The normalized normal constraint method for generating the Pareto frontier

Research paper

Abstract

The authors recently proposed the normal constraint (NC) method for generating a set of evenly spaced solutions on a Pareto frontier – for multiobjective optimization problems. Since few methods offer this desirable characteristic, the new method can be of significant practical use in the choice of an optimal solution in a multiobjective setting. This paper’s specific contribution is two-fold. First, it presents a new formulation of the NC method that incorporates a critical linear mapping of the design objectives. This mapping has the desirable property that the resulting performance of the method is entirely independent of the design objectives scales. We address here the fact that scaling issues can pose formidable difficulties. Secondly, the notion of a Pareto filter is presented and an algorithm thereof is developed. As its name suggests, a Pareto filter is an algorithm that retains only the global Pareto points, given a set of points in objective space. As is explained in the paper, the Pareto filter is useful in the application of the NC and other methods. Numerical examples are provided.

Keywords

design optimization multiobjective optimization normal constraint Pareto generation Pareto filter 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Belegundu, A.; Chandrupatla, T. 1999: Optimization concepts and applications in engineering. New Jersey: Prentice Hall Google Scholar
  2. 2.
    Chen, W.; Wiecek, M.M.; Zhang, J. 1999: Quality utility – a compromise programming approach to robust design. J. Mech. Des. 121, 179–187 Google Scholar
  3. 3.
    Cheng, F.; Li, D. 1998: Genetic algorithm development for multiobjective optimization of structures. AIAA J. 36, 1105–1112 Google Scholar
  4. 4.
    Das, I.; Dennis, J.E. 1997: A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct. Optim. 14, 63–69 Google Scholar
  5. 5.
    Das, I.; Dennis, J.E. 1998: Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J. Optim. 8, 631–657 Google Scholar
  6. 6.
    Grandhi, R.V.; Bharatram, G. 1993: Multiobjective optimization of large-scale structures. AIAA J. 31, 1329–1337 Google Scholar
  7. 7.
    Holland, J.H. 1975: Adaptation in natural and artificial systems. Ann Arbor, MI: The University of Michigan Press Google Scholar
  8. 8.
    Ismail-Yahaya, A.; Messac, A. 2002: Effective generation of the Pareto frontier using the normal constraint method. 40th Aerospace Sciences Meeting and Exhibit (held in Reno, Nevada), Paper No. AIAA 2002-0178 Google Scholar
  9. 9.
    Koski, J. 1985: Defectiveness of weighting methods in multicriterion optimization of structures. Commun. Appl. Numer. Methods 1, 333–337 Google Scholar
  10. 10.
    Messac, A. 1996: Physical programming: effective optimization for computational design. AIAA J. 34, 149–158 Google Scholar
  11. 11.
    Messac, A.; Mattson, C.A. 2002: Generating Well-Distributed Sets of Pareto Points for Engineering Design using Physical Programming. Optim. Eng. 3, 431–450 Google Scholar
  12. 12.
    Messac, A.; Ismail-Yahaya, A. 2001: Required relationship between objective function and Pareto frontier orders: practical implications. AIAA J. 11, 2168–2174 Google Scholar
  13. 13.
    Miettinen, K. 1999: Nonlinear multiobjective optimization. Massachusetts: Kluwer Academic PublishersGoogle Scholar
  14. 14.
    Osyczka, A.; Kundu, S. 1995: New method to solve generalized multicriteria optimization problems using the simple genetic algorithm. Struct. Optim. 10, 94–99 Google Scholar
  15. 15.
    Pareto, V. 1964: Cour d’economie politique, Librarie DrozGeneve (the first edition in 1896) Google Scholar
  16. 16.
    Pareto, V. 1971: Manuale di economica politica, societa editrice libraria. Milano, Italy: MacMillan Press (the first edition in 1906), (translated into English by A. S. Schwier as Manual of Political Economy) Google Scholar
  17. 17.
    Srinivasan, D.; Tettamanzi, A. 1996: Heuristic-guided evolutionary approach to multiobjective generation scheduling. IEE Proc. Generation, Transmission and Distribution 143, 553–559 Google Scholar
  18. 18.
    Steuer, R. 1986: Multiple criteria optimization: theory, computation, and applications. New York: John Wiley & SonsGoogle Scholar

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Multidisciplinary Design and Optimization Laboratory; Department of Mechanical, Aerospace, and Nuclear EngineeringRensselaer Polytechnic InstituteTroyUSA

Personalised recommendations