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Interval-based ranking in noisy evolutionary multi-objective optimization

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Abstract

As one of the most competitive approaches to multi-objective optimization, evolutionary algorithms have been shown to obtain very good results for many real-world multi-objective problems. One of the issues that can affect the performance of these algorithms is the uncertainty in the quality of the solutions which is usually represented with the noise in the objective values. Therefore, handling noisy objectives in evolutionary multi-objective optimization algorithms becomes very important and is gaining more attention in recent years. In this paper we present \(\alpha \)-degree Pareto dominance relation for ordering the solutions in multi-objective optimization when the values of the objective functions are given as intervals. Based on this dominance relation, we propose an adaptation of the non-dominated sorting algorithm for ranking the solutions. This ranking method is then used in a standard multi-objective evolutionary algorithm and a recently proposed novel multi-objective estimation of distribution algorithm based on joint variable-objective probabilistic modeling, and applied to a set of multi-objective problems with different levels of independent noise. The experimental results show that the use of the proposed method for solution ranking allows to approximate Pareto sets which are considerably better than those obtained when using the dominance probability-based ranking method, which is one of the main methods for noise handling in multi-objective optimization.

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References

  1. Pareto, V.: The new theories of economics. J. Polit. Econ. 5(4), 485–502 (1897)

    Article  Google Scholar 

  2. Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Machine Learning, 1st edn. Addison-Wesley Longman Publishing Co., Inc., Boston (1989)

    MATH  Google Scholar 

  3. Abraham, A., Jain, L., Goldberg, R. (eds.): Evolutionary Multiobjective Optimization: Theoretical Advances and Applications. Springer, London (2005)

    Google Scholar 

  4. Coello Coello, C.A., Lamont, G.B., Van Veldhuizen, D.A.: Evolutionary Algorithms for Solving Multi-objective Problems, 2nd edn. Springer, Berlin (2007)

    MATH  Google Scholar 

  5. Deb, K.: Multi-objective Optimization Using Evolutionary Algorithms. Wiley, London (2001)

    MATH  Google Scholar 

  6. Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: empirical results. Evol. Comput. 8(2), 173–195 (2000)

    Article  Google Scholar 

  7. Larrañaga, P., Lozano, J.A. (eds.): Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation. Kluwer Academic Publishers, Norwell (2001)

    Google Scholar 

  8. Lozano, J.A., Larrañaga, P., Inza, I., Bengoetxea, E. (eds.): Towards a New Evolutionary Computation: Advances on Estimation of Distribution Algorithms, vol. 192. Studies in Fuzziness and Soft Computing. Springer (2006)

  9. Mühlenbein, H., Paaß, G.: From recombination of genes to the estimation of distributions I. Binary parameters. In: Fourth International Conference on Parallel Problem Solving from Nature (PPSN IV), vol. 1141. Lecture Notes in Computer Science, pp. 178–187. Springer (1996)

  10. Pelikan, M.: Hierarchical Bayesian optimization algorithm. Toward a new generation of evolutionary algorithms, vol. 170. In: Studies in Fuzziness and Soft Computing, 1st ed. Springer (2005)

  11. Martí, L., Garcia, J., Berlanga, A., Coello Coello, C.A., Molina, J.M.: On current model-building methods for multi-objective estimation of distribution algorithms: shortcommings and directions for improvement. Tech. Rep. GIAA2010E001, Department of Informatics, Universidad Carlos III de Madrid (2010)

  12. Pelikan, M., Sastry, K., Goldberg, D.: Multiobjective estimation of distribution algorithms. In: Scalable Optimization via Probabilistic Modeling, vol. 33. Studies in Computational Intelligence, pp. 223–248. Springer (2006)

  13. Thierens, D., Bosman, P.A.N.: Multi-objective mixture-based iterated density estimation evolutionary algorithms. In: Conference on Genetic and Evolutionary Computation (GECCO ’01), pp. 663–670. Morgan Kaufmann (2001)

  14. Karshenas, H., Santana, R., Bielza, C., Larrañaga, P.: Multi-objective estimation of distribution algorithm based on joint modeling of objectives and variables. IEEE Trans. Evol. Comput. 18(4), 519–542 (2014)

  15. Jin, Y., Branke, J.: Evolutionary optimization in uncertain environments—a survey. IEEE Trans. Evol. Comput. 9(3), 303–317 (2005)

    Article  Google Scholar 

  16. Tan, K., Goh, C.: Handling uncertainties in evolutionary multi-objective optimization. Computational Intelligence: Research Frontiers, vol. 5050. Lecture Notes in Computer Science, pp. 262–292. Springer (2008)

  17. Soares, G., Guimaraes, F., Maia, C., Vasconcelos, J., Jaulin, L.: Interval robust multi-objective evolutionary algorithm. In: IEEE Congress on Evolutionary Computation (CEC’09), pp. 1637–1643 (2009)

  18. Goh, C., Tan, K., Cheong, C., Ong, Y.: An investigation on noise-induced features in robust evolutionary multi-objective optimization. Expert Syst. Appl. 37(8), 5960–5980 (2010)

