, Volume 24, Issue 2, pp 493–513 | Cite as

An external penalty-type method for multicriteria

  • Ellen H. Fukuda
  • L. M. Graña Drummond
  • Fernanda M. P. Raupp
Original Paper


We propose an extension of the classical real-valued external penalty method to the multicriteria optimization setting. As its single objective counterpart, it also requires an external penalty function for the constraint set, as well as an exogenous divergent sequence of nonnegative real numbers, the so-called penalty parameters, but, differently from the scalar procedure, the vector-valued method uses an auxiliary function, which can be chosen among large classes of “monotonic” real-valued mappings. We analyze the properties of the auxiliary functions in those classes and exhibit some examples. The convergence results are similar to those of the scalar-valued method, and depending on the kind of auxiliary function used in the implementation, under standard assumptions, the generated infeasible sequences converge to weak Pareto or Pareto optimal points. We also propose an implementable local version of the external penalization method and study its convergence results.


Constrained multiobjective optimization External penalty method Pareto optimality Scalar representation 

Mathematics Subject Classification

90C29 90C30 



We would like to thank the anonymous referees for their suggestions which improved the original version of the paper. We are also thankful to Alfredo N. Iusem and Benar F. Svaiter for valuable discussions.


  1. Bonnel H, Iusem AN, Svaiter BF (2005) Proximal methods in vector optimization. SIAM J Optim 15(4):953–970CrossRefGoogle Scholar
  2. Carrizosa E, Frenk JBG (1998) Dominating sets for convex functions with some applications. J Optim Theory Appl 96(2):281–295CrossRefGoogle Scholar
  3. Eschenauer H, Koski J, Osyczka A (1990) Multicriteria design optimization. Springer, BerlinCrossRefGoogle Scholar
  4. Fliege J, Graña Drummond LM, Svaiter BF (2009) Newton’s method for multiobjective optimization. SIAM J Optim 20(2):602–626CrossRefGoogle Scholar
  5. Fliege J, Svaiter BF (2000) Steepest descent methods for multicriteria optimization. Math Methods Oper Res 51(3):479–494CrossRefGoogle Scholar
  6. Fu Y, Diwekar U (2004) An efficient sampling approach to multiobjective optimization. Ann Oper Res 132(1–4):109–134CrossRefGoogle Scholar
  7. Fukuda EH, Graña Drummond LM (2011) On the convergence of the projected gradient method for vector optimization. Optimization 60(8–9):1009–1021CrossRefGoogle Scholar
  8. Fukuda EH, Graña Drummond LM (2013) Inexact projected gradient method for vector optimization. Comput Optim Appl 54(3):473–493CrossRefGoogle Scholar
  9. Graña Drummond LM, Iusem AN (2004) A projected gradient method for vector optimization problems. Comput Optim Appl 28(1):5–29CrossRefGoogle Scholar
  10. Graña Drummond LM, Raupp FMP, Svaiter BF (2014) A quadratically convergent Newton method for vector optimization. Optimization 63(5):661–677CrossRefGoogle Scholar
  11. Graña Drummond LM, Svaiter BF (2005) A steepest descent method for vector optimization. J Comput Appl Math 175(2):395–414CrossRefGoogle Scholar
  12. Gravel M, Martel JR, Price W, Tremblay R (1992) A multicriterion view of optimal ressource allocation in job-shop production. Eur J Oper Res 61:230–244CrossRefGoogle Scholar
  13. Jahn J (2003) Vector optimization—theory, applications, and extensions. Springer, ErlangenGoogle Scholar
  14. Leschine TM, Wallenius H, Verdini W (1992) Interactive multiobjective analysis and assimilative capacity-based ocean disposal decisions. Eur J Oper Res 56:278–289CrossRefGoogle Scholar
  15. Luc DT (1989) Theory of vector optimization. In: Lecture notes in economics and mathematical systems, 319. Springer, BerlinGoogle Scholar
  16. Luenberger DG (2003) Linear and nonlinear programming. Kluwer Academic Publishers, BostonGoogle Scholar
  17. Tavana M (2004) A subjective assessment of alternative mission architectures for the human exploration of mars at NASA using multicriteria decision making. Comput Oper Res 31:1147–1164CrossRefGoogle Scholar
  18. White DJ (1984) Multiobjective programming and penalty functions. J Optim Theory Appl 43(4):583–599CrossRefGoogle Scholar
  19. White DJ (1998) Epsilon-dominating solutions in mean-variance portfolio analysis. Eur J Oper Res 105:457–466CrossRefGoogle Scholar
  20. Zangwill WI (1967) Non-linear programming via penalty functions. Manag Sci 13(5):344–358CrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2015

Authors and Affiliations

  • Ellen H. Fukuda
    • 1
  • L. M. Graña Drummond
    • 2
  • Fernanda M. P. Raupp
    • 3
  1. 1.Department of Applied Mathematics and Physics, Graduate School of InformaticsKyoto UniversityKyotoJapan
  2. 2.Faculty of Business and AdministrationFederal University of Rio de JaneiroRio de JaneiroBrazil
  3. 3.National Laboratory for Scientific ComputingPetrópolisBrazil

Personalised recommendations