University of Chicago

Winter 2021

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.

- 12/21/20: This Winter's class will be conducted entirely on Canvas. This webpage would be used only as a mirror repository for course materials.

**Location:** Lectures held online through Canvas

**Times:** Mon & Wed, 4:10–5:30pm

**Instructor:** Lek-Heng
Lim

Office: Jones 122C

`lekheng(at)galton.uchicago.edu`

Tel: (773) 702-4263

Office hours: Held online through Canvas, Tue, 9:00–10:30am.

**Course Assistant I:**
Zhen Dai

`zhen9(at)uchicago.edu`

Office hours: Held online through Canvas, Fri, 4:00–5:30am.

**Course Assistant II:**
Zehua Lai

`laizehua(at)uchicago.edu`

Office hours: Held online through Canvas, Thu, 9:00–10:30am.

- 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
- Methods of penalty function, augmented Lagrangian, and barrier function
- 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: Mar 3, due: Mar 14)

- Problem Set 3 (posted: Feb 19, due: Mar 1)

- Problem Set 2 (posted: Feb 7, due: Feb 17)

- Problem Set 1 (posted: Jan 25, due: Feb 4)

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

- Course homepages from Spring 2020, Spring 2019, Spring 2018, 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:** Option 1 – Four problem sets each
counting towards 25% of grade; Option 2 – Four problem sets and
one final exam each counting towards 20% of grade.

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.