Journal of Global Optimization

, Volume 67, Issue 3, pp 601–619 | Cite as

Covers and approximations in multiobjective optimization

  • Daniel Vanderpooten
  • Lakmali Weerasena
  • Margaret M. Wiecek
Article

Abstract

Due to the growing interest in approximation for multiobjective optimization problems (MOPs), a theoretical framework for defining and classifying sets representing or approximating solution sets for MOPs is developed. The concept of tolerance function is proposed as a tool for modeling representation quality. This notion leads to the extension of the traditional dominance relation to \(t\hbox {-}\)dominance. Two types of sets representing the solution sets are defined: covers and approximations. Their properties are examined in a broader context of multiple solution sets, multiple cones, and multiple quality measures. Applications to complex MOPs are included.

Keywords

Multiobjective optimization Nondominated set Pareto set Approximation Cones Tolerance function 

References

  1. 1.
    Angel, E., Bampis, E., Gourves, L.: Approximating the Pareto curve with local search for the bicriteria TSP (1,2) problem. Theor. Comput. Sci. 310(1–3), 135–146 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Angel, E., Bampis, E., Gourves, L.: Approximation algorithms for the bi-criteria weighted max-cut problem. Discrete Appl. Math. 154(12), 1685–1692 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Angel, E., Bampis, E., Kononov, A.: On the approximate tradeoff for bicriteria batching and parallel machine scheduling problems. Theor. Comput. Sci. 306(1–3), 319–338 (2003)CrossRefMATHGoogle Scholar
  4. 4.
    Bazgan, C., Hugot, H., Vanderpooten, D.: Implementing an efficient fptas for the 0–1 multi-objective knapsack problem. Eur. J. Oper. Res. 198(1), 47–56 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bazgan, C., Jamain, F., Vanderpooten, D.: Approximate Pareto sets of minimal size for multi-objective optimization problems. Oper. Res. Lett. 43(1), 1–6 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Coello, C.A.C., Lamont, G.B., Van Veldhuizen, D.A.: Evolutionary Algorithms for Solving Multi-objective Problems. Springer, Heidelberg (2007)MATHGoogle Scholar
  7. 7.
    Dandurand, B., Guarneri, P., Fadel, G., Wiecek, M.M.: Bilevel multiobjective packaging optimization for automotive design. Struct. Multidiscipl. Optim. 50(4), 663–682 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Diakonikolas, I., Yannakakis, M.: Small approximate Pareto sets for bi-objective shortest paths and other problems. SIAM J. Comput. 39(4), 1340–1371 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Efremov, R.V., Kamenev, G.K.: Properties of a method for polyhedral approximation of the feasible criterion set in convex multiobjective problems. Ann. Oper. Res. 166(1), 271–279 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ehrgott, M.: Approximation algorithms for combinatorial multicriteria optimization problems. Int. Trans. Oper. Res. 7(1), 5–31 (2000)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ehrgott, M., Johnston, R.: Optimisation of beam directions in intensity modulated radiation therapy planning. OR Spectr. 25, 251–264 (2003)CrossRefMATHGoogle Scholar
  12. 12.
    Ehrgott, M., Wiecek, M.M.: Multiobjective programming. In: Figueira, J., Greco, S., Ehrgott, M. (eds.) Multiple Criteria Decision Analysis: State of the Art Surveys, pp. 667–722. Springer, New York (2005)CrossRefGoogle Scholar
  13. 13.
    Engau, A., Wiecek, M.M.: Cone characterizations of approximate solutions in real-vector optimization. J. Optim. Theory Appl. 134(3), 499–513 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Erlebach, T., Kellerer, H., Pferschy, U.: Approximating multiobjective knapsack problems. Manag. Sci. 48(12), 1603–1612 (2002)CrossRefMATHGoogle Scholar
  15. 15.
    Faulkenberg, S.L., Wiecek, M.M.: On the quality of discrete representations in multiple objective programming. Optim. Eng. 11(3), 423–440 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Gardenghi, M., Gómez, T., Miguel, F., Wiecek, M.M.: Algebra of efficient sets for multiobjective complex systems. J. Optim. Theory Appl. 149(2), 385–410 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Goel, T., Vaidyanathan, R., Haftka, R.T., Shyy, W., Queipo, N.V., Tucker, K.: Response surface approximation of Pareto optimal front in multi-objective optimization. Comput. Methods Appl. Mech. Eng. 196, 879–893 (2007)CrossRefMATHGoogle Scholar