    Article  Google Scholar 

  19. Hughes, E.: Evolutionary multi-objective ranking with uncertainty and noise. In: Evolutionary Multi-criterion Optimization (EMO’01), vol. 1993. Lecture Notes in Computer Science, pp. 329–343. Springer (2001)

  20. Teich, J.: Pareto-front exploration with uncertain objectives. In: Evolutionary Multi-Criterion Optimization, vol. 1993. Lecture Notes in Computer Science, pp. 314–328. Springer (2001)

  21. Greiner, D., Galván, B., Aznárez, J., Maeso, O., Winter, G.: Robust design of noise attenuation barriers with evolutionary multiobjective algorithms and the boundary element method. In: Evolutionary Multi-criterion Optimization, vol. 5467. Lecture Notes in Computer Science, pp. 261–274. Springer (2009)

  22. Bui, L.T., Abbass, H.A., Essam, D.: Fitness inheritance for noisy evolutionary multi-objective optimization. In: Conference on Genetic and Evolutionary Computation (GECCO’05), pp. 779–785. ACM (2005)

  23. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)

    Article  Google Scholar 

  24. Bui, L.T., Essam, D., Abbass, H.A., Green, D.: Performance analysis of evolutionary multi-objective optimization methods in noisy environments. Complex. Int. 11, 29–39 (2005)

    Google Scholar 

  25. Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the strength Pareto evolutionary algorithm for multiobjective optimization. In: Evolutionary Methods for Design Optimization and Control with Applications to Industrial Problems (EUROGEN ’01), pp. 95–100. International Center for Numerical Methods in Engineering (2001)

  26. Fieldsend, J., Everson, R.: Multi-objective optimisation in the presence of uncertainty. IEEE Congr. Evol. Comput. (CEC’05) 1, 243–250 (2005)

    Google Scholar 

  27. Büche, D., Stoll, P., Dornberger, R., Koumoutsakos, P.: Multiobjective evolutionary algorithm for the optimization of noisy combustion processes. IEEE Trans. Syst. Man Cybern. Part C: Appl. Rev. 32(4), 460–473 (2002)

    Article  Google Scholar 

  28. Zitzler, E., Thiele, L.: Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans. Evol. Comput. 3(4), 257–271 (1999)

    Article  Google Scholar 

  29. Babbar, M., Lakshmikantha, A., Goldberg, D.E.: A modified NSGA-II to solve noisy multiobjective problems. In: Genetic and Evolutionary Computation Conference (GECCO’03), pp. 21–27 (2003)

  30. Goh, C., Tan, K.: An investigation on noisy environments in evolutionary multiobjective optimization. IEEE Trans. Evol. Comput. 11(3), 354–381 (2007)

    Article  Google Scholar 

  31. Eskandari, H., Geiger, C.: Evolutionary multiobjective optimization in noisy problem environments. J. Heuristics 15(6), 559–595 (2009)

    Article  MATH  Google Scholar 

  32. Bui, L.T., Abbass, H.A., Essam, D.: Localization for solving noisy multi-objective optimization problems. Evol. Comput. 17(3), 379–409 (2009)

    Article  Google Scholar 

  33. Syberfeldt, A., Ng, A., John, R.I., Moore, P.: Evolutionary optimisation of noisy multi-objective problems using confidence-based dynamic resampling. Eur. J. Oper. Res. 204(3), 533–544 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  34. Park, T., Ryu, K.R.: Accumulative sampling for noisy evolutionary multi-objective optimization. In: 13th Annual Conference on Genetic and Evolutionary Computation (GECCO’11), pp. 793–800. ACM (2011)

  35. Mehnen, J., Trautmann, H., Tiwari, A.: Introducing user preference using desirability functions in multi-objective evolutionary optimisation of noisy processes. In: IEEE Congress on Evolutionary Computation (CEC’07), pp. 2687–2694 (2007)

  36. Woźniak, P.: Preferences in evolutionary multi-objective optimisation with noisy fitness functions: hardware in the loop study. In: International Multiconference on Computer Science and Information Technology, pp. 337–346 (2007)

  37. Kaji, H., Ikeda, K., Kita, H.: Uncertainty of constraint function in evolutionary multi-objective optimization. In: IEEE Congress on Evolutionary Computation (CEC’09), pp. 1621–1628 (2009)

  38. Basseur, M., Zitzler, E.: Handling uncertainty in indicator-based multiobjective optimization. Int. J. Comput. Intell. Res. 2(3), 255–272 (2006)

    MathSciNet  Google Scholar 

  39. Boonma, P., Suzuki, J.: A confidence-based dominance operator in evolutionary algorithms for noisy multiobjective optimization problems. In: 21st International Conference on Tools with Artificial Intelligence (ICTAI’09), pp. 387–394 (2009)

  40. Pearl, J.: Bayesian networks: a model of self-activated memory for evidential reasoning. In: 7th Conference of the Cognitive Science Society, pp. 329–334 (1985)

  41. Bielza, C., Li, G., Larrañaga, P.: Multi-dimensional classification with Bayesian networks. Int. J. Approximate Reasoning 52(6), 705–727 (2011)

