Soft Computing

, Volume 21, Issue 9, pp 2283–2295 | Cite as

Nadir point estimation for many-objective optimization problems based on emphasized critical regions

Methodologies and Application


Nadir points play an important role in many-objective optimization problems, which describe the ranges of their Pareto fronts. Using nadir points as references, decision makers may obtain their preference information for many-objective optimization problems. As the number of objectives increases, nadir point estimation becomes a more difficult task. In this paper, we propose a novel nadir point estimation method based on emphasized critical regions for many-objective optimization problems. It maintains the non-dominated solutions near extreme points and critical regions after an individual number assignment to different critical regions. Furthermore, it eliminates similar individuals by a novel self-adaptive \(\varepsilon \)-clearing strategy. Our approach has been shown to perform better on many-objective optimization problems (between 10 objectives and 35 objectives) than two other state-of-the-art nadir point estimation approaches.


Multi-objective evolutionary algorithm Nadir point  Critical region Many-objective optimization problem Decision making 



This work was supported by the National Basic Research Program (973 Program) of China (No.2013CB329402), an EPSRC Grant (No. EP/J017515/1) on “DAASE: Dynamic Adaptive Automated Software Engineering”, the Program for Cheung Kong Scholars and Innovative Research Team in University (No. IRT1170), the National Natural Science Foundation of China (No. 61329302), National Science Foundation of China (Nos. 91438103 and 91438201), and the Fund for Foreign Scholars in University Research and Teaching Programs (the 111 Project) (No. B07048). Xin Yao was supported by a Royal Society Wolfson Research Merit Award.

Compliance with ethical standards

Conflict of interest

The authors declared that they have no conflicts of interest to this work.


