ORF522 2020: Linear and Nonlinear Optimization
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:
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Model decision-making problems across different disciplines as mathematical optimization problems.
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Apply the most appropriate optimization tools when faced with a concrete problem.
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Implement optimization algorithms and prove their convergence.
Office hours
Instructor
Name: Bartolomeo Stellato
Office: Zoom Meeting Room
Time: Wed 2:00pm – 4:00pm EST
Email: bstellato@princeton.edu
Assistant in instruction
Name: Zheng Yu
Office: Zoom Meeting Room
Time: Fri 2:00pm – 4:00pm EST
Email: zhengy@princeton.edu
Schedule
All lectures will be streamed from this Zoom link and recorded with videos available on Canvas.
Linear optimization
| # | Date | Topic | Slides | HW |
|---|---|---|---|---|
| 1 | 09/01 | Introduction | 01_lec.pdf 01_lec_notes.pdf | |
| 2 | 09/03 | Linear optimization | 02_lec.pdf 02_lec_notes.pdf | |
| 3 | 09/08 | Geometry and polyhedra | 03_lec.pdf 03_lec_notes.pdf | 1 Out |
| 4 | 09/10 | Simplex method I | 04_lec.pdf 04_lec_notes.pdf | |
| 5 | 09/15 | Simplex method II | 05_lec.pdf 05_lec_notes.pdf | 1 Due |
| 6 | 09/17 | Linear algebra and simplex implementation | 06_lec.pdf 06_lec_notes.pdf | |
| 7 | 09/22 | Linear optimization duality I | 07_lec.pdf 07_lec_notes.pdf | 2 Out |
| 8 | 09/24 | Linear optimization duality II | 08_lec.pdf 08_lec_notes.pdf | |
| 9 | 09/29 | Sensitivity analysis | 09_lec.pdf 09_lec_notes.pdf | 2 Due |
| 10 | 10/01 | Interior-point methods for linear optimization | 10_lec.pdf 10_lec_notes.pdf | |
| 11 | 10/06 | Interior-point methods implementation | 11_lec.pdf 11_lec_notes.pdf | |
| 10/08 | Midterm |
Nonlinear optimization
| # | Date | Topic | Slides | HW |
|---|---|---|---|---|
| 12 | 10/15 | Introduction to nonlinear optimization | 12_lec.pdf 12_lec_notes.pdf | |
| 13 | 10/20 | Optimality conditions | 13_lec.pdf 13_lec_notes.pdf | 3 Out |
| 14 | 10/22 | Gradient descent | 14_lec.pdf 14_lec_notes.pdf | |
| 15 | 10/27 | Subgradient methods | 15_lec.pdf 15_lec_notes.pdf | 3 Due |
| 16 | 10/29 | Proximal methods and intro to operator theory | 16_lec.pdf 16_lec_notes.pdf | |
| 17 | 11/03 | Operator theory | 17_lec.pdf 17_lec_notes.pdf | 4 Out |
| 18 | 11/05 | Operator splitting algorithms | 18_lec.pdf 18_lec_notes.pdf | |
| 19 | 11/10 | Acceleration schemes | 19_lec.pdf 19_lec_notes.pdf |
Extensions
| # | Date | Topic | Slides | HW |
|---|---|---|---|---|
| 20 | 11/12 | Sequential convex programming | 20_lec.pdf 20_lec_notes.pdf | |
| 21 | 11/17 | Branch and bound algorithms | 21_lec.pdf 21_lec_notes.pdf | 4 Due - 5 Out |
| 22 | 11/19 | Data-driven algorithms | 22_lec.pdf 22_lec_notes.pdf | |
| 23 | 11/24 | The role of optimization | 23_lec.pdf 23_lec_notes.pdf | |
| 12/01 | 5 Due | |||
| 12/10 | Final |
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)
- J. Nocedal, S. J. Wright: Numerical Optimization (available on SpringerLink)
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.
Grading
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). To export a jupyter notebook 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
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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.
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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. 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.
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.