Convex Analysis and Optimization

Convex optimization is essential to a range of current scientific and engineering applications, including machine learning, signal processing, and control systems. It is also forms the backbone for other areas of optimization. The aim of this course is to provide a self-contained introduction to basic concepts in convex analysis and its use in convex optimization.

This course is cross-listed as both CS542F (Topics in Numerical Computation) and MATH 604 (Topics in Optimization).

Syllabus

This list represents a tentative outline of the topics that will be covered.

Part 1: Convex sets

• convexity-preserving operations, derived sets (closure, relative interior, extreme points, faces)
• projection onto convex sets, separation
• conic approximations: tangent and normal cones

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
• conjugate functions

Part 3: Convex optimization

• problem types: linear and quadratic optimization, conic programming (second-order and semi-definite optimization)
• applications of sparse optimization, including compressed sending, matrix completion
• Lagrange and conjugate duality
• first-order methods for unconstrained and nonsmooth problems, including proximal gradient, ADMM, Frank-Wolfe
• interior methods for conic problems

Target audience

This course is intended for students who wish to learn the underpinnings of convex optimization and are considering research in the area. Students looking to gain more practical experience with optimization (e.g., how to use various solvers) may wish to instead consider CPSC 406, which will be taught next year.

Prerequisities

Background in vector calculus, linear algebra, and basic real analysis.

Grading

• Homework assignments. Three homework assignments involving both theoretical and computational exercises.
• Lecture scribe. Every student will transcribe notes for one or more lectures (depending on the class size). These notes will be made available to others in the class.
• Grade distribution. 25% each homework; 25% lecture scribe.

Auditors and undergraduates

Auditors are welcome. Graduate students who wish to audit, please bring a graduate registration form to the first lecture. Undergraduate students who wish to take the course for credit should fill out an undergraduate registration form.

References

The course isn’t based on any one particular text. These references should be helpful for further reading.

• 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.

Piazza

Sign up for the Piazza course page.