STAT 28000/CAAM 28000. Optimization

Department of Statistics
University of Chicago
Spring 2020

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.

Announcements

04/03/20: This year's class will be conducted entirely on Canvas. This webpage would be used only as a mirror repository for course materials.

Lectures

Location: Lectures held online through Canvas

Times: Mon & Wed, 3:00–4:20pm

Course staff

Instructor: Lek-Heng Lim
Office: Jones 122C
lekheng(at)galton.uchicago.edu
Tel: (773) 702-4263
Office hours: Held online through Canvas, Fri, 3:00–4:20pm.

Course Assistant I: Zhen Dai
zhen9(at)uchicago.edu
Office hours: Held online through Canvas, Thu, 4:00–5:20pm.

Course Assistant II: Zehua Lai
laizehua(at)uchicago.edu
Office hours: Held online through Canvas, Tue, 4:00–5:20pm.

Syllabus

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

Problem Sets

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 25, due: Jun 4)

Problem Set 3 (posted: May 11, due: May 24)

Problem Set 2 (posted: Apr 27, due: May 10)

Problem Set 1 (posted: Apr 13, due: Apr 24)

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

Supplementary materials

Course homepages from 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.

Implicit function theorems and Lagrange multipliers.

Grades and quizzes

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.

References

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.