Abstract
Convergence analyses of evolutionary multiobjective optimization algorithms typically deal with the convergence in limit (stochastic convergence) or the run time. Here, for the first time concrete results for convergence rates of several popular algorithms on certain classes of continuous functions are presented. We consider the algorithms in the version of using a (1+1) selection scheme. Then, SMS-EMOA and IBEA ε + achieve linear convergence rate, proved by showing algorithmic equivalence to the single-objective (1+1)-EA with self-adaptation, whereas NSGA-II and SPEA2 have a sub-linear convergence rate, proved by reducing them to a multiobjective algorithm with known properties.
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References
Rudolph, G.: On a multi–objective evolutionary algorithm and its convergence to the Pareto set. In: Proceedings of the 1998 IEEE International Conference on Evolutionary Computation, pp. 511–516. IEEE Press, Piscataway (1998)
Hanne, T.: On the convergence of multiobjective evolutionary algorithms. European Journal of Operational Research 117(3), 553–564 (1999)
Teytaud, O.: On the hardness of offline multi-objective optimization. Evolutionary Computation 15(4), 475–491 (2007)
Jägersküpper, J.: How the (1+1) ES using isotropic mutations minimizes positive definite quadratic forms. Theoretical Computer Science 361(1), 38–56 (2006)
Jägersküpper, J.: Algorithmic analysis of a basic evolutionary algorithm for continuous optimization. Theoretical Computer Science 379(3), 329–347 (2007)
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 1998. LNCS, vol. 1498, pp. 292–304. Springer, Heidelberg (1998)
Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., Grunert da Fonseca, V.: Performance Assessment of Multiobjective Optimizers: An Analysis and Review. IEEE Transactions on Evolutionary Computation 7(2), 117–132 (2003)
Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: Multiobjective selection based on dominated hypervolume. European Journal of Operational Research 181(3), 1653–1669 (2007)
Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6(2), 182–197 (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 2004. LNCS, vol. 3242, pp. 832–842. Springer, Heidelberg (2004)
Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the Strength Pareto Evolutionary Algorithm for Multiobjective Optimization. In: Evolutionary Methods for Design, Optimisation and Control with Application to Industrial Problems (EUROGEN 2001), CIMNE, pp. 95–100 (2002)
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Beume, N., Laumanns, M., Rudolph, G. (2010). Convergence Rates of (1+1) Evolutionary Multiobjective Optimization Algorithms. In: Schaefer, R., Cotta, C., Kołodziej, J., Rudolph, G. (eds) Parallel Problem Solving from Nature, PPSN XI. PPSN 2010. Lecture Notes in Computer Science, vol 6238. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15844-5_60
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DOI: https://doi.org/10.1007/978-3-642-15844-5_60
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