Advertisement

A Parallel Tabu Search Heuristic to Approximate Uniform Designs for Reference Set Based MOEAs

  • Alberto Rodríguez SánchezEmail author
  • Antonin Ponsich
  • Antonio López Jaimes
  • Saúl Zapotecas Martínez
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11411)

Abstract

Recent Multi-objective Optimization (MO) algorithms such as MOEA/D or NSGA-III make use of an uniformly scattered set of reference points indicating search directions in the objective space in order to achieve diversity. Apart from the mixture-design based techniques such as the simplex lattice, the mixture-design based techniques, there exists the Uniform Design (DU) approach, which is based on based on the minimization of a discrepancy metric, which measures how well equidistributed the points are in a sample space. In this work, this minimization problem is tackled through the \(L_2\) discrepancy function and solved with a parallel heuristic based on several Tabu Searches, distributed over multiple processors. The computational burden does not allow us to perform many executions but the solution technique is able to produce nearly Uniform Designs. These point sets were used to solve some classical MO test problems with two different algorithms, MOEA/D and NSGA-III. The computational experiments proves that, when the dimension increases, the algorithms working with a set generated by Uniform Design significantly outperform their counterpart working with other state-of-the-art strategies, such as the simplex lattice or two layer designs.

Keywords

NSGA-III MOEA/D Uniform design Weight vector Multi-objective evolutionary algorithm (MOEA) 

References

  1. 1.
    Cornell, J.A.: Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data, vol. 403. Wiley, Hoboken (2011)CrossRefGoogle Scholar
  2. 2.
    Das, I., Dennis, J.E.: Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J. Optim. 8(3), 631–657 (1998).  https://doi.org/10.1137/S1052623496307510MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Deb, K., Jain, H.: An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part i: solving problems with box constraints. IEEE Trans. Evol. Comput. 18(4), 577–601 (2014).  https://doi.org/10.1109/TEVC.2013.2281535CrossRefGoogle Scholar
  4. 4.
    Fang, K.T.: Uniform design: application of number-theoretic methods in experimental design. Acta Math. Appl. Sin. 3, 363–372 (1980)Google Scholar
  5. 5.
    Fang, K.T., Qin, H.: A note on construction of nearly uniform designs with large number of runs. Stat. Prob. Lett. 61(2), 215–224 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fang, K., Lin, D.: J. Uniform designs and their application in industry. In: Handbook on Statistics in Industry, pp. 131–170. Elsevier, Amsterdam (2003)Google Scholar
  7. 7.
    Glover, F.: Future paths for integer programming and links to artificial intelligence. Comput. Oper. Res. 13(5), 533–549 (1986).  https://doi.org/10.1016/0305-0548(86)90048-1MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hua, L.K., Wang, Y.: Applications of Number Theory to Numerical Analysis. Springer, Heidelberg (2012)Google Scholar
  9. 9.
    Ma, C., Fang, K.T.: A new approach to construction of nearly uniform designs. Int. J. Mater. Product Technol. 20(1–3), 115–126 (2004)CrossRefGoogle Scholar
  10. 10.
    Roth, K.F.: Rational approximations to algebraic numbers. Mathematika 2(1), 1–20 (1955)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Scheffé, H.: Experiments with mixtures. J. R. Stat. Soc. Ser. B (Methodol.) 344–360 (1958)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Tan, Y.Y., et al.: MOEA/D+ uniform design: a new version of MOEA/D for optimization problems with many objectives. Comput. Oper. Res. 40(6), 1648–1660 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Wang, Y., Hickernell, F.J.: An historical overview of lattice point sets. In: Fang, K.T., Niederreiter, H., Hickernell, F.J. (eds.) Monte Carlo and Quasi-Monte Carlo Methods, pp. 158–167. Springer, Heidelberg (2002).  https://doi.org/10.1007/978-3-642-56046-0_10CrossRefGoogle Scholar
  14. 14.
    Xie, M.Y., Fang, K.T.: Admissibility and minimaxity of the uniform design measure in nonparametric regression model. J. Stat. Plan. Inference 83(1), 101–111 (2000)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zapotecas-Martínez, S., Aguirre, H.E., Tanaka, K., Coello Coello, C.A.: On the low-discrepancy sequences and their use in MOEA/D for high-dimensional objective spaces. In: 2015 IEEE Congress on Evolutionary Computation (CEC), pp. 2835–2842. IEEE (2015)Google Scholar
  16. 16.
    Zapotecas-Martínez, S., Coello, C.A.C., Aguirre, H.E., Tanaka, K.: A review of features and limitations of existing scalable multi-objective test suites. IEEE Trans. Evol. Comput. 1 (2018).  https://doi.org/10.1109/TEVC.2018.2836912CrossRefGoogle Scholar
  17. 17.
    Zapotecas-Martínez, S., López-Jaimes, A., García-Nájera, A.: LIBEA: a Lebesgue indicator-based evolutionary algorithm for multi-objective optimization. Swarm Evol. Comput. (2018).  https://doi.org/10.1016/j.swevo.2018.05.004, http://www.sciencedirect.com/science/article/pii/S2210650217307216CrossRefGoogle Scholar
  18. 18.
    Zhang, Q., Li, H.: MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans. Evol. Comput. 11(6), 712–731 (2007).  https://doi.org/10.1109/TEVC.2007.892759CrossRefGoogle Scholar
  19. 19.
    Zhang, Q., et al.: Multiobjective optimization test instances for the CEC 2009 special session and competition. University of Essex, Colchester, UK and Nanyang technological University, Singapore, special session on performance assessment of multi-objective optimization algorithms, Technical report 264 (2008)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alberto Rodríguez Sánchez
    • 1
    Email author
  • Antonin Ponsich
    • 1
  • Antonio López Jaimes
    • 2
  • Saúl Zapotecas Martínez
    • 2
  1. 1.Dpto. de SistemasUniversidad Autónoma Metropolitana AzcapotzalcoMexico CityMexico
  2. 2.Dpto. de Matemáticas Aplicadas y SistemasUniversidad Autónoma Metropolitana CuajimalpaMexico CityMexico

Personalised recommendations