Bartolomeo Stellato

ORF522 2021: Linear and Nonlinear Optimization

Previous years: 2020. Latest year.

Description

This course introduces analytical and computational tools for linear and nonlinear optimization.
Topics include linear optimization modeling, duality, the simplex method, degeneracy, sensitivity analysis and interior point methods.
Nonlinear optimality conditions, KKT conditions, first order and operator splitting methods for nonlinear optimization, real-time optimization and data-driven algorithms.
A broad spectrum of applications in engineering, finance and statistics is presented.

Learning objectives

This course introduces analytical and computational tools for linear and nonlinear optimization.
Upon successful completion of this course you should be able to:

• Model decision-making problems across different disciplines as mathematical optimization problems.

• Apply the most appropriate optimization tools when faced with a concrete problem.

• Implement optimization algorithms and prove their convergence.

Office hours

Instructor

Name: Bartolomeo Stellato
Oﬀice: Sherrerd 323
Time: Thu 2:00pm – 4:00pm EST
Email: bstellato@princeton.edu

Assistant in instruction

Name: Rajiv Sambharya
Oﬀice hours: Sherrerd 003
Time: Fri 11am - 1pm
Email: rajivs@princeton.edu

Name: Scander Mustapha
Oﬀice hours: Sherrerd 003
Time: Mon 1:30pm - 3:30pm
Email: mustapha@g.princeton.edu

Schedule

All lectures will take place in Friend Center 008

Linear optimization

# Date Topic Slides HW References
1 09/02 Introduction 01_lec.pdf
2 09/07 Linear optimization 02_lec.pdf 02_lec_notes.pdf [Ch 1, LO]
3 09/09 Geometry and polyhedra 03_lec.pdf 03_lec_notes.pdf 1 Out [Ch 2, LO]
4 09/14 Simplex method I 04_lec.pdf 04_lec_notes.pdf [Ch 3, LO]
5 09/16 Simplex method II 05_lec.pdf 05_lec_notes.pdf 1 Due [Ch 3, LO]
6 09/21 Linear algebra and simplex implementation 06_lec.pdf 06_lec_notes.pdf [Ch 3, LO] [Ch 13, NO] [Ch 8, LP]
7 09/23 Linear optimization duality I 07_lec.pdf 07_lec_notes.pdf 2 Out [Ch 4, LO] [Ch 5, LP]
8 09/28 Linear optimization duality II 08_lec.pdf 08_lec_notes.pdf [Ch 4, LO] [Ch 11, LP]
9 09/30 Sensitivity analysis 09_lec.pdf 09_lec_notes.pdf 2 Due [Ch 5, LO]
10 10/05 Interior-point methods for linear optimization 10_lec.pdf 10_lec_notes.pdf [Ch 14, NO] [Ch 17 and 18, LP]
11 10/07 Interior-point methods implementation 11_lec.pdf 11_lec_notes.pdf [Ch 14, NO] [Ch 22, LP]

Nonlinear optimization

# Date Topic Slides HW References
12 10/12 Introduction to nonlinear optimization 12_lec.pdf 12_lec_notes.pdf [Ch 2 to 4 and 6, CO] [Ch A and B, FCA]
10/14 Midterm (only linear optimization)
13 10/26 Optimality conditions 13_lec.pdf 13_lec_notes.pdf [Ch 2 and 12, NO] [Ch 4 and 5, CO]
14 10/28 Gradient descent 14_lec.pdf 14_lec_notes.pdf 3 Out [Ch 1 and 2, ICLO] [Ch 9, CO] [Ch 5, FMO]
15 11/02 Subgradient methods 15_lec.pdf 15_lec_notes.pdf [Ch 3 and 8, FMO] [ee364b] [Ch 3, ILCO]
16 11/04 Proximal methods and intro to operator theory 16_lec.pdf 16_lec_notes.pdf 3 Due [Ch 3 and 6, FMO] [PA] [PMO]
17 11/09 Operator theory 17_lec.pdf 17_lec_notes.pdf [Ch 4, FMO] [PA] [PMO] [LSMO]
18 11/11 Operator splitting algorithms 18_lec.pdf 18_lec_notes.pdf 4 Out [PMO] [PA] [LSMO] [ADMM]
19 11/16 Acceleration schemes 19_lec.pdf 19_lec_notes.pdf [Ch 1, FMO] [Ch 2, ILCO] [Ch 3, COAC]

Extensions

# Date Topic Slides HW References
20 11/18 Sequential convex programming 20_lec.pdf 20_lec_notes.pdf [Ch 4 and 17, NO] [ee364b]
11/25 4 Due - 5 Out
21 11/30 Branch and bound algorithms 21_lec.pdf 21_lec_notes.pdf [ee364b] [MINLO]
22 12/02 Conclusion 22_lec.pdf
12/09 5 Due
12/16 Final

