University of Chicago

Spring 2018

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.

- 05/23/18: Review session on Thu, Apr 24, 3:30–4:50, Eckhart 133.

- 05/23/18: Quiz II on Tue, May 29, 3:30–4:50pm, Eckhart 133.

- 05/23/18: Lecture notes 17 posted. All lecture notes + appendix combined into a single file.

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

- 05/17/18: Office hours on Mon, May 21, 3:00–5:00, Jones 122B.

- 05/15/18: Lecture notes 15 posted.

- 05/10/18: Lecture notes 14 and Homework 4 posted.

- 05/08/18: Lecture notes 13 posted.

- 05/03/18: Office hours on Wed, May 9, 1:30–3:30, Jones 122B.

- 05/03/18: Lecture notes 12 posted.

- 05/01/18: Lecture notes 11 and Homework 3 posted.

- 04/19/18: Lecture notes 10 posted.

- 04/17/18: Lecture notes 9 posted.

- 04/17/18: Office hours on Wed, Apr 18, 1:30–3:30, Jones 122B.

- 04/17/18: Review session on Tue, Apr 24, 3:30–4:50, Eckhart 133.

- 04/16/18: Quiz I on Thu, Apr 26, 3:30–4:50pm, Eckhart 133.

- 04/13/18: Lecture notes 8 posted.

- 04/10/18: Lecture notes 7 and Homework 2 posted.

- 04/06/18: Lecture notes 6 posted.

- 04/06/18: Office hours on Mon, Apr 9, 2:30–4:30pm.

- 04/05/18: Lecture notes 5 posted.

- 04/04/18: Lecture notes 4 posted.

- 03/29/18: Lecture notes 3 and Homework 1 posted.

- 03/29/18: Make-up lecture 2 on Wed, Apr 4, 5:00–7:00pm in Saieh 146.

- 03/29/18: Lecture notes 2 posted.

- 03/27/18: Make-up lecture 1 on Wed, Mar 28, 5:00–7:00pm in Kent 107.

- 03/27/18: Lecture notes 1 posted (see email announcement for url).

- 03/27/18: Check back regularly for announcements.

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

**Times:** Tue & Thu, 3:30–4:50pm

**Instructor:** Lek-Heng
Lim

Office: Jones 122B

`lekheng(at)galton.uchicago.edu`

Tel: (773) 702-4263

Office hours: Two-hour session the day before problem set is due, Jones
122B

**Course Assistant I:** Minzhe Wang

Office: Jones 203/204

`minzhew(at)uchicago.edu`

Office hours: Mon, 6:00–7:00pm, in Jones 304

**Course Assistant II:** Ken
Sze-Wai Wong

Office: Jones 203/204

`kenwong(at)uchicago.edu`

Office hours: Wed, 7:00–8:00pm, in Jones 308

**Grader:** Xinyi Ge

`xinyige(at)uchicago.edu`

- 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
- Methods of penalty function, augmented Lagrangian, and barrier function
- 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 10, due: May 22)

- Problem Set 3 (posted: May 1, due: May 10)

- Problem Set 2 (posted: Apr 10, due: Apr 19)

- Problem Set 1 (posted: Mar 29, due: Apr 10)

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

- Course homepages from Spring 2017, 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.

**Quizzes:** Quiz I on Thu, Apr 26, 3:30–4:50pm,
Eckhart 133. Quiz II on Tue, May 29, 3:30–4:50pm, 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.