CS 726: Nonlinear Optimization I (Fall 2022)
Class is also listed as ISYE 726/MATH 726/STAT 726.
Instructor: Jelena Diakonikolas
Email: jelena at cs dot wisc dot edu
Office hours: Friday 12-2pm, or by appointment, in CS 4369.
Communication policy: I try to respond to all emails, but during the semester my email load may become too high, in which case I may miss responding to some emails. If your question is urgent and I do not respond promptly, please send me a reminder. For all non-urgent class-related questions, please use the class Piazza (accessible from Canvas) and/or one of the office hours slots.
Teaching Assistant: Puqian Wang
Email: pwang333 at wisc dot edu
Office hours: Tuesday and Thursday 4-5pm (after class), in CS 4331.
This class meets in Engineering Hall, Rm 2317.
The class is scheduled for Tue-Thu 2.30-3.45pm (75 min).
General Course Information
Most of the class is theoretical and assumes mathematical maturity: you need to be comfortable with reading, understanding, and writing proofs. Basic background in linear algebra, real analysis, and probability is expected.
Some of the homework problems will require coding in Python, for which basic knowledge is assumed.
We will use the following two textbooks for some of the topics:
For the topics not covered by the textbooks additional lecture materials will be shared on Canvas.
Additional books and resources that you may find useful:
- Y. Nesterov, Lectures on Convex Optimization, Springer, 2018.
- A. Beck, First-order Methods in Optimization. Vol. 25. SIAM, 2017.
- S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
- R.T. Rockafellar, R. J-B Wets, Variational Analysis, Springer, 1998.
- D. P. Bertsekas, with A. Nedic and A. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, 2003.
- Dmitriy Drusvyatskiy's course notes on convex analysis and optimization, 2019.
This class focuses on foundations of iterative (first- and second-order) optimization algorithms. In particular, the focus is on understanding and rigorously characterizing when, why, and how well different optimization methods work. The coding assignments are used for illustrating the performance of different optimization methods on some characteristic examples. While we will occasionally mention some interesting applications, this class does not focus on applications or modeling. For these two topics, students may consider CS 524 and different machine learning classes.
This is a tentative list of topics that will be covered in class. Most of the topics listed here will be covered, and some other topics may be added.
- Introduction and background: general continuous optimization background; p-normed real spaces; convex sets; convex functions; Lipschitz continuous and smooth functions; strongly convex functions; local error bounds; Taylor theorem; optimality conditions.
- First-order methods: gradient descent for convex and nonconvex optimization, line search methods, projected gradient descent, mirror descent, Nesterov acceleration for convex optimization, conjugate gradients, conditional gradients (Frank-Wolfe methods), basic coordinate descent, stochastic gradient descent.
- Second-order methods: Newton method, trust-region Newton, inexact Newton methods and Newton-CG, cubic regularization, quasi-Newton methods (DFP, BFGS, SR-1, general Broyden class), limited-memory quasi-Newton (L-BFGS).
This is a graduate-level class that teaches fundamentals of nonlinear optimization, and, as such, will have a high load, requiring strong commitment to mastering the material. Your focus should not be on the grade -- it should be on learning. The instructor's point of view is that if you go through the class without feeling challenged at all, then you are working below your potential. However, if the class becomes overwhelming and is causing you distress, you are encouraged to come talk to us, and we will look into possible accommodations.
All grades will be posted on Canvas. The information provided here is tentative and is subject to change.
Homework: There will be 5-6 homework assignments, accounting for ~50% of the grade. You may discuss problems with other students, but you need to declare it on your homework submission. Any discussion can be verbal only: you are required to work out and write the solutions on your own. Submitting someone else's work as your own constitutes academic misconduct. Academic honesty is taken very seriously in this class, and any breach of it will be treated according to the University Policy.
Homework assignments and solutions will be posted on Canvas.
Midterm: Date and Time: TBD. Held in class. Accounts for ~20% of the grade.
Final: Scheduled for 12/20/2021 10.05am-12.05pm. Location: TBD. Accounts for ~30% of the grade.
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