, P. Hammond, A. Seierstand, and A. Strom (SHSS).

We will also be using two texts which are available online (the first is free, and from the second we will only be using chapters which are freely available online):

-Pinkham, Analysis, Convexity, and Optimization, and

-Larson, Edwards, Falvo, (LEF) Elementary linear algebra.

Most course topics will be covered in both SHSS and Pinkham.

Course Goals:

This course focuses on a basic mathematical question which arises in many contexts in economics, engineering, and the social and physical sciences: given a function of several variables, how can one find an assignment of those variables, subject to some constraints, which minimizes or maximizes the value of the function. This is called the “optimization problem.”

We will cover aspects of multi-variable calculus, linear algebra, and mathematical analysis which are most relevant to this question, and discuss several techniques for solving problems in both constrained and unconstrained optimization, specifically:

- Linear programming;
- Implicit function theorem, Lagrange multipliers;
- Convex optimization, Kuhn-Tucker conditions;
- Calculus of variations (if time permits).

Prerequisites: Calculus III and Linear Algebra. We will review some of the relevant background, but at a pace and level that assumes you have seen it before. Please speak with me if you are not sure if you have the background for for the course.

Exams: There will be two 75-minute midterm exams and a final. The dates for the midterms are as follows:

- First Midterm February 16 in class
- Second Midterm March 28 in class
- Final: This is scheduled by the university. It must be taken at the scheduled time.

If you have two final examinations scheduled at the same time, it is the responsibility of the other department to provide an alternate exam. Examinations will not be rescheduled because of travel arrangements -- it is your responsibility to schedule travel appropriately. Makeup midterms will be given only under exceptional circumstances and you will need a note from a doctor or a dean.

Homework: Problem sets, every 7-10 days, due in the homework box of the fourth floor of the Mathematics building. Collaboration and discussion with your classmates is encouraged, but I also encourage you to think about the problems on your own before you discuss them with friends. You must write up assignments individually. I will drop the lowest two homework grades from your average. Late homework will not be accepted.

Grading: The final course grade will be determined by:

Homework: 15%

Midterm 1: 20%

Midterm 2: 25%

Final exam: 40%.

Help: Besides my office hours, help is also available without appointment in the Mathematics Help Room (406 Mathematics) whenever it is open.The Help Room is staffed both by faculty and teaching assistants, who will be able to help you with questions related to this course.