Abstract
In this work, we present a new set-oriented numerical method for the numerical solution of multiobjective optimization problems. These methods are global in nature and allow to approximate the entire set of (global) Pareto points. After proving convergence of an associated abstract subdivision procedure, we use this result as a basis for the development of three different algorithms. We consider also appropriate combinations of them in order to improve the total performance. Finally, we illustrate the efficiency of these techniques via academic examples plus a real technical application, namely, the optimization of an active suspension system for cars.
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References
K. Miettinen (1999) Nonlinear Multiobjective Optimization Kluwer Academic Publishers Boston, Massachusetts
K. Deb (2001) Multiobjective Optimization Using Evolutionary Algorithms Wiley New York, NY
I. Das J. Dennis (1998) ArticleTitleNormal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems. SIAM Journal on Optimization 8 631–657
E. Zitzler K. Deb L. Thiele C. A. Coello Coello D. Corne (2001) Evolutionary Multicriterion Optimization Springer Berlin, Germany
S. M. Fonseca P. J. Fleming E. Zitzler K. Deb L. Thiele (2003) Evolutionary Multicriterion Optimization Springer Berlin, Germany
C. Hillermeier (2001) Nonlinear Multiobjective Optimization : A Generalized Homotopy Approach Birkhäuser Basel, Switzerland
S. Schäffler R. Schultz K. Weinzierl (2002) ArticleTitleA Stochastic Method for the Solution of Unconstrained Vector Optimization Problems Journal of Optimization Theory and Applications 114 209–222
M. Dellnitz A. Hohmann (1997) ArticleTitleA Subdivision Algorithm for the Computation of Unstable Manifolds and Global Attractors Numerische Mathematik 75 293–317
Dellnitz, M., and Hohmann, A., The Computation of Unstable Manifolds Using Subdivision and Continuation, Nonlinear Dynamical Systems and Chaos, Edited by H. W. Broer, S. A. Van Gils, I. Hoveijn, and F. Takens, Birkhäuser, Basel, Switzerland, Vol. 19, pp. 449--459, 1996.
M. Dellnitz O. Schütze S. Sertl (2002) ArticleTitleFinding Zeros by Multilevel Subdivision Techniques IMA Journal of Numerical Analysis 22 167–185
R. Horst H. Tuy (1993) Global Optimization: Deterministic Approaches. Springer Berlin, Germany
V. Pareto (1964) Cours d’Economie Politique Libraire Droz Genève, Switzerland
Kuhn, H., and Tucker, A., Nonlinear Programming, Proceedings of the Berkeley Symposium on Mathematical and Statistical Probability, Edited by J. Neumann, University of California at Berkeley, Berkeley, California, pp. 481–492, 1951.
Junge, O., Mengenorientierte Methoden zur numerischen Analyse dynamischer Systeme, PhD Thesis, University of Paderborn, 1999.
J. E. Dennis R. B. Schnabel (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations Prentice-Hall Englewood Cliffs, New Jersey
Schütze, O., Mostaghim, S., Dellnitz, M., and Teich, J., Covering Pareto Sets by Multilevel Evolutionary Subdivision Techniques, Evolutionary Multicriterion Optimization, Edited by C. M. Fonseca, P. J. Fleming, E. Zitzler, D. Deb, and L. Thiele, Springer, Berlin, Germany, Vol. 2, pp. 118--132, 2003.
Sertl, S., and Dellnitz, M., Global Optimization Using a Dynamical Systems Approach, Journal of Global Optimization (submitted).
Chauduri, I., Sertl, S., Zoltán, H., Dellnitz, M., and Fraunheim, T., Global Optimization of Silicon Nanoclusters, Applied Surface Science (submitted).
Schütze, O., A New Data Structure for the Nondominance Problem in Multiobjective Optimization, Evolutionary Multicriterion Optimization, Edited by C. M. Fonseca, P. J. Fleming, E. Zitzler, K. Deb, and L. Thiele, Springer, Berlin, Germany, Vol. 2, pp. 509--518, 2003.
M. Mitschke (1990) Dynamik der Kraftfahrzeuge, Volume C: Fahrverhalten EditionNumber2 Springer Berlin, Germany
G. Castiglioni K. Jäker F. Schlüter (1996) ArticleTitleDas aktive Fahrwerk mit elektrischen Aktuatoren AT Automatisierungstechnik 7 345–350
T. Hestermeyer O. Oberschelp (2003) ArticleTitleSelbstoptimierende Fahrzeugregelung- Verhaltensbasierte Adaption. Intelligente mechatronische Systeme HNI- Verlagsschriftenreihe Heinz Nixdorf Institut 122 231–241
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The authors thank Joachim Lückel for his suggestion to get into the interesting field of multiobjective optimization. Katrin Baptist as well as Frank Scharfeld helped the authors with fruitful discussions. This work was partly supported by the Deutsche Forschungsgemeinschaft within SFB 376 and SFB 614.
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Dellnitz, M., Schütze, O. & Hestermeyer, T. Covering Pareto Sets by Multilevel Subdivision Techniques. J Optim Theory Appl 124, 113–136 (2005). https://doi.org/10.1007/s10957-004-6468-7
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DOI: https://doi.org/10.1007/s10957-004-6468-7