ProgramTo be announced.
|Wed, Oct 11||16:00-16:30||Welcome address (M. Hintermüller)|
|16:30-18:00||M. Brokate 1|
|Thu, Oct 12||9:00-10:30||R. Bot 1|
|10:45-12:15||J. Outrata 1|
|13:30-15:00||M. Brokate 2|
|15:15-16:45||R. Bot 2|
|Fri, Oct 13||9:00-10:30||J. Outrata 2|
|10:45-12:15||M. Brokate 3|
|13:30-15:00||R. Bot 3|
|15:15-16:45||M. Brokate 4|
|Sat, Oct 14||9:00-10:30||R. Bot 4|
|11:00-12:30||J. Outrata 3|
Abstracts of the lecture series
Radu Ioan Bot (Faculty of Mathematics, University of Vienna): Splitting algorithms in nonsmooth convex and nonconvex optimizationThe aim of the lecture series is to present the state of the art in the field of numerical algorithms for solving nonsmooth optimization problems. We will start by presenting some elements of convex analysis and monotone operator theory, will go through standard numerical methods for nonsmooth optimiza- tion problems, and will close with an overview on the latest developments. We will point out the role of the splitting paradigm in the context of solving nons- mooth convex and nonconvex optimization problems with complex structures, with a particular emphasis on primal-dual algorithms, proximal ADMM, and proximal AMA.
Martin Brokate (Technische Universität München): Systems with rate independence: models, sensitivity, optimizationThis part of the autumn school is devoted to the modeling and analysis of systems which are rate independent, or which include rate independent elements. These systems are inherently nonsmooth. We present some examples together with a formal framework for rate independent systems, and discuss existence and uniqueness results. Then we proceed to more refined sensitivity results and discuss in particular the existence of generalized differentials for special situations. Finally we derive optimality conditions for control problems for such systems.
Jiri Outrata (UTIA, Czech Academy of Science): Problems with equilibrium constraints: Theory and numericsUnder problems with equilibrium constraints we understand variational problems in which various types of equilibria arise among the constraints.These problems encompass a wide range starting with bilevel programming over various types of MPECs (mathematical programs with equilibrium constraints) up to EPECs (equilibrium problems with equilibrium constraints) and MOPECs (multiple optimization problems with equilibrium constraints) and enable us to model rather complicated hierarchical games. The lecture is divided into three parts. In the first part we start with proper formulations of the main problems of our interest and their possible classification. Thereafter we make an overview of the notions and tools from the generalized differential calculus which are indispensable for our aims and end up with the basic reformulations and properties of the equilibria arising in the constrains. The second lecture starts with a fine analysis of stabiity and sensitivity of these equilibria. The obtained results have numerous applications, but we will employ them in deriving stationarity/ optimality conditions for some of the considered problems with equilibrium constraints. The final third lecture is devoted to numerical issues. We briefly recall the bundle idea in nonsmooth optimization and the semismooth Newton method and present two approaches used currently for the numerical solution of MPECs. Thereafter the performance of the presented techniques will be demonstrated with the help of some source problems coming from mechanics and economy.