University of Chicago

Spring 2016

This is an introductory course on optimization that will cover the rudiments of unconstrained and constrained optimization of a real-valued multivariate function. The focus is on the settings where this function is, respectively, linear, quadratic, convex, or differentiable. Time permitting, topics such as nonsmooth, integer, vector, and dynamic optimization may be briefly addressed. Materials will include basic duality theory, optimality conditions, and intractability results, as well as algorithms and applications.

- 06/08/16: Office hours 2:00–3:30pm on Thu, Jun 09.

- 05/30/16: Vivak will hold office hours 3:00–3:50pm today in the Regenstein cafe.

- 05/25/16: Office hours 1:00–2:30pm on Fri, May 27.

- 05/17/16: Homework 4 posted, due May 31 in class.

- 05/17/16: Reminder: Midterm Exam this Thursday, May 19, 3:00–4:20pm, Eckhart 133. Closed book, closed notes, no cheat sheet.

- 05/12/16: Office hours 2:30–4:00pm on Fri, May 13 and 1:00–2:30pm on Mon, May 16.

- 05/05/16: Homework 3 posted, due May 17 in class.

- 05/05/16: Make-up lecture on Mon, May 9, 5:30–7:30pm in E133.

- 04/26/16: Office hours 1:00–3:00pm on Mon, May 2 in Jones 122B.

- 04/25/16: Reminder: Midterm Exam this Thursday, Apr 28, 3:00–4:20pm, Eckhart 133. Closed book, closed notes, no cheat sheet.

- 04/24/16: Handout 1 providing some linear algebra background posted.

- 04/19/16: Homework 2 posted, due May 3 in class.

- 04/14/16: Office hours 3:30–5:00pm on Fri, Apr 15 and 2:00–3:30pm on Mon, Apr 18.

- 04/06/16: Homework 1 posted, due Apr 19 in class.

- 04/06/16: Class will meet as usual on Thu, Apr 7.

- 04/01/16: Exam dates: Midterm I on Apr 28, Midterm II on May 19, 3:00–4:20pm, in E133.

- 03/30/16: Make-up lectures on Apr 4 and Apr 11, 5:30–7:30pm in E133.

- 03/30/16: Make-up lecture originally scheduled for today (Mar 30) has been rescheduled.

- 03/29/16: Class will not meet on Thu, Apr 21.

- 03/29/16: Check back regularly for announcements.

**Location:** Eckhart
Hall, Room 133.

**Times:** Tue & Thu, 3:00–4:20pm

**Instructor:** Lek-Heng
Lim

Office: Jones 122B

`lekheng(at)galton.uchicago.edu`

Tel: (773) 702-4263

Office hours: Two 1.5-hour sessions before problem set is due, Jones
122B

**Course Assistant:** Vivak
Patel

Office: Ryerson N375

`vpatel(at)galton.uchicago.edu`

Office hours: 3:00–3:50pm, Harper Memorial 150

- Optimization of a univariate real-valued function
- Multivariate real-valued functions
- Higher-order derivatives of multivariate functions
- Multivariate Taylor expansion with remainder in mean-value form and integral form
- Unconstrained optimization
- Newton, quasi-Newton, conjugate gradient, and steepest descent methods
- Constrained optimization
- Implicit function theorem and Lagrange multipliers
- KKT conditions
- Nonlinear optimization
- Convex sets and convex functions
- Convex optimization
- Newton and steepest descent methods revisited
- Linear programming
- Basic duality theory
- Brief overview of nonsmooth, integer, vector, and dynamic optimization

Collaborations are permitted but you will need to write up your own solutions and declare your collaborators. The problem sets are designed to get progressively more difficult. You will get at least six days for each problem set.

You are required to implement your own programs for problems that require some amount of simple coding (using Matlab, Mathematica, R, or SciPy).

- Problem Set 4 (posted: May 17, due: May 31)

- Problem Set 3 (posted: May 7, due: May 17)

- Problem Set 2 (posted: Apr 19, due: May 5)

- Problem Set 1 (posted: Apr 6, due: Apr 19)

**Bug report** on the problem sets:
`lekheng(at)galton.uchicago.edu`

- Course homepage from Spring 2015.

- These are slightly more advanced versions of this course, intended primarily for graduate students: Stat 31015 Winter 2015, Stat 31020 Winter 2009–2012.

**Grade composition:** 60% Problem Sets, 40% Midterms.

**Midterm exams:** Midterm I on Apr 28, 3:00–4:20pm, Eckhart
133. Midterm II on May 19, 3:00–4:20pm, Eckhart 133. Closed book,
closed notes, no cheat sheet.

We will not use any specific textbook but will use selected material from the following references, all of which would be accessible to undergraduates.

You may download all these books online from an UChicago IP address or via ProxyIt! if you are off-campus.

- E. Çinlar and R. J. Vanderbei, Real and Convex Analysis, Springer, 2013.

- J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, 1996.

- G. Hurlbert, Linear Optimization: The Simplex Workbook, Springer 2010.

- J. Nocedal and S. J. Wright, Numerical Optimization, 2nd Ed, Springer, 2006.

- P. Pedregal, Introduction to Optimization, Springer, 2004.