  1. Amiri M, Ekhtiari M, Yazdani M (2010) Nadir compromise programming: a model for optimization of multi-objective portfolio problem. Expert Syst Appl 36(6):7222–7226Google Scholar
  2. Bechikh S, Ben Said L, Ghedira K (2010) Estimating nadir point in multi-objective optimization using mobile reference points. In: Evolutionary computation (CEC), 2010 IEEE congress on, IEEE Press, pp 1–9Google Scholar
  3. Ben Said L, Bechikh S, Ghédira K (2010) The r-dominance: a new dominance relation for interactive evolutionary multicriteria decision making. IEEE Trans Evol Comput 14(5):801–818CrossRefGoogle Scholar
  4. Benayoun R, De Montgolfier J, Tergny J, Laritchev O (1971) Linear programming with multiple objective functions: step method (STEM). Math Program 1(1):366–375MathSciNetCrossRefMATHGoogle Scholar
  5. Branke J, Kaußler T, Smidt C, Schmeck H (2000) A multi-population approach to dynamic optimization problems. In: Evolutionary design and manufacture. Springer, pp 299–307Google Scholar
  6. Branke J, Deb K, Dierolf H, Osswald M (2004) Finding knees in multi-objective optimization. In: Parallel problem solving from nature-PPSN VIII. Springer, pp 722–731Google Scholar
  7. Branke J, Deb K, Miettinen K (2008) Multiobjective optimization: interactive and evolutionary approaches. Springer, New YorkCrossRefMATHGoogle Scholar
  8. Chen N, Chen WN, Gong YJ, Zhan ZH, Zhang J, Li Y, Tan YS (2015) An evolutionary algorithm with double-level archives for multiobjective optimization. IEEE Trans Cybern 45(9):1851–1863CrossRefGoogle Scholar
  9. Dai J, Wang W, Xu Q (2013) An uncertainty measure for incomplete decision tables and its applications. IEEE Trans Cybern 43(4):1277–1289CrossRefGoogle Scholar
  10. Deb K, Jain H (2014) 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):577–601CrossRefGoogle Scholar
  11. Deb K, Kumar A (2007) Interactive evolutionary multi-objective optimization and decision-making using reference direction method. In: Proceedings of the genetic and evolutionary computation conference (GECCO 2007). ACM Press, pp 781–788Google Scholar
  12. Deb K, Miettinen K (2009) A review of nadir point estimation procedures using evolutionary approaches: a tale of dimensionality reduction. In: Proceedings of the multiple criterion decision making (MCDM-2008) conference. Springer, pp 1–14Google Scholar
  13. Deb K, Pratap A, Agarwal S, Meyarivan T (2002a) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197CrossRefGoogle Scholar
  14. Deb K, Thiele L, Laumanns M, Zitzler E (2002b) Scalable multi-objective optimization test problems. In: Evolutionary computation (CEC), 2002 IEEE congress on, vol 1. IEEE Press, pp 825–830Google Scholar
  15. Deb K, Chaudhuri S, Miettinen K (2006) Towards estimating nadir objective vector using evolutionary approaches. In: Proceedings of the genetic and evolutionary computation conference (GECCO 2006). ACM Press, pp 643–650Google Scholar
  16. Deb K, Miettinen K, Sharma D (2009) A hybrid integrated multi-objective optimization procedure for estimating nadir point. In: Evolutionary multi-criterion optimization. Springer, pp 569–583Google Scholar
  17. Deb K, Miettinen K, Chaudhuri S (2010) Toward an estimation of nadir objective vector using a hybrid of evolutionary and local search approaches. IEEE Trans Evol Comput 14(6):821–841CrossRefGoogle Scholar
  18. Derrac J, García S, Molina D, Herrera F (2011) A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evol Comput 1(1):3–18CrossRefGoogle Scholar
  19. Dessouky M, Ghiassi M, Davis W (1986) Estimates of the minimum nondominated criterion values in multiple-criteria decision-making. Eng Costs Prod Econ 10(1):95–104CrossRefGoogle Scholar
  20. Ehrgott M, Tenfelde-Podehl D (2003) Computation of ideal and nadir values and implications for their use in MCDM methods. Eur J Oper Res 151(1):119–139MathSciNetCrossRefMATHGoogle Scholar
  21. Fan Z, Liu Y (2010) A method for group decision-making based on multi-granularity uncertain linguistic information. Expert Syst Appl 37(5):4000–4008CrossRefGoogle Scholar
  22. He Y, He Z, Chen H (2015) Intuitionistic fuzzy interaction bonferroni means and its application to multiple attribute decision making. IEEE Trans Cybern 45(1):116–128CrossRefGoogle Scholar
  23. Hsu S, Wang T (2011) Solving multi-criteria decision making with incomplete linguistic preference relations. Expert Syst Appl 38(9):10882–10888CrossRefGoogle Scholar
  24. Huband S, Hingston P, Barone L, While L (2006) A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans Evol Comput 10(5):477–506CrossRefMATHGoogle Scholar
  25. Hughes E (2005) Evolutionary many-objective optimisation: many once or one many? In: Evolutionary computation (CEC), 2005 IEEE congress on, vol 1. IEEE Press, pp 222–227Google Scholar
