Course content

This course is an introduction to optimisation techniques for the solution of non-smooth optimisation problems on Hilbert spaces, with a focus on convex problems. Topics to be discussed include: convex functions on Hilbert spaces; subdifferential calculus; Fenchel duality; monotone operators and proximal points; splitting methods for optimisation. In addition, further specialisation in a selected topic in optimisation: possible topics include optimal control for partial differential equations, optimisation on manifolds, inverse problems, or set and vector optimisation.

Learning outcome

After meeting the learning objectives of the course, the student will be able to:

• understand the challenges of non-smooth optimisation problems;
• explain the underlying principles of modern methods for non-smooth optimisation;
• apply subdifferential calculus for the investigation of convex functions;
• calculate the Fenchel dual of a convex function;
• derive optimality conditions for a given optimisation problem;
• recognise advantages and disadvantages of different types of splittings;
• assess convergence speed and computational complexity of optimisation algorithms;
• implement optimisation algorithms on a computer;
• apply optimisation algorithms to the solution of model problems;
• have a deeper knowledge of the specialisation topic.

Learning methods and activities

Lecture and mandatory project work.

Compulsory assignments

• Project

Recommended previous knowledge

The course TMA4145 Linear Methods or equivalent. The courses TMA4180 Optimisation 1 and TMA4215 Numerical Mathematics (or equivalent) can be an advantage.

Course materials

Will be announced at the start of the semester.