German original

# Non-smooth Optimization (dt. Nichtglatte Optimierung)

 Level, degree of commitment in original study programme Advanced module, compulsory elective module 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: 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 study course M.Sc. Mathematics. Original study programme M.Sc. Mathematik / Vertiefungsbereich Mathematik Duration,frequency One semester, Im Wechsel mit anderen advanced moduleen zur Optimierung Person in charge of the module's outline Prof. Dr. Thomas Surowiec

## Contents

I. Convex Analysis and Geometry

• Basic concepts of convex analysis, in particular generalized derivatives, tangent and normal cones, calulus rules and relationships between the analytical and geometric concepts.

II. Nonconvex Nonsmooth Analysis and Geometry

• Nonsmooth analysis in the sense of F. Clarke, Clarke's directional derivative, subdifferential, tangent and normal cones, calculus rules and relations between the analytical and geometric concepts.

III. Numerical Methods of Nonsmooth Optimization

• Numerical solution algorithms for nonsmooth optimization problems, in particular subgradient methods and bundle methods for convex and nonconvex problems
• The semi-smooth Newton method for non-smooth operator equations

## Qualification Goals

The students shall

• to receive a thorough introduction to the necessary concepts of convex analysis in finite dimensions, which are especially important for the development of numerical optimization algorithms for non-smooth convex problems,
• learn the non-smooth analysis from the point of view of F. Clarke in finite dimensions (directional derivation, subdifferentials, calculation rules) and their application in the development of efficient numerical optimization algorithms for non-smooth nonconvex problems,
• learn the formulation, implementation and convergence analysis of important algorithms in non-smooth optimization,
• 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,
• 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. In addition, knowledge of nonlinear optimization is an advantage.