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Nonlinear Optimization II - 553.763 (Spring 2025)

This course builds upon Nonlinear Optimization I to explore advanced topics in optimization theory and algorithms. Our primary focus will be on constrained optimization and duality theory. Topics include: convex analysis fundamentals and optimality conditions for constrained problems; Fenchel and Lagrangian duality theory; classical algorithmic solutions, such as, the simplex method and interior point methods, and modern splitting methods, e.g., ADMM and PDHG; and elements of variational analysis for handling nonsmooth and nonconvex problems.

Coordinates #

Time: MW 12:00M - 1:15PM
Location: Bloomberg 278

Personnel #

Instructor:
Mateo Díaz (mateodd at jhu dot edu)
OH Th 3:00PM - 4:30PM Wyman S429

Teaching Assistants:
Pedro Izquierdo Lehmann (pizquie1 at jhu dot edu)
OH Tu 9:15-10:00am Wyman S425

Thabo Samakhoana (tsamakh1 at jhu dot edu)
OH Fr 3:30-5:00pm Wyman S425

Lecture notes #

The lecture notes will be posted here.

Homework #

Assignments (approximately five) will be posted here and on the course Canvas. Most homework assignments include at least one question that involves the writing and testing of code. Python is preferred, and I will use it in my demos.

Textbook #

We will not be following any particular textbook. Notes will be posted on this website. Other potentially useful textbooks as references (but not required):

  • J. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization, Springer, (2006),
  • J. Nocedal and S. Wright, Numerical Optimization, Second Edition, Springer, (2006),
  • R. T. Rockafellar, Convex Analysis, Princeton University Press (1970)
  • J.-F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer (2000)
  • R. T. Rockafellar and R. J-B. Wets, Variational Analysis, Springer (1998)
  • H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer (2011)
  • D. Drusvyatskiy, Convex Analysis and Nonsmooth Optimization, Lecture notes, (2020) PDF.

Grading system #

We will use an ingenious grading scheme invented by Ben Grimmer for one of the previous iterations of this class. Course grades will be based on four components: Homework, Midterm, Final, Participation. These are described individually below. I will maximize the course score that I give each of you. To optimize each student’s course score, I will solve the following optimization problem.

Denote the student’s performance in each of these four components as

\[ \begin{array}{rl} C_H &= \text{Homework score},\\ C_M &= \text{Midterm score},\\ C_F &= \text{Final score, and}\\ C_P &= \text{Participation score}. \end{array}\] All of the above are scores out of \(100\). The optimization problem will be solved individually for each student to give them the best rubric and highest course score I can justify. Then, a rubric for grading this student is given by selecting weights for these four components as \[ \begin{array}{rl} H &= \text{Homework weight}, \\ M &= \text{Midterm exam weight}, \\ F &= \text{Final exam weight}, \\ P &= 100 - H - M - F. \end{array}\] Notice the participation weight is determined by the other three since they must sum to 100. Each student’s score is given by maximizing over the set of all reasonable rubrics \((H, M, F)\) by solving \[\begin{array}{rl} \text{max} & C_H H + C_M M + C_F F + C_P (100 - H - M - F)&&\\ \text{subject to} & \\ & \begin{array}{rllll} && (H, M, F) &\in \mathbb{R}^3&\\ && H + M + F &\leq 100 &\text{(Percentages are at most 100)}\\ 15 &\leq& H, M & &\text{(Homework and Midterm matter)}\\ M &\leq& F &&\text{(Final is more important than Midterm)}\\ 50 & \leq& M + F & \leq 80 & \text{(Exams are most, but not all of the score)}\\ 90 & \leq& H + M + F && \text{(H, M, and F are the majority of the score)}. \end{array} \end{array}\]

Midterm and Final #

Additional assessment will be based on one midterm exam and the final exam, both will be take home. The dates of the midterm and final exams will be posted on the course Canvas as they become available, although the date of the final exam is determined by the official JHU final exam schedule.

Participation #

The grading program described above will assign between 0 and 10 percent of the course to be a participation grade. As a result, you can get full marks in the course without any participation. Students will receive full points in participation for engaging during lectures, engaging in office hours, and/or asking insightful questions.