It is the aim of the subproject A1 to lay the mathematical foundations upon which the principles of self-optimization for mechatronical systems can be successfully realized. Based on models designed in engineering -- i. e. in close cooperation with the subprojects D1 and D2 -- new mathematical methods are developed and brought from their theoretical conception to an implementation and application.  In particular we consider continuous (group of Prof. Dellnitz) and discrete (group of Prof. Monien) optimization problems.

Continuous optimization methods are applied whenever the objective functions to be considered depend only on continuous variables (in ⁿ or ⁿ) .  Within the framework of self-optimization it is often especially important to consider several objective functions simultaneously, so that in a natural way one is lead into the mathematical field of multiobjective optimization. For a multiobjective optimization problem, one typically does not obtain  a single (global) optimum as a solution. Instead it consists of a set of optimal compromises -- the so-called Pareto set.

In particular, in the second step of the self-optimization process these methods allow the generation of new objective functions by varying the priorities of the objective functions included in the model.

The algorithms developed within the subproject A1 have already been applied to engineering problems in manifold ways. For example, in the case of the operation point assignment for a doubly-fed linear motor considered in the subproject D1 it was on the one hand important to maximize the degree of efficiency and on the other hand also of interest to have an optimal power factor. Additionally, in this example time-dependencies came into play. Using special pathfollowing techniques that were developed in our subproject, it was possible to design an algorithm that computes solutions to this problem during operation of the linear motor.

Apart from classical multiobjective optimization problems, the fundamental questions considered in the collaborative research center also lead to optimal control problems with several objectives. An example for this is the guidance module of a railbound vehicle that was considered in cooperation with the subproject D1. As the guidance module is modeled as a differentially flat system, i. e. all feasible trajectories of the system can be written as a function of the flat output and its derivatives, it is possible to transform the optimal control problem into a (classical) nonlinear, constrained multiobjective optimization problem, which in turn can be solved numerically using e. g. the image-oriented "recovering algorithms" developed in our subproject.

In the field of robust planning we successfully applied the method of intelligent lookahead in applications like e.g. airplane scheduling in the presence of disruptions, energy management for hybrid energy supply modules, and resource scheduling in self-optimizing real time operating systems.

For distributed systems with selfish agents we developed efficient algorithms based on the best response dynamics to compute Nash equilibria. The algorithms have been implemented to compute equilibria in road traffic systems. First experiments have been conducted to compute equilibria in the highway network of North Rhine-Westfalia.  In contrast to road traffic systems where latency functions are increasing with the traffic due to congestion, shuttles of the NBP to a certain extend benefit from larger traffic by forming convoys. This is modelled by decreasing latency functions. The algorithms developed within this project can be used to handle systems with decreasing latencies as well.

Directors of the Subproject:

 Prof. Dr. rer. nat. Michael Dellnitz

 Jun. Prof. Dr. rer. nat. Sina Ober-Blöbaum

Coordinator of the Subproject:

 Katrin Witting

Publications (since 6/2005)

Reviewed Publications

Dell'Aere, A.: Multi-Objective Optimization in Self-Optimizing Systems. In: Proceedings of the IEEE 32nd Annual Conference on Industrial Electronics (IECON), Paris, 2006, pp. 4755-4760

Dumrauf, D.; Gairing, M.: Price of Anarchy for Polynomial Wardrop Games. In: Proceedings of the WINE 2006, 2nd Workshop on Internet and Network Economics. Patra, Greece, 2006

Ehrhoff, J.; Grothklags, S.; Lorenz, U.: Parallelism for Perturbation Management and Robust Plans. In: Cunha, J. C., Medeiros, P. D. (Eds.): Euro-Par 2005 Parallel Processing: 11th International Euro-Par Conference, Lisbon, Portugal, August 30 - September 2, 2005, Proceedings (Lecture Notes in Computer Science), Springer, Volume 3648/2005, Berlin, 2005, pp. 1265-1274