Material

The lecture notes are available from the course website and intended to be self contained. The following books and monographs are useful as reference texts.
They are either free or digitally available via Princeton University library:

• [LP] R. J. Vanderbei: Linear Programming: Foundations & Extensions (available on SpringerLink)
• [LO] D. Bertsimas, J. Tsitsiklis: Introduction to Linear Optimization (available Princeton Controlled Digital Lending)
• [NO] J. Nocedal, S. J. Wright: Numerical Optimization (available on SpringerLink)
• [FMO] A. Beck: First-order methods in optimization (available on SIAM)
• [FCA] J. B. Hiriart-Hrruty, C. Lemarechal: Fundamentals of Convex Analysis (available on SpringerLink)
• [ILCO] Y. Nesterov: Introductory Lectures to Convex Optimization (available on SpringerLink)
• [e364b] S. Boyd: Convex Optimization II Lecture Notes (available online)
• [PMO] E. K. Ryu, S. Boyd: A primer on monotone operators (available for free)
• [LSMO] E. K. Ryu and W. Yin: Large-Scale Convex Optimization via Monotone Operators (Draft) (available for free)
• [ADMM] S. Boyd, N. Parikh, B. Peleato, J. Eckstein: Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers (available for free)
• [COAC] S. Bubeck: Convex Optimization: Algorithms and Complexity (available for free)
• [MINLO] P. Belotti, C. Kirches, S. Leyffer, J. Linderoth, J. Luedtke, A. Mahajan: Mixed-integer nonlinear optimization (available online)

In this course we strictly follow the crimes against matrices laws!

Code used in the lectures will be available on the github companion repo.

Prerequisites

• Good knowledge of linear algebra and calculus. For a refresher, we recommend reading Appendices A and C of the S. Boyd and L. Vandenberghe Convex Optimization (available online).
• Familiarity with Python.

Software

Students will use the Python-based modeling software CVXPY (cvxpy.org) to solve optimization problems arising in several applications in operations research, finance, machine learning and engineering.

The assignments will be using jupyter notebooks. We suggest you edit them with Google Colab linked with your Google Drive (recommended way). Alternatively, you can run them locally by installing jupyter lab and the required packages on your machine. Please follow these instructions to complete the setup.

All submissions should take place on Gradescope (accessible from the canvas website).

• 25% Homeworks. 5 bi-weekly homeworks. Almost all of them will include a computational component. Homeworks are due at 11:59pm EST of the date specified. Requests for extension on homework will not be accepted, unless there is an extremely valid reason. Homeworks must always be submitted as a single pdf file which includes your written exercises (typed or handwritten) and code (pdf-exported jupyter notebook). You have three options to export your notebooks:
• On Colab: To export as pdf just click on: File → Print → Save as PDF. If this does not work or your some of your images are cut use the following
• On Colab: To export as pdf run the following commands:
# Install required packages!apt-get install texlive texlive-xetex texlive-latex-extra pandoc cm-super dvipng!pip install pypandoc# Mount Google Drivefrom google.colab import drivedrive.mount('/content/drive')
Then you can go to the notebook directory on your drive and export it, for example
%cd drive/MyDrive/orf522/homeworks/01_homework/!jupyter nbconvert --to PDF "ORF522_HW1.ipynb"
• Local jupyter: To export as pdf just click on: File → Download as → PDF via LaTeX (.pdf).
• 25% Midterm. 90 minutes written exam. No coding required.
• 40% Final. Take-home final exam with written and computational questions.
• 10% Participation. Students are expected to submit one question or note on each lecture on Ed. The note should summarize what you learned in the last lecture, and highlight the concepts that were most confusing or that you would like to review. A note will receive full credit if: it is submitted before the beginning of next lecture, it is related to the content of the lecture, and it is understandable and coherent. You can make the note private (visible only by you and the course staff) or public, as you choose.

Questions and discussions

Students are encouraged to discuss and ask questions on Ed.
Please make sure to specify if questions are about General information of the course, about the Lectures or about Homeworks by assigning them to the related category.

Collaboration policy

• Homeworks. Students are allowed, and even encouraged, to collaborate on homeworks. When submitting your homework, you are required to list the name of the students you worked with. Also, please write the textbooks, notes or websites that were helpful to you.

• Midterm and final. No collaborations allowed.

Honor code

All work in this course must uphold the University’s commitment to academic integrity. This includes the Honor Code (for written examinations, tests, and quizzes) and guidelines for the submission of original work on all other assignments. More guidance can be found in Rights, Rules, and Responsibilities as well as the handbook Academic Integrity at Princeton.

Attendance

Students are expected to attend each scheduled class on time and ready to participate fully. An excused absence will only be granted in the case of a religious observance, an ODS-approved accommodation, or -- as verified by your residential college -- a serious illness or an exceptional circumstance.