Mathematical Institute, University of Freiburg, Teaching

Dr. Sebastian Sager (Heidelberg):
Trends in Nichtlinearer Gemischt-Ganzzahliger Optimalsteuerung

Time and place

Thursday, 14.1.10, 17:00-18:00, Hörsaal II, Albertstr. 23b

Abstract

In model-based nonlinear optimal control switching decisions that can be \noptimized often play an important role. Prominent examples of such hybrid \nsystems are gear switches for transport vehicles, traffic lights, or on/off \nvalves in engineering. Optimization algorithms need to take the discrete \nnature of the variables that model these switching decisions into account.\n

\nMixed-integer optimal control problems (MIOCPs) include features related to \ndifferent mathematical disciplines. Hence, it is not surprising that distinct \napproaches have been proposed to analyze and solve them. There are at least \nthree generic approaches to solve model-based optimal control problems: first, \nsolution of the Hamilton-Jacobi-Bellman equation and in a discrete setting \nDynamic Programming, second indirect methods, also known as the first \noptimize, then discretize approach, and third direct methods (first optimize, \nthen discretize) and in particular all--at--once approaches that solve the \nsimulation and the optimization task simultaneously. The combination with the \ncombinatorial restrictions on control functions comes at different levels: for \nfree in dynamic programming, as the full control space is explored anyhow, by \nmeans of an enumeration in the inner optimization problem of the necessary \nconditions of optimality in Pontryagin's maximum principle, or by various \nmethods from integer programming in the direct methods. We will survey some of \nthese approaches.\n

\nWe will mention several extensions that have been made possible by recent \nadvances. They include an extension to multiple objective optimization, \nnonlinear model predictive control in real time, and the efficient treatment \nof switching constraints.\n

\nWe conclude by pointing out future challenges for process control with \nswitching decisions, among them the availability of well-defined test \ninstances for algorithm developpers.

Prof. Dr. Roland Fried, Daniel Vogel:
Robust partial correlation graphs

Time and place

Friday, 29.1.10, 11:15-12:15, Raum 404, Eckerstr. 1

Abstract

Graphical models allow a simple graphical visualization of the (conditional) dependence structure among multiple variables: each variable is represented by a vertex, and conditional dependence between a pair of variables given all the other variables is illustrated by connecting the corresponding pair of vertices by an edge. Within the Gaussian framework, conditional independence is equivalent to zero partial correlation, i.e. we just need to estimate the partial correlation between each pair of variables and test whether it is zero or not in order to decide on the inclusion of edges in the graph. The arising diagram can thus be called a partial correlation graph.\n\nIn this talk we treat two extensions of Gaussian graphical models. Brillinger (1996) and Dahlhaus (2000) suggest to explore the linear dependence structure among multivariate time series by analyzing the partial spectral coherences between the component processes. These are a natural generalization of the partial correlations to the frequency domain. Fried and Didelez (2003) show how to perform stepwise model selection in this context by estimating the partial spectral coherences from suitably chosen subsets of the component processes.\n\nAnother generalization of Gaussian graphical models are elliptical graphical models, that is, we allow the population distribution to be elliptical instead of normal. We examine the class of affine equivariant scatter estimators and show how they can be used to derive generalizations of classical Gaussian graphical modelling tools derived from the empirical covariance matrix and the adjusted deviance tests. We demonstrate the feasibility of our approach by a simulation study, using, among others, Tyler's scatter estimator (Tyler, 1987), which is distribution-free within the elliptical model. This technique is in particular suited to robustify the established, likelihood-based Gaussian graphical modelling methods, which are known to be very sensitive to model misspecifications and outlying observations. Some of the results are summarized in Vogel and Fried (2009).\n\nThe robust fitting of partial correlation graphs to multivariate time series data, e.g. by extending the ideas of elliptical graphical modelling to the time series context, is the scope of future research.\n\n \nReferences\n\n[1] D. R. Brillinger. Remarks concerning graphical models for time series and point processes. Revista de Econometria, 16:1-23, 1996.\n\n[2] R. Dahlhaus. Graphical interaction models for multivariate time series. Metrika, 51:157-172, 2000.\n\n[3] R. Fried and V. Didelez. Decomposability and selection of graphical models for multivariate time series. Biometrika, 90: 251-267, 2003.\n\n[4] D. E. Tyler. A distribution-free Mestimator of multivariate scatter. Annals of Statistics, 15:234-251, 1987.\n\n[5] D. Vogel and R. Fried. On robust Gaussian graphical modelling. Discussion Paper 36/2009, SFB 823, Technische UniversitÄat Dortmund, 2009.

Talks at the mathematical institute

Veranstaltungen und Vorträge

Winter term 2009