Feldmann, R.; Mavronicolas, M.; Pieris, A.: Facets of the Fully Mixed Nash Equilibrium Conjecture. In: Monien, B., Schroeder, U. (Eds.): Algorithmic Game Theory, First International Symposium, Proceedings of SAGT 2008, April 30-May 2, 2008, Paderborn, Germany (Lecture Notes in Computer Science), Springer, Volume 4997, 2008, pp. 145-157

Grüne, L.; Junge, O.: Constructing robust feedback laws by set oriented numerical methods. In: Proceedings in Applied Mathematics and Mechanics, Special Issue: GAMM Annual Meeting 2005 - Luxembourg, WILEY-VCH Verlag, Volume 5 (1), Weinheim, 2005, pp. 157-160

Grüne, L.; Junge, O.: Optimal stabilization of hybrid systems using a set oriented approach. In: Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Japan, 2006, 2006, pp. 2089-2093

Grüne, L.; Junge, O.: Approximately optimal nonlinear stabilization with preservation of the Lyapunov function property. In: Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, Louisiana, 2007, 2007, pp. 702-707

Grüne, L.; Junge, O.: Global optimal control of perturbed systems. Journal of Optimization Theory and Applications, Volume 136 (3), Springer Netherlands, 2008

Gairing, M.; Lücking, T.; Mavronicolas, M.; Monien, B.: The Price of Anarchy for Restricted Parallel Links. Parallel Processing Letters (PPL), Volume 16(1), World Scientific Publishing, Singapore, 2006, pp. 117-131

Gairing, M.; Lücking, T.; Mavronicolas, M.; Monien, B.; Rode, M.: Nash Equilibria in Discrete Routing Games with Convex Latency Functions. Journal of Computer and System Sciences, Volume 74, Springer, Berlin, 2008, pp. 1199-1225

Gairing, M.; Monien, B.; Tiemann, K.: Routing (Un-) Splittable Flow in Games with Player-Specific Linear Latency Functions. In: Proceedings of the ICALP 2006, 33rd International Colloquium on Automata, Languages and Programming, July 9 - 16, 2006, Venice, Italy, Springer-Verlag, Venice, Volume 4051, 2006, pp. 501-512

Gairing, M.; Monien, B.; Tiemann, K.: Selfish Routing with Incomplete Information. Theory of Computing Systems, Volume 42 (1), ACM, New York, 2007, pp. 91-130

Gairing, M.; Schoppmann, F.: Total Latency in Singleton Congestion Games. In: Deng, X., Graham, F. C. (Eds.): Internet and Network Economics, Proceedings of the Third International Workshop, WINE 2007, December 12-14, 2007, San Diego, CA, USA (Lecture Notes in Computer Science), Springer, Volume 4858, Berlin, 2007, pp. 381-387

Geisler, J.; Witting, K.; Trächtler, A.; Dellnitz, M.: Multiobjective Optimization of Control Trajectories for the Guidance of a Rail-bound Vehicle. In: 17th IFAC World Congress, July 6-11, 2008, Seoul, Korea, 2008

Geisler, J.; Witting, K.; Trächtler, A.; Dellnitz, M.: Self-Optimization of the Guidance Module of a Rail-bound Vehicle. 7th International Heinz Nixdorf Symposium "Selfoptimizing Mechatronic Systems: Design the Future", February 20-21, 2008, Paderborn, HNI-Verlagsschriftenreihe, 2008, pp. 85-100

Gausemeier, J.; Znamenshchykov, O.; Oberthür, S.; Podlogar, H.: An Approach for Achieving Self-Optimization in Mechatronic Systems supported by Active Patterns. In: Proceedings of the 8th International Conference on Intelligent Systems Design and Applications, 2008

Junge, O.; Marsden, J. E.; Ober-blöbaum, S.: Discrete Mechanics and Optimal Control. In: Proceedings of the 16th IFAC World Congress, July 3-8, 2005, Praha, Czech Republic, Elsevier Science Ltd, 2005

