Abstract
In multi-objective optimization, set-based performance indicators have become the state of the art for assessing the quality of Pareto front approximations. As a consequence, they are also more and more used within the design of multi-objective optimization algorithms. The R2 and the Hypervolume (HV) indicator represent two popular examples. In order to understand the behavior and the approximations preferred by these indicators and algorithms, a comprehensive knowledge of the indicator’s properties is required. Whereas this knowledge is available for the HV, we presented a first approach in this direction for the R2 indicator just recently. In this paper, we build upon this knowledge and enhance the considerations with respect to the integration of preferences into the R2 indicator. More specifically, we analyze the effect of the reference point, the domain of the weights, and the distribution of weight vectors on the optimization of μ solutions with respect to the R2 indicator. By means of theoretical findings and empirical evidence, we show the potentials of these three possibilities using the optimal distribution of μ solutions for exemplary setups.
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References
Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Theory of the Hypervolume Indicator: Optimal μ-Distributions and the Choice of the Reference Point. In: Foundations of Genetic Algorithms (FOGA 2009), pp. 87–102. ACM, New York (2009)
Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Hypervolume-based Multiobjective Optimization: Theoretical Foundations and Practical Implications. Theoretical Computer Science 425, 75–103 (2012)
Auger, A., Hansen, N.: A Restart CMA Evolution Strategy With Increasing Population Size. In: Congress on Evolutionary Computation (CEC 2005), pp. 1769–1776. IEEE Press (2005)
Bader, J., Zitzler, E.: HypE: An Algorithm for Fast Hypervolume-Based Many-Objective Optimization. Evolutionary Computation 19(1), 45–76 (2011)
Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: Multiobjective selection based on dominated hypervolume. European Journal of Operational Research 181(3), 1653–1669 (2007)
Beume, N., Naujoks, B., Preuss, M., Rudolph, G., Wagner, T.: Effects of 1-Greedy \(\mathcal{S}\)-Metric-Selection on Innumerably Large Pareto Fronts. In: Ehrgott, M., Fonseca, C.M., Gandibleux, X., Hao, J.-K., Sevaux, M., et al. (eds.) EMO 2009. LNCS, vol. 5467, pp. 21–35. Springer, Heidelberg (2009)
Bozkurt, B., Fowler, J.W., Gel, E.S., Kim, B., Köksalan, M., Wallenius, J.: Quantitative Comparison of Approximate Solution Sets for Multicriteria Optimization Problems with Weighted Tchebycheff Preference Function. Operations Research 58(3), 650–659 (2010)
Brockhoff, D., Wagner, T., Trautmann, H.: On the Properties of the R2 Indicator. In: Genetic and Evolutionary Computation Conference (GECCO 2012), pp. 465–472 (2012)
Byrd, R.H., Lu, P., Nocedal, J., Zhu, C.: A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing 16, 1190–1208 (1994)
Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable Multi-Objective Optimization Test Problems. In: Congress on Evolutionary Computation (CEC 2002), pp. 825–830. IEEE Press (2002)
Hansen, M.P., Jaszkiewicz, A.: Evaluating The Quality of Approximations of the Non-Dominated Set. Technical report, Institute of Mathematical Modeling, Technical University of Denmark, IMM Technical Report IMM-REP-1998-7 (1998)
Hansen, N.: The CMA Evolution Strategy: A Comparing Review. In: Lozano, J., Larrañaga, P., Inza, I., Bengoetxea, E. (eds.) Towards a New Evolutionary Computation. STUDFUZZ, vol. 192, pp. 75–102. Springer, Heidelberg (2006)
Hansen, N.: CMA Evolution Strategy Source Code (2012), http://www.lri.fr/~hansen/cmaes_inmatlab.html
Igel, C., Hansen, N., Roth, S.: Covariance Matrix Adaptation for Multi-objective Optimization. Evolutionary Computation 15(1), 1–28 (2007)
Knowles, J.: ParEGO: A Hybrid Algorithm With On-Line Landscape Approximation for Expensive Multiobjective Optimization Problems. IEEE Transactions on Evolutionary Computation 10(1), 50–66 (2005)
Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer, Boston (1999)
Zhang, Q., Li, H.: MOEA/D: A multi-objective evolutionary algorithm based on decomposition. IEEE Transactions on Evolutionary Computation 11(6), 712–731 (2007)
Zitzler, E.: Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications. PhD thesis, ETH Zurich, Switzerland (1999)
Zitzler, E., Deb, K., Thiele, L.: Comparison of Multiobjective Evolutionary Algorithms: Empirical Results. Evolutionary Computation 8(2), 173–195 (2000)
Zitzler, E., Knowles, J., Thiele, L.: Quality Assessment of Pareto Set Approximations. In: Branke, J., Deb, K., Miettinen, K., Słowiński, R. (eds.) Multiobjective Optimization. LNCS, vol. 5252, pp. 373–404. Springer, Heidelberg (2008)
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., 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)
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Wagner, T., Trautmann, H., Brockhoff, D. (2013). Preference Articulation by Means of the R2 Indicator. In: Purshouse, R.C., Fleming, P.J., Fonseca, C.M., Greco, S., Shaw, J. (eds) Evolutionary Multi-Criterion Optimization. EMO 2013. Lecture Notes in Computer Science, vol 7811. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37140-0_10
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DOI: https://doi.org/10.1007/978-3-642-37140-0_10
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