ORF307: Optimization

Description

This course focuses on analytical and computational tools for optimization. We will introduce least squares optimization with multiple objectives and constraints. We will also discuss linear optimization modeling, duality, the simplex method, degeneracy, interior point methods and network flow optimization. Finally, we will cover integer programming and branch-and-bound algorithms. A broad spectrum of real-world applications in engineering, finance and statistics is presented.

Learning objectives

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

  • Model decision-making problems across different disciplines as least squares, linear and integer optimization problems.
  • Apply the most appropriate optimization tools when faced with a concrete problem.
  • Understand which algorithms are slower or faster, and which problems are easier or harder to solve.

Office hours

Instructors

Name: Bartolomeo Stellato
Office hours: Thu 5:00pm – 6:30pm EST
Website: https://stellato.io
Email: bstellato@princeton.edu

Name: Paul Goulart
Office hours: Wed 11:00am - 12:30pm EST
Website: http://users.ox.ac.uk/~engs1373/
Email: paul.goulart@eng.ox.ac.uk

Assistants in instruction

Name: Rajiv Sambharya (Head AI)
Office hours: Mon 4:30pm-6pm
Email: rajivs@princeton.edu

Name: Brian H. Cheung
Office hours: Fri 9:00am-10:30am
Email: bcheung@princeton.edu

Name: Hao Lu
Office hours: Mon 1:30pm-3:00pm
Email: haolu@princeton.edu

Schedule

Lectures

Lectures will be on Tuesdays and Thursdays 1:30pm-2:30pm. We will stream live on Zoom with link and recordings available on Canvas. The schedule is as follows (subject to change):

Least squares

# Date Topic Slides Homeworks
1 02/02 Introduction 01_lec.pdf 01_lec_ann.pdf
2 02/04 Solving linear systems in practice 02_lec.pdf 02_lec_ann.pdf 1 Out
3 02/09 Least squares 03_lec.pdf 03_lec_ann.pdf
4 02/11 Least squares data-fitting 04_lec.pdf 04_lec_ann.pdf 2 Out
5 02/16 Multiobjective least squares 05_lec.pdf 05_lec_ann.pdf
6 02/18 Constrained least squares 06_lec.pdf 06_lec_ann.pdf 3 Out

Linear optimization

# Date Topic Slides Homeworks
7 02/23 Linear optimization 07_lec.pdf 07_lec_ann.pdf
8 02/25 Piecewise linear optimization 08_lec.pdf 08_lec_ann.pdf 4 Out
9 03/02 Geometry and polyhedra 09_lec.pdf 09_lec_ann.pdf
10 03/04 Applications: data science, control, finance 10_lec.pdf
11 03/09 Simplex method 11_lec.pdf 11_lec_ann.pdf
03/11 Midterm 1
12 03/18 Simplex method implementation 12_lec.pdf 12_lec_ann.pdf 5 Out
13 03/23 Duality 13_lec.pdf 13_lec_ann.pdf
14 03/25 Duality II 14_lec.pdf 14_lec_ann.pdf 6 Out
15 03/30 Sensitivity analysis 15_lec.pdf 15_lec_ann.pdf
16 04/01 Network optimization 16_lec.pdf 16_lec_ann.pdf 7 Out
17 04/06 Interior point methods 17_lec.pdf 17_lec_ann.pdf
18 04/08 Interior point methods II 18_lec.pdf 18_lec_ann.pdf
19 04/13 Linear optimization review 19_lec.pdf 19_lec_ann.pdf
04/15 Midterm 2

Integer Optimization

# Date Topic Slides Homeworks
19 04/20 Integer optimization
20 04/22 Branch-and-bound algorithms 8 Out
21 04/27 Conclusions: the role of optimization
05/06 Final project out
05/14 Final project deadline

Precepts

There will be weekly 50 minutes long precepts on Zoom. The focus will be on problem solving and Python programming. There are 4 available time slots:

  • P01: Tuesdays 7:30pm - 8:20pm
  • P02: Tuesdays 7:30pm - 8:20pm
  • P05: Wednesdays 8:00am - 8:50am
  • P03: Wednesdays 7:30pm - 8:20pm

Material

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

  • R. J. Vanderbei: Linear Programming: Foundations & Extensions (available on SpringerLink)
  • D. Bertsimas, J. Tsitsiklis: Introduction to Linear Optimization (available Princeton Controlled Digital Lending)
  • S. Boyd, L. Vandenberghe: Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares (available online)

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

Precepts material and homework templates are available on the github companion repo.

Prerequisites

  • Linear algebra MAT202 and/or MAT204.
  • Basic computer programming knowledge suggested.

Software

Students will use the Python-based modeling software CVXPY (www.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.

Grading

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

  • 25% Homeworks. 8 weekly homeworks. Almost all of them will include a computational component. Homeworks are due at 9pm 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
      !pip install pypandoc
      # Mount Google Drive
      from google.colab import drive
      drive.mount('/content/drive')

      Then you can go to the notebook directory on your drive and export it, for example

      %cd drive/MyDrive/orf307/homeworks/01_homework/
      !jupyter nbconvert --to PDF "ORF307_HW1.ipynb"
    • Local jupyter: To export as pdf just click on: File → Download as → PDF via LaTeX (.pdf).
  • 40% Midterms. Two 120 minutes written exams at weeks 6 and 11. No coding required.
  • 25% Final Project. 24 hours take-home final project with written and computational questions.
  • 10% Participation. Students are expected to submit one question or note on each lecture on Ed Forum. 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 Forum (link available on Canvas). 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.
  • Midterms and final project. 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. Please have your video (with whatever background you prefer to use, there are many Princeton backgrounds here) turned on, except during class breaks, unless you have an unexpected difficulty or have arranged with me otherwise. 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.