Optimization problems related to industrial resource

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Optimization problems related to industrial resource scheduling

Collaborator: I. Bremer, R. Henrion

Cooperation with: W. Römisch (Humboldt-Universität (HU) zu Berlin), T. Szántai (Technical University of Budapest, Hungary), J. Outrata (Institute of Information Theory and Automation (UTIA) Prague, Czech Republic), A. Jourani (Université de Bourgogne, Dijon, France), D. Dentcheva (Stephens Institute of Technology, New Jersey, USA), A. Seeger (Université d'Avignon, France)

Supported by: DFG-Forschungszentrum ``Mathematik für Schlüsseltechnologien'' (Research Center ``Mathematics for Key Technologies''), project C7;

Rücker Ges.m.b.H, Graz, Austria

Description:1. Path planning for industrial robots and human models in automotive industry (I. Bremer).

The VR tool of Rücker for production planning in automotive industry uses a large number of externally created 3D models in different formats. The models which are usually generated from CAD are not optimized with respect to memory and/or speed performance. Hence, real-time animations of complete shop floors with a lot of moving materials, robots, and humans become very difficult.

On the basis of OpenGL Performer [5], which offers several possibilities to improve performance, i.e. the frame rate, we provided a tool for automatic model conversion such that the visual impression for the user is not affected but the cost of rendering is reduced as much as possible.

In particular, we have considered the problem of removing holes and the development and implementation of active surface definitions (ASD). There ASDs are hierarchical multigrid structures. Based on the distance between viewer and object each object is visualized on a certain grid level. For distance changes, the corresponding change in visualization is realized by a smooth morphing between the respective grid levels.

See also the results of the year before ^, .

2. Mean risk models for electricity portfolio management (R. Henrion).

A typical feature of optimization problems arising in engineering sciences is the presence of random and nonsmooth parameters. In this research project, modeling, solution procedures, and investigations on structure and stability of such problems are the main objective. It is embedded into a cooperation with other scientific institutions in Berlin via the DFG Research Center (with HU), a joint research seminar (with HU and Konrad-Zuse-Zentrum für Informationstechnik (ZIB)), and the joint organization of a yearly course for chemical engineers supported by DECHEMA (with Technische Universität (TU) and ZIB). The focus of research is on stochastic optimization problems.

The following subjects were considered:

Scenario reduction: The solution of power management problems is typically based on a scenario tree formulation. The complexity of such trees requires a drastical reduction to subtrees without much loss of information. A general theoretical framework to do so on the basis of suitable probability metrics was developed in [1]. For mixed-integer models to be considered in this project, certain discrepancy distances have to be used. As a first step, an explicit formula for scenario reduction could be obtained in case of the ``closed-set'' discrepancy and a linear programming reformulation in case of the interval discrepancy.
Chance constraints: One possibility to model risk in stochastic optimization is to use probabilistic (chance) constraints. Due to the discretization process in power management, possibly continuous distributions are approximated by discrete ones. This raises the question of solution stability for problems with chance constraints. An extensive analysis was provided in [2] and [3].
Error bounds: The concept of error bounds is a key concept in nonlinear optimization (numerical solution, constraint qualifications, stability) and is itself a special instance of the more general calmness concept for multifunctions. A detailed characterization in the framework of nonsmooth optimization including relations to the above research project is contained in [4].

References:

J. DUPACOVA, N. GRÖWE-KUSKA, W. RÖMISCH, Scenario reduction in stochastic programming: An approach using probability metrics, Math. Program., 95 (2003), pp. 493-511.
R. HENRION, Perturbation analysis of chance-constrained programs under variation of all constraint data, in: Dynamic Stochastic Optimization, K. Marti et al., eds., vol. 532 of Lect. Notes Econ. Math. Syst., Springer, Heidelberg, 2004, pp. 257-274.
R. HENRION, W. RÖMISCH, Hölder and Lipschitz stability of solution sets in programs with probabilistic constraints, to appear in: Math. Program.
R. HENRION, J. OUTRATA, Calmness of constraint systems with applications, in preparation.
SILICON GRAPHICS, INC., OpenGL Performer, http://www.sgi.com/software/performer/ .