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Convex Optimization & applications (Summer term 2022)

We will give with this lecture an introduction to basics of convex optimization theory in infinite dimensional spaces. In particular, the following properties are covered

Part 1

• Convex functions
• Constrained minimization problems
• Convex conjugates
• Proximal maps
• Primal and dual problem formulation
• Minimization schemes, in particular splitting approaches

Part 2

• Algorithms based on forward backward splitting
• Algorithms based on primal dual splitting
• Semi-smooth Newton methods
• Application to variational regularization in imaging
• Outlook to non-convex optimization

In the first 7 weeks, the course will be given together with the course "Optimization" (part 1). You thus need to decide for one of the two course options!

Exercises:

• One exercises sheet per week;
• Minimum 60 % of the exercises required for participating at the final exam.

Final exam:

Literature:

V. Barbu & Th. Precupanu, Convexity and optimization in Banach spaces

I. Ekeland & R. Teman, Convex analysis and variational problems

H. Bauschke & P. Combettes, Convex analysis and monotone operator theory in Hilbert spaces

J. Peypouquet, Convex optimization in normed spaces: theory, methods and examples

M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, Optimization with PDE Constraints (only used for Descent methods)

Other useful material

Convex analysis Script of Prof. G. Wanka

Convex optimization Script of Prof. J. Peypouquet