Convex Analysis



The course introduces basic and advanced concepts of convex analysis. After definition, generation, and relations of convexity for sets and functions are studied, slightly more advanced tools such as the Moreau envelope, the subdifferential, and Fermat's rule are developed. These tools are fundamental for the study of convex optimization problems, optimality conditions, and algorithms. The second part of the lecture is devoted to the analysis of first order convex optimization algorithms that are ubiquitious in data science applications such as Image Processing, Computer Vision, Machine Learning and Statistics. Then, the study of convex duality allows us to introduce widely used primal-dual algorithms.


This course is devoted to the mathematical fundaments. However, convex optimization techniques are widely applied in machine learning and computer vision and will allow a more theoretical understanding of many aspects of these fields. Application examples will illustrate its utility.



Course Information

Semester:  SS

Year:  2022

Lecture start: Wednesday April 13

Tutorial start:  Wednesday April 13



lecture: Wednesdays 08:00 - 10:00 and Fridays 08:00 - 10:00 (start at 08:15)

tutorial: Wednesdays 12:00 - 14:00


Location: E024 in the building of MPI-INF.


Zoom room: send an email to ahmed.abbas[at] for information.


Registration:  send an email with your matriculation number and full name to ahmed.abbas[at] with [convex analysis-subscribe] in the subject.


Lecturer(s):   Dr. Paul Swoboda

TA(s):              Ahmed Abbas (office hour: Fridays 3:30 pm - 5:30 pm, contact by email ahmed.abbas[at]




  • Rockafellar: Convex Analysis. Princeton University Press, 1970
  • Y. Nesterov: Introductory Lectures on Convex Optimization - A Basic Course. Kluwer Academic Publishers, 2004
  • D. P. Bertsekas: Convex Analysis and Optimization. Athena Scientific, 2003
  • S. Boyd: Convex Optimization. Cambridge Univeristy Press, 2004
  • H. H. Bauschke and P. L. Combettes: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, 2011
  • T. Rockafellar and R. J.-B. Wets: Variational Analysis, Springer-Verlag Berlin Heidelberg, 1998