Annals of Operations Research

, Volume 249, Issue 1–2, pp 5–15 | Cite as

An equivalent transformation of multi-objective optimization problems

Article

Abstract

A new equivalent definition of proper efficiency is presented. With the aid of the new definition of properness, a transformation technique is proved to transform a multi-objective problem to a more convenient one. Some conditions are determined under which the original and the transformed problems have the same Pareto and properly efficient solutions. This transformation could be employed for the sake of convexification and simplification in order to improve the computational efficiency for solving the given problem. Moreover, some existing results about the weighted sum method in the multi-objective optimization literature are generalized using the special case of the proposed transformation scheme.

Keywords

Multi-objective optimization Vector optimization Proper efficiency  Pareto optimality 

Notes

Acknowledgments

P.M. Pardalos is partially supported by NSF, and by LATNA Laboratory, NRU HSE, RF government grant, ag. 11.G34.31.0057.

References

  1. Boyd, S., & Vandenberghe, L. (2004). Convex optimization. Cambridge: Cambridge university press.CrossRefGoogle Scholar
  2. Chinchuluun, A., Pardalos, P., Migdalas, A., & Pitsoulis, L. (2008). Pareto optimality, game theory and equilibria (Vol. 17). Berlin: Springer.CrossRefGoogle Scholar
  3. Ehrgott, M. (2005). Multicriteria optimization (Vol. 2). Berlin: Springer.Google Scholar
  4. Gearhart, W. (1979). Compromise solutions and estimation of the noninferior set. Journal of Optimization Theory and Applications, 28(1), 29–47.CrossRefGoogle Scholar
  5. Geoffrion, A. M. (1968). Proper efficiency and the theory of vector maximization. Journal of Mathematical Analysis and Applications, 22(3), 618–630.CrossRefGoogle Scholar
  6. Goh, C., & Yang, X. (1998). Convexification of a noninferior frontier. Journal of Optimization Theory and Applications, 97(3), 759–768.CrossRefGoogle Scholar
  7. Guerraggio, A., Molho, E., & Zaffaroni, A. (1994). On the notion of proper efficiency in vector optimization. Journal of Optimization Theory and Applications, 82(1), 1–21.CrossRefGoogle Scholar
  8. Li, D. (1996). Convexification of a noninferior frontier. Journal of Optimization Theory and Applications, 88(1), 177–196.CrossRefGoogle Scholar
  9. Li, D., & Biswal, M. (1998). Exponential transformation in convexifying a noninferior frontier and exponential generating method. Journal of Optimization Theory and Applications, 99(1), 183–199.CrossRefGoogle Scholar
  10. Miettinen, K. (1999). Nonlinear multiobjective optimization. Boston: Kluwer Academic Publishers.Google Scholar
  11. Romeijn, H., Dempsey, J., & Li, J. (2004). A unifying framework for multi-criteria fluence map optimization models. Physics in Medicine and Biology, 49(10), 1991–2013.CrossRefGoogle Scholar
  12. Sawaragi, Y., Nakayama, H., & Tanino, T. (1985). Theory of multiobjective optimization (Vol. 176). New York: Academic Press.Google Scholar
  13. Wallenius, J., Dyer, J. S., Fishburn, P. C., Steuer, R. E., Zionts, S., & Deb, K. (2008). Multiple criteria decision making, multiattribute utility theory: Recent accomplishments and what lies ahead. Management Science, 54(7), 1336–1349. doi: 10.1287/mnsc.1070.0838.CrossRefGoogle Scholar
  14. Zarepisheh, M., Khorram, E., & Pardalos, P. (2012). Generating properly efficient points in multi-objective programs by the nonlinear weighted sum scalarization method. Optimization: A Journal of Mathematical Programming and Operations Research, 63(3), 473–486.Google Scholar
  15. Zarepisheh, M., Uribe-Sanchez, A. F., Li, N., Jia, X., & Jiang, S. B. (2014). A multicriteria framework with voxel-dependent parameters for radiotherapy treatment plan optimizationa. Medical Physics, 41(4), 041,705. doi: 10.1118/1.4866886.CrossRefGoogle Scholar
  16. Zopounidis, C., & Pardalos, P. (2010). Handbook of multicriteria analysis (Vol. 103). Berlin: Springer.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA

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