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A note on approximate proper efficiency in linear fractional vector optimization

Abstract

In this note, we show that for linear fractional vector optimization problems with bounded constraint sets there is no difference between the \(\epsilon \)-efficiency and the \(\epsilon \)-proper efficiency.

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References

  1. 1.

    Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71, 232–241 (1979)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Borwein, J.: Proper efficient points for maximizations with respect to cones. SIAM J. Control Optim. 15, 57–63 (1977)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Choo, E.U., Atkins, D.R.: Bicriteria linear fractional programming. J. Optim. Theory Appl. 36, 203–220 (1982)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Choo, E.U., Atkins, D.R.: Connectedness in multiple linear fractional programming. Manag. Sci. 29, 250–255 (1983)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Choo, E.U.: Proper efficiency and the linear fractional vector maximum problem. Oper. Res. 32, 216–220 (1984)

    Article  Google Scholar 

  6. 6.

    Chuong, T.D., Kim, D.S.: Approximate solutions of multiobjective optimization problems. Positivity 20, 187–207 (2016)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Geoffrion, A.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Gutiérrez, C., Huerga, L., Novo, V., Sama, M.: Limit behavior of approximate proper solutions in vector optimization. SIAM J. Optim. 29, 2677–2696 (2019)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36, 387–407 (1982)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. I. Fundamentals. Springer, Berlin (1993)

    Book  Google Scholar 

  11. 11.

    Hoa, T.N., Phuong, T.D., Yen, N.D.: Linear fractional vector optimization problems with many components in the solution sets. J. Ind. Manag. Optim. 1, 477–486 (2005)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Hong, Z., Piao, G.R., Kim, D.S.: On approximate solutions of nondifferentiable vector optimization problems with cone-convex objectives. Optim. Lett. 13, 891–906 (2019)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Hong, Z., Jiao, L.G., Kim, D.S.: On a class of nonsmooth fractional robust multi-objective optimization problems. Part I: Optimality conditions. Appl. Set-Valued Anal. Optim. 2, 109–121 (2020)

    Google Scholar 

  14. 14.

    Huong, N.T.T., Yao, J.C., Yen, N.D.: Geoffrion’s proper efficiency in linear fractional vector optimization with unbounded constraint sets. J. Glob. Optim. 78, 545–562 (2020)

  15. 15.

    Huong, N.T.T., Yao, J.C., Yen, N.D.: New results on proper efficiency for a class of vector optimization problems. Appl. Anal. (2020). https://doi.org/10.1080/00036811.2020.1712373

    Article  Google Scholar 

  16. 16.

    Isermann, H.: Proper efficiency and the linear vector maximum problem. Oper. Res. 22, 189–191 (1974)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kim, D.S., Mordukhovich, B.S., Pham, T.S., Tuyen, N.V.: Existence of efficient and properly efficient solutions to problems of constrained vector optimization. Math. Program. (2020). https://doi.org/10.1007/s10107-020-01532-y

    Article  Google Scholar 

  18. 18.

    Kutateladze, S.: Convex \(\epsilon \)-programming. Sov. Math. Dokl. 20, 391–393 (1979)

    Google Scholar 

  19. 19.

    Lee, G.M., Tam, N.N., Yen, N.D.: Quadratic Programming and Affine Variational Inequalities: A Qualitative Study. Springer, New York (2005)

    MATH  Google Scholar 

  20. 20.

    Li, Z., Wang, S.: \(\epsilon \)-approximate solutions in multiobjective optimization. Optimization 44, 161–174 (1998)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Liu, J.C.: \(\epsilon \)-properly efficient solutions to nondifferentiable multiobjective programming problems. Appl. Math. Lett. 12, 109–113 (1999)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Long, X.J., Li, X.B., Zeng, J.: Lagrangian conditions for approximate solutions on nonconvex set-valued optimization problems. Optim. Lett. 7, 1847–1856 (2013)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Loridan, P.: \(\epsilon \)-solutions in vector minimization problems. J. Optim. Theory Appl. 43, 265–276 (1984)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)

    Book  Google Scholar 

  25. 25.

    Malivert, C.: Multicriteria fractional programming. In: Sofonea, M., Corvellec, J.N. (eds.) Proceedings of the 2nd Catalan Days on Applied Mathematics, pp. 189–198. Presses Universitaires de Perpinan (1995)

  26. 26.

    Mastroeni, G., Pappalardo, M., Raciti, F.: Some topics in vector optimization via image space analysis. J. Nonlinear Var. Anal. 4, 5–20 (2019)

    MATH  Google Scholar 

  27. 27.

    Pareto, V.: Course d’Economie Politique. Rouge, Lausanne (1896)

  28. 28.

    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  Google Scholar 

  29. 29.

    Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, Orlando (1985)

    MATH  Google Scholar 

  30. 30.

    Son, T.Q., Tuyen, N.V., Wen, C.F.: Optimality conditions for approximate Pareto solutions of a nonsmooth vector optimization problem with an infinite number of constraints. Acta Math. Vietnam 45, 435–448 (2020)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Steuer, R.E.: Multiple Criteria Optimization: Theory, Computation and Application. Wiley, New York (1986)

    MATH  Google Scholar 

  32. 32.

    Tigan, S.: Sur le problème de la programmation vectorielle fractionnaire. Rev. Anal. Numér. Théor. Approx. 4, 99–103 (1975)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Tuyen, N.V.: Approximate solutions of interval-valued optimization problems. Investigación Oper. 42, 223–237 (2021)

    Google Scholar 

  34. 34.

    Yen, N.D.: Linear fractional and convex quadratic vector optimization problems. In: Ansari, Q.H., Yao, J.C. (eds.) Recent Developments in Vector Optimization, pp. 297–328. Springer, Berlin (2012)

    Chapter  Google Scholar 

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Acknowledgements

This research is funded by Hanoi Pedagogical University 2 under Grant Number HPU2.UT-2021.15.

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Correspondence to Nguyen Van Tuyen.

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Tuyen, N.V. A note on approximate proper efficiency in linear fractional vector optimization. Optim Lett (2021). https://doi.org/10.1007/s11590-021-01806-0

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Keywords

  • Linear fractional vector optimization
  • \(\epsilon \)-efficient solution
  • \(\epsilon \)-properly efficient