  26. Ishibuchi H, Tsukamoto N, Nojima Y (2008) Evolutionary many-objective optimization: a short review. In: Evolutionary computation (CEC), 2008 IEEE congress on. IEEE Press, pp 2419–2426Google Scholar
  27. Ishibuchi H, Masuda H, Tanigaki Y, Nojima Y (2014) Review of coevolutionary developments of evolutionary multi-objective and many-objective algorithms and test problems. In: Computational intelligence in multi-criteria decision-making (MCDM), 2014 IEEE symposium on. IEEE, pp 178–184Google Scholar
  28. Jiang S, Ong YS, Zhang J, Feng L (2014) Consistencies and contradictions of performance metrics in multiobjective optimization. IEEE Trans Cybern 44(12):2391–2404CrossRefGoogle Scholar
  29. Ke L, Zhang Q, Battiti R (2013) MOEA/D-ACO: a multiobjective evolutionary algorithm using decomposition and antcolony. IEEE Trans Cybern 43(6):1845–1859CrossRefGoogle Scholar
  30. Khare V, Yao X, Deb K (2003) Performance scaling of multi-objective evolutionary algorithms. In: Fonseca C, Fleming P, Zitzler E, Thiele L, Deb K (eds) Evolutionary multi-criterion optimization. Lecture notes in computer science, vol 2632. Springer, Berlin, pp 376–390CrossRefGoogle Scholar
  31. Klamroth K, Miettinen K (2008) Integrating approximation and interactive decision making in multicriteria optimization. Oper Res 56(1):222–234MathSciNetCrossRefMATHGoogle Scholar
  32. Korhonen P, Salo S, Steuer R (1997) A heuristic for estimating nadir criterion values in multiple objective linear programming. Oper Res 45(5):751–757CrossRefMATHGoogle Scholar
  33. Li K, Kwong S, Zhang Q, Deb K (2015) Interrelationship-based selection for decomposition multiobjective optimization. IEEE Trans Cybern 45(10):2076–2088Google Scholar
  34. Ma X, liu f, Qi Y, Wang X, Li L, Jiao L, Yin M, Gong M (2015) A multiobjective evolutionary algorithm based on decision variable analyses for multi-objective optimization problems with large scale variables. IEEE Trans Evolut Comput PP(99):1–1. doi: 10.1109/TEVC.2015.2455812
  35. Metev B, Vassilev V (2003) A method for nadir point estimation in MOLP problems. Cybern Inf Technol 3(2):15–24MathSciNetGoogle Scholar
  36. Miettinen K (1999) Nonlinear multiobjective optimization. Springer, New YorkMATHGoogle Scholar
  37. Miettinen K, Eskelinen P, Ruiz F, Luque M (2010) NAUTILUS method: an interactive technique in multiobjective optimization based on the nadir point. Eur J Oper Res 206(2):426–434MathSciNetCrossRefMATHGoogle Scholar
  38. Praditwong K, Yao X (2006) A new multi-objective evolutionary optimisation algorithm: the two-archive algorithm. In: Computational intelligence and security, 2006 international conference on, vol 1. IEEE Press, pp 286–291Google Scholar
  39. Praditwong K, Yao X (2007) How well do multi-objective evolutionary algorithms scale to large problems. In: Evolutionary computation (CEC), 2007 IEEE congress on. IEEE Press, pp 3959–3966Google Scholar
  40. Purshouse R, Fleming P (2003) Evolutionary many-objective optimisation: an exploratory analysis. In: Evolutionary computation (CEC), 2003 IEEE congress on, vol 3. IEEE Press, pp 2066–2073Google Scholar
  41. Szczepanski M, Wierzbicki A (2003) Application of multiple criteria evolutionary algorithms to vector optimisation, decision support and reference point approaches. J Telecommun Inf Technol 3:16–33Google Scholar
  42. Thiele L, Miettinen K, Korhonen P, Molina J (2009) A preference-based evolutionary algorithm for multi-objective optimization. Evol Comput 17(3):411–436CrossRefGoogle Scholar
  43. Wang H, Jiao L, Yao X (2015) Two\_Arch2: an improved two-archive algorithm for many-objective optimization. IEEE Trans Evol Comput 19(4):524–541CrossRefGoogle Scholar
  44. Yuan Y, Xu H, Wang B, Yao X (2015a) A new dominance relation based evolutionary algorithm for many-objective optimization. doi: 10.1109/TEVC.2015.2420112
  45. Yuan Y, Xu H, Wang B, Zhang B, Yao X (2015b) Balancing convergence and diversity in decomposition-based many-objective optimizers. doi: 10.1109/TEVC.2015.2443001
  46. Zitzler E, Künzli S (2004) Indicator-based selection in multiobjective search. In: Parallel problem solving from nature-PPSN VIII. Springer, pp 832–842Google Scholar
  47. Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength pareto approach. IEEE Trans Evol Comput 3(4):257–271CrossRefGoogle Scholar
  48. Zitzler E, Deb K, Thiele L (2000) Comparison of multiobjective evolutionary algorithms: empirical results. Evol Comput 8(2):173–195CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Key Lab of Intelligent Perception and Image Understanding of Ministry of Education, International Research Center of Intelligent Perception and ComputationXidian UniversityXi’anChina
  2. 2.Department of Computer ScienceUniversity of SurreyGuildfordUK
  3. 3.CERCIA, School of Computer ScienceThe University of BirminghamBirminghamUK

Personalised recommendations