    Article  MATH  Google Scholar 

  42. de Waal, P., van der Gaag, L.: Inference and learning in multi-dimensional Bayesian network classifiers. In: Symbolic and Quantitative Approaches to Reasoning with Uncertainty, pp. 501–511 (2007)

  43. Ahn, C.W., Ramakrishna, R.S.: Multiobjective real-coded Bayesian optimization algorithm revisited: diversity preservation. In: 9th Annual Conference on Genetic and Evolutionary Computation (GECCO ’07), pp. 593–600. ACM (2007)

  44. Katsumata, Y., Terano, T.: Bayesian optimization algorithm for multi-objective solutions: application to electric equipment configuration problems in a power plant. IEEE Congr. Evol. Comput. (CEC ’03) 2, 1101–1107 (2003)

    Google Scholar 

  45. Khan, N., Goldberg, D.E., Pelikan, M.: Multiple-objective Bayesian optimization algorithm. In: Conference on Genetic and Evolutionary Computation (GECCO ’02), p. 684. Morgan Kaufmann (2002)

  46. Laumanns, M., Očenášek, J.: Bayesian optimization algorithms for multi-objective optimization. In: 7th International Conference on Parallel Problem Solving from Nature (PPSN VII), vol. 2439. Lecture Notes in Computer Science, pp. 298–307. Springer (2002)

  47. Pelikan, M., Sastry, K., Goldberg, D.E.: Multiobjective hBOA, clustering, and scalability. In: Conference on Genetic and Evolutionary Computation (GECCO ’05), pp. 663–670. ACM (2005)

  48. Schwarz, J., Očenášek, J.: Multiobjective Bayesian optimization algorithm for combinatorial problems: theory and practice. Neural Netw. World 11(5), 423–442 (2001)

    Google Scholar 

  49. Corani, G., Antonucci, A., Zaffalon, M.: Bayesian networks with imprecise probabilities: theory and application to classification. Tech. Rep. IDSIA-02-10, Dalle Molle Institute for Artificial Intelligence (2010)

  50. Buntine, W.: Theory refinement on Bayesian networks. In: 7th Annual Conference on Uncertainty in Artificial Intelligence (UAI ’91), pp. 52–60. Morgan Kaufmann (1991)

  51. Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6(2), 461–464 (1978)

    Article  MATH  Google Scholar 

  52. Henrion, M.: Propagating uncertainty in Bayesian networks by probabilistic logic sampling. In: Second Annual Conference on Uncertainty in Artificial Intelligence (UAI ’86), vol. 2, pp. 149–163. Elsevier (1986)

  53. Arnold, D., Beyer, H.G.: A general noise model and its effects on evolution strategy performance. IEEE Trans. Evol. Comput. 10(4), 380–391 (2006)

    Article  Google Scholar 

  54. Huband, S., Hingston, P., Barone, L., While, L.: A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans. Evol. Comput. 10(5), 477–506 (2006)

    Article  Google Scholar 

  55. Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable multi-objective optimization test problems. IEEE Congr. Evol. Comput. (CEC ’02) 1, 825–830 (2002)

    Google Scholar 

  56. Okabe, T., Jin, Y., Olhofer, M., Sendhoff, B.: On test functions for evolutionary multi-objective optimization. In: 8th International Conference on Parallel Problem Solving from Nature (PPSN VIII), vol. 3242. Lecture Notes in Computer Science, pp. 792–802. Springer (2004)

  57. Deb, K., Agrawal, B.: Simulated binary crossover for continuous search space. Complex Syst. 9(2), 115–148 (1995)

    MATH  MathSciNet  Google Scholar 

  58. Deb, K., Goyal, M.: A combined genetic adaptive search (GeneAS) for engineering design. Comput. Sci. Inf. 26(4), 30–45 (1996)

    Google Scholar 

  59. Coello Coello, C.A., Cortés, N.C.: Solving multiobjective optimization problems using an artificial immune system. Genet. Program. Evol. Mach. 6, 163–190 (2005)

    Article  Google Scholar 

  60. Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C., da Fonseca, V.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evol. Comput. 7(2), 117–132 (2003)

    Article  Google Scholar 

  61. Schott, J.R.: Fault Tolerant Design Using Single and Multicriteria Genetic Algorithm Optimization. Master’s thesis, Massachusetts Institute of Technology, Department of Aeronautics and Astronautics (1995)

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Acknowledgments

This work has been partially supported by TIN2010-20900-C04-04 and Cajal Blue Brain projects (Spanish Ministry of Economy and Competitiveness).

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  1. Computational Intelligence Group, Facultad de Informática, Universidad Politécnica de Madrid, Campus de Montegancedo, 28660, Boadilla del Monte, Madrid, Spain

    Hossein Karshenas, Concha Bielza & Pedro Larrañaga

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  1. Hossein Karshenas
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Correspondence to Hossein Karshenas.

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Karshenas, H., Bielza, C. & Larrañaga, P. Interval-based ranking in noisy evolutionary multi-objective optimization. Comput Optim Appl 61, 517–555 (2015). https://doi.org/10.1007/s10589-014-9717-1

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