Structural and Multidisciplinary Optimization

, Volume 46, Issue 6, pp 839–852 | Cite as

On improving normal boundary intersection method for generation of Pareto frontier

  • S. SiddiquiEmail author
  • S. Azarm
  • S. A. Gabriel
Research Paper


Gradient-based methods, including Normal Boundary Intersection (NBI), for solving multi-objective optimization problems require solving at least one optimization problem for each solution point. These methods can be computationally expensive with an increase in the number of variables and/or constraints of the optimization problem. This paper provides a modification to the original NBI algorithm so that continuous Pareto frontiers are obtained “in one go,” i.e., by solving only a single optimization problem. Discontinuous Pareto frontiers require solving a significantly fewer number of optimization problems than the original NBI algorithm. In the proposed method, the optimization problem is solved using a quasi-Newton method whose history of iterates is used to obtain points on the Pareto frontier. The proposed and the original NBI methods have been applied to a collection of 16 test problems, including a welded beam design and a heat exchanger design problem. The results show that the proposed approach significantly reduces the number of function calls when compared to the original NBI algorithm.


Normal Boundary Intersection (NBI) Multi-objective optimization Continuous nonlinear optimization Pareto solutions Quasi-Newton methods 



The authors wish to thank the anonymous referees and Mr. Amir Mortazavi for their constructive ideas to improve the paper. The work presented in this paper was supported in part by the Office of Naval Research Contract N000140810384. Such support does not constitute an endorsement by the funding agency of the opinions expressed in this paper. The opinions and ideas expressed in this paper are the authors and do not necessarily represent the opinions and ideas of their affiliated institutions.


  1. Bazaraa M, Sherali H, Shetty C (1993) Nonlinear programming: theory and algorithms, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  2. Becerra R, Coello CAC (2006) Solving hard multi-objective optimization problems using epsilon-constraint with cultured differential equation. In: Runarsson TP et al (eds) PPSN IX LNCS, vol 4193, pp 543–552Google Scholar
  3. Cohon J (2004) Multi-objective programming and planning. Dover Publications, MineolaGoogle Scholar
  4. Collette Y, Siarry P (2004) Multiobjective optimization: principles and case studies. Springer, New YorkzbMATHGoogle Scholar
  5. Das I (1999) On characterizing the “knee” of the Pareto curve based on normal-boundary intersection. Struct Multidisc Optim 18(2–3):107–115. doi: 10.1007/BF01195985 Google Scholar
  6. Das I, Dennis J (1997) A closer look at drawbacks of minimizing weighted sums of objectives for Pareto Set Generation in Multicriteria Optimization problems. Struct Optim 14:63–69CrossRefGoogle Scholar
  7. Das I, Dennis J (1998) Normal boundary intersection: a new method for generating Pareto surface in nonlinear multicriteria optimization problems. SIAM J Optim 8(3):631–657MathSciNetzbMATHCrossRefGoogle Scholar
  8. Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197CrossRefGoogle Scholar
  9. Goel T, Vaidyanathan R, Haftka R, Shyy W, Queipo NV, Tucker K (2007) Response surface approximation of Pareto optimal front in multi-objective optimization. Comput Methods Appl Math Eng 196(4–6):879–893zbMATHCrossRefGoogle Scholar
  10. Gunawan S, Azarm S (2004) Non-gradient-based parameter sensitivity estimation for single-objective robust design optimization. J Mech Des 126(3):395–402CrossRefGoogle Scholar
  11. Isaacs A, Ray T, Smith W (2008) Blessings of maintaining infeasible solutions for constrained multi-objective optimization problems. IEEE Congress Evol Comput 2780–2787Google Scholar
  12. Jia Z, Ierapetritou MG (2007) Generate Pareto optimal solutions of scheduling problems using normal boundary intersection technique. Comp Chem Eng 31:268CrossRefGoogle Scholar
  13. Li G, Li M, Azarm S, Al Hashimi S, Al Ameri T, Al Qasas N (2009) Improving multi-objective genetic algorithms with adaptive design of experiments and online metamodeling. Struct Multidisc Optim 37(5):447–461CrossRefGoogle Scholar
  14. Magrab E, Azarm S, Balachandran B, Duncan J, Herold K, Walsh G (2004) An engineer’s guide to Matlab. Prentice Hall, New YorkGoogle Scholar
  15. Marler R, Arora J (2004) Survey of multi-objective optimization methods for engineering. Struct Multidisc Optim 26:369–395MathSciNetzbMATHCrossRefGoogle Scholar
  16. MATLAB (2008) MATLAB and Simulink for technical computing. Mathworks, Version 2008bGoogle Scholar
  17. Messac A, Ismail-Yahaya A, Mattson CA (2003) The normalized normal constraint method for generating the Pareto frontier. Struct Multidisc Optim 25(2):86–98MathSciNetzbMATHCrossRefGoogle Scholar
  18. Messac A, Mattson CA (2004) Normal constraint method with guarantee of even representation of complete Pareto frontier. AIAA J 42(10):2101–2111CrossRefGoogle Scholar
  19. Mueller-Gritschneider D, Graeb H, Schlichtmann U (2009) A successive approach to compute the bounded Pareto front of practical multi-objective problems. SIAM J Optim 20(2):915–934MathSciNetCrossRefGoogle Scholar
  20. Ragsdell K, Phillips D (1976) Optimal design of a class of welded structures using geometric programming. J Eng Ind 98(3):1021–1025CrossRefGoogle Scholar
  21. Rakowska J, Haftka R, Watson L (1993) Multiobjective control-structure optimization via homotopy methods. SIAM J Optim 3(3):654–667MathSciNetzbMATHCrossRefGoogle Scholar
  22. Schoenberg R (2001) Optimization with the quasi-Newton method. Aptech Systems Inc., Maple Valley, WAGoogle Scholar
  23. Steur R (1985) Multicriteria optimization – theory, computation and application. Wiley, New YorkGoogle Scholar
  24. Siddiqui S, Azarm S, Gabriel S (2011) A modified benders decomposition method for efficient robust optimization under interval uncertainty. Struct Multidisc Optim 44(2):259–275CrossRefGoogle Scholar
  25. Venter G, Haftka R (2010) Constrained particle swarm optimization using a bi-objective formulation. Struct Multidisc Optim 40(1–6):65–76MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.ICF InternationalFairfaxUSA
  2. 2.Department of Mechanical EngineeringUniversity of MarylandCollege ParkUSA
  3. 3.Department of Civil and Environmental EngineeringUniversity of MarylandCollege ParkUSA

Personalised recommendations