Klöpper, B.; Podlogar, H.; Gausemeier, J.; Witting, K.: Domain Spanning Search for Solution Patterns for the Conceptual Design of Self-Optimizing Systems. In: Bhowmick, S. S., Küng, J., (Eds.), R. W.: 19th International Conference on Database and Expert Systems Applications (DEXA 2008), September 1-5, 2008, Turin, Springer Verlag, Volume 5181, Berlin, 2008

Knoke, T.; Romaus, C.; Böcker, J.; Dell'Aere, A.; Witting, K.: Energy Management for an Onboard Storage System Based on Multi-Objective Optimization. In: Proceedings of the IEEE 32nd Annual Conference on Industrial Electronics (IECON), Paris, 2006, pp. 4677-4682

Mavronicolas, M.; Milchtaich, I.; Monien, B.; Tiemann, K.: Congestion Games with Player-Specific Constants. In: Proceedings of the MFCS 2007, 32nd International Symposium on Mathematical Foundations of Computer Science. Cesky Krumlov, Czech Republic, 2007, pp. 633-644

Monien, B.; Tiemann, K.: Routing and Scheduling with Incomplete Information. In: Proceedings of the DISC 2007, 21st International Symposium on Distributed Computing, September 24-26, 2007, Lemesos, Cyprus (Lecture Notes in Computer Science), Springer-Verlag, Berlin, 2007, pp. 1-2

Oberthür, S.; Znamenshchykov, A.; Klöpper, B.; Vöcking, H.: Improved Flexible Resource Management by Means of Look-Ahead Scheduling and Bayesian Forecasting. 7th International Heinz Nixdorf Symposium "Self-optimizing Mechatronic Systems: Design the Future", February 20-21, 2008, Paderborn, HNI-Verlagsschriftenreihe, Paderborn, 2008, pp. 361-376

Schütze, O.; Coello Coello, C. A.; Mostaghim, S.; Talbi, E.-g.; Dellnitz, M.: Hybridizing Evolutionary Strategies with Continuation Methods for Solving Multi-Objective Problems. Engineering Optimization, Volume 40(5), 2008, pp. 383-402

Schütze, O.; Dell'Aere, A.; Dellnitz, M.: On Continuation Methods for the Numerical Treatment of Multi-Objective Optimization Problems. In: Branke, J., Deb, K., Miettinen, K., Steuer, R.: Practical Approaches to Multi-Objective Optimization, Internationales Begegnungs- und Forschungszentrum (IBFI), Schloss Dagstuhl, Germany, Dagstuhl, Germany, 2005

Schütze, O.; Laumanns, M.; Coello Coello, C. A.; Dellnitz, M.; Talbi, E.-g.: Convergence of Stochastic Search Algorithms to Finite Size Pareto Set Approximations. Journal of Global Optimization, Volume 41, Kluwer Academic Publishers, 2007, pp. 559-577

Schütze, O.; Vasile, M.; Junge, O.; Dellnitz, M.; Izzo, D.: Designing Optimal Low Thrust Gravity Assist Trajectories Using Space Pruning and a Multi-Objective Approach. Engineering Optimization (Accepted), 2008

Witting, K.; Schulz, B.; Dellnitz, M.; Böcker, J.; Fröhleke, N.: A new approach for online multiobjective optimization of mechatronic systems. International Journal on Software Tools for Technology Transfer STTT, Volume 10(3), 2008, pp. 223-231

Ph.D.-Theses

Dell'Aere, A.: Numerical Methods for the Solution of Bi-Level Multi-Objective Optimization Problems. Dissertation, Fakultät für Elektrotechnik, Informatik und Mathematik, Universität Paderborn, 2008

Gairing, M.: Selfish routing in networks. Dissertation, Fakultät für Elektrotechnik, Informatik und Mathematik, Universität Paderborn, 2006

Meyerhenke, H.: Disturbed Diffusive Schemes for Solving Partitioning Problems on Graphs. Dissertation, Fakultät für Elektrotechnik, Informatik und Mathematik, Universität Paderborn, 2008

Ober-blöbaum, Sina: Discrete mechanics and optimal control. Dissertation, Fakultät für Elektrotechnik, Informatik und Mathematik, Universität Paderborn, 2008