  18. 18.
    Goh, C.J., Yang, X.Q.: Analytic efficient solution set for multi-criteria quadratic programs. Eur. J. Oper. Res. 92(1), 166–181 (1996)CrossRefMATHGoogle Scholar
  19. 19.
    Gomez, T., Gonzalez, M., Luque, M., Miguel, M., Ruiz, F.: Multiple objectives decomposition coordination methods for hierarchical organizations. Eur. J. Oper. Res. 133(2), 323–341 (2001)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Guarneri, P., Wiecek, M.M.: Pareto-based negotiation in distributed multidisciplinary design. Struct. Multidiscipl. Optim. (2015). doi:10.1007/s00158-015-1348-3 Google Scholar
  21. 21.
    Haftka, R.T., Watson, L.T.: Multidisciplinary design optimization with quasiseparable subsystems. Optim. Eng. 6(1), 9–20 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Hansen, P.: Bicriterion path problems. In: Fandel, G., Gal, T. (eds.) Multiple Criteria Decision Making: Theory and Applications, pp. 109–127. Springer, Berlin (1980)CrossRefGoogle Scholar
  23. 23.
    Hartikainen, M., Miettinen, K., Wiecek, M.M.: Constructing a Pareto front approximation for decision making. Math. Methods Oper. Res. 73(2), 209–234 (2011)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Hartikainen, M., Miettinen, K., Wiecek, M.M.: PAINT: Pareto front interpolation for nonlinear multiobjective optimization. Comput. Optim. Appl. 52(3), 845–867 (2012)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Helbig, S., Pateva, D.: On several concepts for \(\varepsilon \text{- }\)efficiency. OR Spectr. 16(2), 179–186 (1994)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Henig, M.I.: The domination property in multicriteria optimization. J. Math. Anal. Appl. 114, 7–16 (1986)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Hong, S.P., Chung, S.J., Park, B.H.: A fully polynomial bicriteria approximation scheme for the constrained spanning tree problem. Oper. Res. Lett. 32(3), 233–239 (2004)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Huang, C.H., Galuski, J., Bloebaum, C.L.: Multi-objective Pareto concurrent subspace optimization for multidisciplinary design. AIAA J. 45(8), 1894–1906 (2007)CrossRefGoogle Scholar
  29. 29.
    Hunt, B.J., Blouin, V.Y., Wiecek, M.M.: Relative importance of design criteria: a preference modeling approach. J. Mech. Des. 129(9), 907–914 (2007)CrossRefGoogle Scholar
  30. 30.
    Hunt, B.J., Wiecek, M.M., Hughes, C.: Relative importance of criteria in multiobjective programming: a cone-based approach. Eur. J. Oper. Res. 207(2), 936–945 (2010)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Jilla, C.D., Miller, D.W.: Multi-objective, multidisciplinary design optimization methodology for distributed satellite systems. J. Spacecr. Rockets 41, 39–50 (2004)CrossRefGoogle Scholar
  32. 32.
    Kang, N., Kokkolaras, M., Papalambros, P.Y.: Solving multiobjective optimization problems using quasi-separable MDO formulations and analytical target cascading. Struct. Multidiscipl. Optim. (2014). doi:10.1007/s00158-014-1144-5 MathSciNetGoogle Scholar
  33. 33.
    Klimova, O.N., Noghin, V.D.: Using interdependent information on the relative importance of criteria in decision making. Comput. Math. Math. Phys. 46(12), 2178–2190 (2006)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Koltun, V., Papadimitriou, C.: Approximately dominating representatives. Theor. Comput. Sci. 371(3), 148–154 (2007)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Kutateladze, S.S.: Convex \(\varepsilon \text{- }\)programming. Sov. Math. Doklady 20(2), 391–393 (1979)MATHGoogle Scholar
  36. 36.
    Laumanns, M., Zenklusen, R.: Stochastic convergence of random search methods to fixed size Pareto front approximations. Eur. J. Oper. Res. 213(2), 414–421 (2011)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Legriel J., Guernic C.L., Cotton S., & Maler O.: Approximating the Pareto front of the multi-criteria optimization problems. In: 19th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS), pp. 69–83 (2010)Google Scholar
  38. 38.
