FMA285F

Temporary

Convex Analysis

15

Third-cycle course

7151 (Centre of Mathematical Sciences / Mathematics)

2025-06-16

Maria Sandsten

English

If sufficient demand

On completion of the course, participants shall be able to:
• Describe modern algorithms for minimization of a convex functional, such as e.g. FBS, Dykstra, Douglas-Rachford, ADMM, Chambolle-Pock, and get an overview of pros and cons as well as different areas of application, for each respective algorithm.
• Be able to clarify under which circumstances the above algorithms converge, and in basic terms understand the underlying proof.
• Understand how minimization problems can be reformulated in terms of maximally monotone operators, and its connection to various fixpoint theorems.
• Understand how duality and the Fenchel transform is used to reformulate minimization problems.
• Clarify basic properties of convex functions and its connections to lower semicontinuous functions and subdifferential calculus.
• Understand the role of proximal operators for convex optimization routines.

Theory for classes of non-expansive operators, fixpoint iterations, Fejer monotinicity, Krasnoelskii-Mann’s theorem. Classes of convex functions and their properties, semicontinuous functions. The Fenchel-transform and convex hulls, the Fenchel-Moreau theorem. Subdifferentiability of convex funktions. Monotone operators and proximal operators. Convergence theorems for known algorithms such as those mentioned above.

Describe modern algorithms for minimization of a convex functional, such as e.g. FBS, Dykstra, Douglas-Rachford, ADMM, Chambolle-Pock, and get an overview of pros and cons as well as different areas of application, for each respective algorithm.

Understand how duality and the Fenchel transform is used to reformulate minimization problems.

Clarify basic properties of convex functions and its connections to lower semicontinuous functions and subdifferential calculus.

Understand the role of proximal operators for convex optimization routines.

Determine under which circumstances the above algorithms converge, and in basic terms understand the underlying proof.

Determine when minimization problems can be reformulated in terms of maximally monotone operators, and its connection to various fixpoint theorems.

Lectures

Seminars

Miscellaneous

Self studies followed by presentations of the material by the students.

Oral exam

Miscellaneous

Presentations by the students

Failed, pass

Functional analysis MATP15 or equivalent

Heinz H. Bauschke, Patrick L. Combettes: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer.

FMA285F

2025-06-16

Maria Sandsten