This entry is from Winter semester 2019/20 and might be obsolete. No current equivalent could be found.

# Convex Optimization in Banach Spaces (dt. Konvexe Optimierung in Banachräumen)

 Level, degree of commitment Specialization module, depends on importing study program Forms of teaching and learning,workload Lecture (3 SWS), recitation class (1 SWS), 180 hours (60 h attendance, 120 h private study) Credit points,formal requirements 6 CP Course requirement(s): Successful completion of at least 50 percent of the points from the weekly exercises. Examination type: Written or oral examination Language,Grading German,The grading is done with 0 to 15 points according to the examination regulations for the degree program M.Sc. Mathematics. Subject, Origin Mathematics, M.Sc. Mathematics Duration,frequency One semester, Im Wechsel mit anderen specialization moduleen zur Optimierung Person in charge of the module's outline Prof. Dr. Thomas Surowiec

## Contents

I. Infinite-Dimensional Optimization

• Semi-continuous functions in topological vector spaces
• Variational problems in (compact) metric and topological vector spaces
• Ekeland's Variational Principle
• First-order necessary and approximate conditions for variational problems
• Continuity of integral functionals on L-p spaces, Krasnoselskii's Theorem
• The role of the weak topology in existence theory

II. Convex analysis and Optimization

• Convex sets, convex functionals, Fenchel-Legendre conjugates (theorem of Fenchel-Moreau-Rockafellar, the Fenchel-Young-inequality)
• Generalized derivatives, e.g., directional differentiability, subdifferentials
• Calculation Rules for Convex Subdifferentials, Applications in Optimization

III. Numerical solution methods

• First-order numerical solution methods, e.g., projected (sub)gradients, mirror descent
• Second order numerical solution methods: Semismooth-Newton

## Qualification Goals

The students shall

• learn classical propositions of existence of the calculus of variations as well as some important concepts and results from nonlinear functional analysis, such as Nemytski operators and their role in the optimization and analysis of nonlinear partial differential equations,
• learn the extension of concepts from finite-dimensional convex analysis to infinite-dimensional problems; here a focus is placed on duality theory and subdifferentials,
• learn the formulation, implementation and convergence analysis of important algorithms in function spaces,
• Reassess knowledge from the basic modules and some advanced modules, e.g. from the modules for analysis and linear algebra as well as the optimization modules,
• learn the application of concepts from functional analysis, e.g. Dual Spaces, Hahn-Banach Set and Separation Sets,
• recognise relations with other areas of mathematics and other sciences,
• practice mathematical working methods (development of mathematical intuition and its formal justification, training of the ability to abstract, proof techniques),
• improve their oral communication skills in the exercises by practicing free speech in front of an audience and during discussion.

## Prerequisites

None. The competences taught in the following modules are recommended: either Linear Algebra I and Linear Algebra II and Analysis I and Analysis II or Basic Linear Algebra and Basic Real Analysis and Basics of Advanced Mathematics, Measure and Integration Theory . In addition, knowledge of functional analysis is an advantage.