    Loridan, P.: \(\varepsilon \)-solutions in vector minimization problems. J. Optim. Theory Appl. 43(2), 265–276 (1984)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Manthey, B.: On approximating multicriteria TSP. ACM Trans. Algorithms 8(2), 1–18 (2012)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Martin, J., Bielza, C., Insua, D.R.: Approximating nondominated sets in continuous multiobjective optimization problems. Naval Res. Logist. 52(5), 469–480 (2005)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Myung, Y.-S., Kim, H.-G., Tcha, D.-W.: A bi-objective uncapacitated facility location problem. Eur. J. Oper. Res. 100(3), 608–616 (1997)CrossRefMATHGoogle Scholar
  42. 42.
    Naderi, B., Aminnayeri, M., Piri, M., Haeri Yazdi, M.H.: Multi-objective no-wait flowshop scheduling problems: models and algorithms. Int. J. Prod. Res. 50(10), 2592–2608 (2012)CrossRefGoogle Scholar
  43. 43.
    Noghin, V.D.: Relative importance of criteria: a quantitative approach. J. Multicrit. Decis. Anal. 6(6), 355–363 (1997)CrossRefMATHGoogle Scholar
  44. 44.
    Office of Naval Research: Special Notice 13-SN-0009 Special Program Announcement for 2013 Office of Naval Research Research Opportunity: Computational Methods for Decision Making (2013). http://www.onr.navy.mil/. Accessed May 2015
  45. 45.
    Papadimitriou C.H., Yannakakis M.: On the approximability of trade-offs and optimal access of web sources. In: Proceedings 41st Annual Symposium on Foundations of Computer Science, pp. 86–92 (2000)Google Scholar
  46. 46.
    Peri, D., Campana, E.F.: Multidisciplinary design optimization of a naval surface combatant. J. Ship Res. 47, 1–12 (2003)Google Scholar
  47. 47.
    Roy, B., Figueira, J.R., Almeida-Dias, J.: Discriminating thresholds as a tool to cope with imperfect knowledge in multiple criteria decision aiding: theoretical results and practical issues. Omega 43(11), 9–20 (2014)CrossRefGoogle Scholar
  48. 48.
    Roy, B., Vincke, Ph: Relational systems of preference with one or more pseudo-criteria: some new concepts and results. Manag. Sci. 30(11), 1323–1335 (1984)CrossRefMATHGoogle Scholar
  49. 49.
    Ruzika, S., Wiecek, M.M.: Survey paper: approximation methods in multiobjective programming. J. Optim. Theory Appl. 126(3), 473–571 (2005)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, Orlando (1985)MATHGoogle Scholar
  51. 51.
    Sayin, S.: Measuring the quality of discrete representations of efficient sets in multiple-objective mathematical programming. Math. Program. 87(3), 543–560 (2000)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Stein, C., Wein, J.: On the existence of schedules that are near-optimal for both makespan and total weighted completion time. Oper. Res. Lett. 21(3), 115–122 (1997)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Steuer, R.E.: Multiple Criteria Optimization: Theory, Computation, and Application. Wiley, New York (1986)MATHGoogle Scholar
  54. 54.
    Tsaggouris, G., Zaroliagis, C.: Multiobjective optimization: improved FPTAS for shortest paths and non-linear objectives with applications. Theory Comput. Syst. 45(1), 162–186 (2009)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Vassilvitskii, S., Yannakakis, M.: Efficiently computing succinct trade-off curves. Theor. Comput. Sci. 348(2–3), 334–356 (2005)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Warburton, A.: Approximation of Pareto optima in multiple-objective, shortest-path problems. Oper. Res. 35(1), 70–79 (1987)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    White, D.J.: Epsilon efficiency. J. Optim. Theory Appl. 49(2), 319–337 (1986)MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Wiecek, M.M.: Advances in cone-based preference modeling for decision making with multiple criteria. Decis. Mak. Manuf. Serv. 1(1–2), 153–173 (2007)MathSciNetMATHGoogle Scholar
  59. 59.
    Yu, P.L.: Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14(3), 319–377 (1974)MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Zhang, K.S., Han, Z.H., Li, W.J., Song, W.P.: Bilevel adaptive weighted sum method for multidisciplinary multi-objective optimization. AIAA J. 46(10), 2611–2622 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York (outside the USA) 2016

Authors and Affiliations

  • Daniel Vanderpooten
    • 1
  • Lakmali Weerasena
    • 2
  • Margaret M. Wiecek
    • 2
  1. 1.LAMSADEPSL, Université Paris DauphineParis Cedex 16France
  2. 2.Department of Mathematical SciencesClemson UniversityClemsonUSA

Personalised recommendations