We will give with this lecture an introduction to basics of convex optimization theory in finite dimensional spaces, where we are mainly interested in non-smooth minimization problems (i.e. no classical derivatives excist). Such proplems arise in many modern applications such as imaging problems or machine learnign applications. 

In particular, the following properties are covered: 

• Convex funtions and subderivates
• Constrained miminimzation problems
• Convex conjugates
• Proximal maps
• Primal and dual problem formulation
• Minimization schemes, in particular splitting approaches

I will adapt the content to. the needs of the participants, i.e. we could for example discuss acceleration strategies for minimization schemes such that stochasic minimization etc.

Note that is not mandatory that you have heared any other course about optimization. However, it is an advantage to be familar with analysis (Hilbert and Banach spaces, opterator norm, convergence etc).  

Literature:

• V. Barbu and Th. Precupanu, Convexity and Optimization in Banach Spaces
• I. Ekeland and R. Teman, Convex Analysis and Variational Problems
• H. Bauschke and 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