CPSC 536M:
Optimization Theory

The convex geometry of structured optimization

Convex optimization is a key tool for analyzing and solving a range of computational problems that arise in machine learning, statistical signal and image processing, theoretical computer science, and other fields. It’s also the backbone for other areas of optimization, including algorithms for nonconvex problems. This course aims to provide a self-contained introduction to a few of the many geometric and intuitive ideas in convex analysis and their usefulness for developing and understanding computationally-efficient algorithms for a range of scientific and engineering applications.

In addition to convexity, a recurring theme in the course is the convex geometry of structured optimization, which involves problems whose solutions exhibit a latent structure, and are assembled using a prescribed set of atoms. As has been recognized in the optimization and related communities for at least the past decade, significant computational efficiencies are made available by exploiting this latent structure, coupled with the overarching structure provided by convexity. The course will emphasize the rich convex geometry that underlies atomic decomposition and its algorithmic manifestations.

Syllabus

Here is a broad overview of the topics covered.

Part 1: Convex sets

convexity-preserving operations, generated sets, atomic sets
projection and separation of sets
cones and polarity

Part 2: Convex functions

epigraphical view
convexity-preserving operations, infimal convolution, perspective transform, smoothing
differentiable and nondifferentiable functions
support functions: correspondence with convex sets; norms and duals; gauges and polars
dual operations

Part 3: Convex optimization

problem types: linear and quadratic optimization, conic programming
applications of structured optimization, including compressed sensing, matrix completion, superresolution
Lagrange and conjugate duality
first-order methods for unconstrained and nonsmooth problems, including subgradient, proximal gradient, and conditional gradient methods

Target Audience

This course is intended for students who wish to learn the underpinnings of optimization and are considering research related to this area. Students looking to gain more practical experience with optimization (e.g., how to use various solvers) should instead consider CPSC 406 (Computational Optimization), taught during Term 2.

Prerequisites

Background in vector calculus, numerical linear algebra, and basic real analysis. The course is unapologetically mathematical, and a casual background in these topics will not be sufficient. Consider taking this online self-assesment on numerical linear algebra.

Grading

Homework assignments. Several assignments involving both theoretical and computational exercises.
Course project. A project will be assigned that will require deriving, implementing, and applying an optimization algorithm to solve a problem numerically. A short report (4-6 pages) will be required and will be evaluated for mathematical correctness, quality of the software implementation, and presentation. This will be due in the last week of term.
Grade distribution (tentative). 50% homework; 50% class project.

Auditors and Undergraduates

Auditors welcome. Graduate students who wish to audit should file a graduate registration form with the CS front office. Undergraduate students who wish to take the course for credit should request permission from the instructor and file an undergraduate registration form to the CS front office.

References

These references should be helpful to supplement course notes:

Bertsekas, D. P., Convex optimization theory. Athena Scientific, 2009
Boyd, S., and Vandenberghe, L. Convex optimization. Cambridge University Press, 2004
Hiriart-Urruty, J.-B., and Lemaréchal, C., Fundamentals of convex analysis. Springer, 2012
Rockafellar, R. T., Convex analysis. Princeton University Press, 2015
Rockafellar, R. T., and Wets, R. J-B., Variational analysis. Springer, 2009