We offer students a profound and broad education in the field of continuous optimization. This includes introductory courses on the basics of optimization for both, students of mathematics as well as students of engineering or computer science, as well as a broad range of advanced and in-depth courses. Those cover more theoretical topics like optimization in infinite-dimensional vector spaces, but also more applied topics like algorithmic methods of optimization. In addition, we offer lectures on the basics of mathematics for business and engineering courses. For us, a high quality of teaching is important, which spans the arc from theoretical principles and scientific approaches to practical application.

Video on studying mathematics in Ilmenau with Ms. Kästner.

Video on studying mathematics in Ilmenau with Ms. Bold.

TU Ilmenau

### Lecture on Optimization

Within this course, we study properties and solution approaches for convex optimization problems, i.e., for nonlinear optimization problems with a convex objective function and convex constraints. Besides optimality conditions of unconstrained and constrained optimization problems, numerical methods like descent methods, Newton method, or penalty approaches are discussed. Moreover, numerical methods for linear optimization problems like the Simplex method and the interior-point method will be introduced.

Expected audience: Students of mathematics in the 4th semester of their bachelor studies.

Lecturer: Prof. Eichfelder, M. Sc. Hoff

https://en.wikipedia.org/wiki/Subderivative#/media/File:Subderivative_illustration.png

### Theory of Nonlinear Optimization

This lecture is on theoretical results for general optimization problems on real linear spaces, such as optimization problems on the space of continuous functions or on matrix spaces. After an introduction, the lecture proves a modification of Weierstrass' theorem for weakly lower semicontinuous functions on weakly sequentially compact sets. The necessary basic concepts are introduced or repeated. The results, including those under convexity assumptions, are applied to approximation problems. In addition to that, the lecture covers generalized derivative concepts, including directional, Gateaux, and Frechet derivatives, as well as the subdifferential of convex analysis and Clarke's subdifferential for non-differentiable functions. All these concepts are used to formulate optimality conditions.

Expected audience: Students of mathematics and business mathematics who have already attended the basic course Optimization and, preferably, the lecture Functional Analysis.

Lecturer:  Prof. Eichfelder und Dr. T. Gerlach

G. Eichfelder

T. Gerlach

### Lecture on Applied Optimization

The lecture provides an introduction to the field of continuous mathematical optimization. Besides basics (such as convex sets, convex functions, optimality conditions of unconstrained and constrained optimization) the lecture focuses on linear optimization problems.

Expected audience: Students of computer science with mathematics as minor subject in the 3rd semester, as well as students of other courses of study with sufficient basics from introductory mathematics lectures (linear algebra and analysis)

## Further lectures offered by the group

G. Eichfelder

### Lecture on Global Optimization

Numerical methods of nonlinear optimization find in general only locally optimal solutions. However, for non-convex optimization problems, there may be locally optimal points which are not also globally optimal. In global optimization approaches are developed which can also guarantee to find the global minimum value of a non-convex optimization problem. Further topics of the lecture are the reformulation of combinatorial or mixed-integer optimization problems as continuous optimization problems or as linear optimization problems over the cone of copositive matrices, and how this can be used for numerical methods.

Lecturer: Prof. Gabriele Eichfelder, Dr. T. Gerlach

G. Eichfelder

## Numerical Methods of Convex Optimization

In this lecture, we will consider numerical solution methods for nonlinear optimization problems. For most methods we will assume differentiability of the functions. However, in the case that the objective function is not differentiable, the lecture will also consider, under suitable further conditions, methods based on subgradients. Depending on the time remaining, the lecture will be supplemented by a consideration of methods in current research for special classes of problems.

Lecturer:  Prof. Gabriele Eichfelder, Dr. T. Gerlach

G. Eichfelder

### Lecture on Vector Optimization

The topic of the lecture are optimization problems where the objective function maps into a partially ordered linear space. This also includes the class of multiobjective optimization problems, where several objective functions are optimized simultaneously. A main focus of the lecture is the examination of scalarization approaches, i.e., the formulation of parameter-dependent replacement problems, which can be used to calculate optimal solutions of vector optimization problems by numerical methods. Further topics are existence results, optimality conditions, duality results, robust approaches to uncertain optimization, and set optimization.

Expected audience: Students of mathematics and business mathematics at the end of the bachelor’s programme or in the master's programme; especially suitable for preparation of a master's thesis at the department.

Lecturer: Prof. Eichfelder, M. Sc. Hoff

TU Ilmenau

### Lecture on Mathematics for Economics and Social Sciences 1

In this lecture, the mathematical basics for further studies in economics and social science are taught, like basic concepts of logic, properties of functions of one variable, sequences and continuity of functions, differentiation, also for multivariate functions. The lecture is continued with Mathematics for Economics and Social Sciences 2 .

Expected audience: First year students of Wirtschaftsinformatik, Betriebswirtschaftslehre mit technischer Orientierung or Medienwirtschaft in the bachelor's program

Lecturer:  Prof. Eichfelder, Dr. R. Gerlach, Dr. T. Gerlach

TU Ilmenau

### Lecture on Mathematics for Economics and Social Sciences 2

In this lecture, the mathematical basics for further studies in economics and social science are taught, like basics of linear algebra, differentiation and the integral of multivariate functions. The lecture is based on Mathematics for Economics and Social Sciences 1.

Expected audience: First year students of Wirtschaftsinformatik, Betriebswirtschaftslehre mit technischer Orientierung or Medienwirtschaft in the bachelor's program

TU Ilmenau

### Seminar

In this seminar the participants will present a selected topic from the current scientific literature from the field of nonlinear optimization. Each student will be given a research paper, a recent publication from a journal, and will present a selection from its content (including a proof) in a 60-90 minutes seminar talk. These presentations will be supplemented by guest lectures and presentations from the working group.

Expected audience: Students of mathematics

Lecturer: Prof. Eichfelder