Abstract
The problem of computing a good approximation set of the Pareto front of a multiobjective optimization problem can be recasted as the maximization of its S-metric value, which measures the dominated hypervolume. In this way, the S-metric has recently been applied in a variety of metaheuristics. In this work, a novel high-precision method for computing approximation sets of a Pareto front with maximal S-Metric is proposed as a high-level relay hybrid of an evolutionary algorithm and a gradient method, both guided by the S-metric. First, an evolutionary multiobjective optimizer moves the initial population close to the Pareto front. The gradient-based method takes this population as its starting point for computing a local maximal approximation set with respect to the S-metric. Thereby, the population is moved according to the gradient of the S-metric.
This paper introduces expressions for computing the gradient of a set of points with respect to its S-metric on basis of the gradients of the objective functions. It discusses singularities where the gradient is vanishing or differentiability is one sided. To circumvent the problem of vanishing gradient components of the S-metric for dominated points in the population a penalty approach is introduced.
In order to test the new hybrid algorithm, we compute the precise maximizer of the S-metric for a generalized Schaffer problem and show, empirically, that the relay hybrid strategy linearly converges to the precise optimum. In addition we provide first case studies of the hybrid method on complicated benchmark problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Zitzler, E., Thiele, L.: Multiobjective Optimization Using Evolutionary Algorithms—A Comparative Case Study. In: Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.-P. (eds.) PPSN V. LNCS, vol. 1498, pp. 292–301. Springer, Heidelberg (1998)
Zitzler, E., Brockhoff, D., Thiele, L.: The hypervolume indicator revisited: On the design of pareto-compliant indicators via weighted integration. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 862–876. Springer, Heidelberg (2007)
Fleischer, M.: The measure of pareto optima. Applications to multi-objective metaheuristics. In: Fonseca, C.M., Fleming, P.J., Zitzler, E., Deb, K., Thiele, L. (eds.) EMO 2003. LNCS, vol. 2632, pp. 519–533. Springer, Heidelberg (2003)
Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., Grunert da Fonseca, V.: Performance assessment of multiobjective optimizers: An analysis and review. IEEE TEC 7(2), 117–132 (2003)
Deb, K.: Multi-Objective Optimization using Evolutionary Algorithms. Wiley, Chichester, UK (2001)
Coello Coello, C.A., Van Veldhuizen, D.A., Lamont, G.B.: Evolutionary Algorithms for Solving Multi-Objective Problems. Kluwer Academic Publishers, New York (2002)
Zitzler, E., Künzli, S.: Indicator-based selection in multiobjective search. In: Yao, X., Burke, E.K., Lozano, J.A., Smith, J., Merelo-Guervós, J.J., Bullinaria, J.A., Rowe, J.E., Tiňo, P., Kabán, A., Schwefel, H.-P. (eds.) PPSN VIII. LNCS, vol. 3242, pp. 832–842. Springer, Heidelberg (2004)
Huband, S., Hingston, P., While, L., Barone, L.: An evolution strategy with probabilistic mutation for multi-objective optimisation. In: CEC 2003, vol. 4, pp. 2284–2291. IEEE Computer Society Press, Los Alamitos (2003)
Knowles, J.: Local-Search and Hybrid Evolutionary Algorithms for Pareto Optimization. Phd thesis, Department of Computer Science, University of Reading, UK (2002)
Emmerich, M., Beume, N., Naujoks, B.: An EMO Algorithm Using the Hypervolume Measure as Selection Criterion. In: Coello Coello, C.A., Hernández Aguirre, A., Zitzler, E. (eds.) EMO 2005. LNCS, vol. 3410, pp. 62–76. Springer, Heidelberg (2005)
Talbi, E.G.: A Taxonomy of Hybrid Metaheuristics. Journal of Heuristics 8(5), 541–564 (2002)
Timmel, G.: Ein stochastisches Suchverfahren zur Bestimmung der Optimalen Kompromißlösungen bei statistischen polykriteriellen Optimierungsaufgaben. Journal TH Ilmenau 6, 139–148 (1980)
Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Mathematical Methods of Operations Research 51(3), 479–494 (2000)
Shukla, P., Deb, K., Tiwari, S.: Comparing Classical Generating Methods with an Evolutionary Multi-objective Optimization Method. In: Coello Coello, C.A., Hernández Aguirre, A., Zitzler, E. (eds.) EMO 2005. LNCS, vol. 3410, pp. 311–325. Springer, Heidelberg (2005)
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. (eds.) Practical Approaches to Multi-Objective Optimization. Dagstuhl Seminar Proceedings, IBFI, Schloss Dagstuhl, Germany, vol. 04461 (2005)
Bosman, P.A., de Jong, E.D.: Combining gradient techniques for numerical multi-objective evolutionary optimization. In: Keijzer, M., et al. (eds.) GECCO 2006, vol. 1, pp. 627–634. ACM Press, Seattle, USA (2006)
Emmerich, M., Deutz, A.: Test Problems based on Lamé Superspheres. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 922–936. Springer, Heidelberg (2007)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge, UK (2006)
Wagner, T., Beume, N., Naujoks, B.: Pareto-, Aggregation-, and Indicator-based Methods in Many-objective Optimization. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 742–756. Springer, Heidelberg (2007)
Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: Multiobjective Selection Based on Dominated Hypervolume. European Journal of Operational Research 181(3), 1653–1669 (2007)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Emmerich, M., Deutz, A., Beume, N. (2007). Gradient-Based/Evolutionary Relay Hybrid for Computing Pareto Front Approximations Maximizing the S-Metric. In: Bartz-Beielstein, T., et al. Hybrid Metaheuristics. HM 2007. Lecture Notes in Computer Science, vol 4771. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75514-2_11
Download citation
DOI: https://doi.org/10.1007/978-3-540-75514-2_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-75513-5
Online ISBN: 978-3-540-75514-2
eBook Packages: Computer ScienceComputer Science (R0)