University of Chicago

Spring 2017

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.

- 05/26/17: Reminder: Quiz II from 3:00–4:20pm on Tue, May 30, in Eckhart 133.

- 05/26/17: Lecture 18 notes posted.

- 05/23/17: Office hours from 3:00–5:00pm on Wed, May 24, in Jones 122B. Greg has moved his office hours this week to 1:00–3:00pm on Fri, May 26, in Gates-Blake 401.

- 05/23/17: Lecture 17 notes posted.

- 05/18/17: Lecture 16 notes posted.

- 05/17/17: Lecture 15 notes posted.

- 05/12/17: Lecture 14 notes and Homework 4 posted.

- 05/09/17: Office hours from 1:00–3:00pm on Mon, May 10, in Jones 122B. Greg will also hold his regular office hours from 4:00–5:00pm in Jones 226.

- 05/09/17: Lecture 13 notes posted.

- 05/05/17: Lecture 12 notes posted.

- 05/02/17: Lecture 11 notes and Homework 3 posted.

- 04/28/17: Lecture 10 notes posted. Greg will hold office hours from 2:00–4:00pm on Fri, Apr 28, in Cobb 102.

- 04/27/17: Office hours from 1:00–3:00pm on Mon, May 1, in Jones 122B.

- 04/25/17: Lecture 9 notes posted.

- 04/18/17: Lecture 8 notes and Homework 2 posted.

- 04/18/17: Reminder: Quiz I from 3:00–4:20pm on Thu, Apr 20, in Eckhart 133.

- 04/14/17: Lecture 7 notes posted.

- 04/14/17: Office hours from 2:00–4:00pm on Mon, Apr 17, in Jones 122B.

- 04/11/17: Lecture 6 notes posted. Greg will hold office hours from 3:00–5:00pm on Fri, Apr 14, in Cobb 101.

- 04/10/17: Lecture 5 notes posted.

- 04/06/17: Homework 1 and Lecture 4 notes posted.

- 04/04/17: Lecture 3 notes posted.

- 03/31/17: Lecture 2 notes posted.

- 03/29/17: Lecture 1 notes posted.

- 03/28/17: Make-up lecture Friday, Apr 7, 5:00–8:00pm in Eckhart 133.

- 03/27/17: Quiz I on Thu, Apr 20. Quiz II on Tue, May 30.

- 03/27/17: 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-hour session before problem set is due, Jones
122B

**Course Assistant:** Greg
Naisat

Office: Jones 226

`gregn(at)galton.uchicago.edu`

Office hours: Wed, 4:00–5:00pm

- 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 12, due: May 25)

- Problem Set 3 (posted: May 02, due: May 11)

- Problem Set 2 (posted: Apr 18, due: May 02)

- Problem Set 1 (posted: Apr 06, due: Apr 18)

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

- Course homepages from Spring 2016, 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% Quizzes.

**Midterm exams:** Quiz I on Thu, Apr 20, 3:00–4:20pm,
Eckhart 133. Quiz II on Tue, May 30, 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.