Skip to main content
SpringerLink
Log in

Geoffrion’s proper efficiency in linear fractional vector optimization with unbounded constraint sets

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

Choo (Oper Res 32:216–220, 1984) has proved that any efficient solution of a linear fractional vector optimization problem with a bounded constraint set is properly efficient in the sense of Geoffrion. This paper studies Geoffrion’s properness of the efficient solutions of linear fractional vector optimization problems with unbounded constraint sets. By examples, we show that there exist linear fractional vector optimization problems with the efficient solution set being a proper subset of the unbounded constraint set, which have improperly efficient solutions. Then, we establish verifiable sufficient conditions for an efficient solution of a linear fractional vector optimization to be a Geoffrion properly efficient solution by using the recession cone of the constraint set. For bicriteria problems, it is enough to employ a system of two regularity conditions. If the number of criteria exceeds two, a third regularity condition must be added to the system. The obtained results complement the above-mentioned remarkable theorem of Choo and are analyzed through several interesting examples, including those given by Hoa et al. (J Ind Manag Optim 1:477–486, 2005).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Benoist, J.: Connectedness of the efficient set for strictly quasiconcave sets. J. Optim. Theory Appl. 96, 627–654 (1998)

    Article  MathSciNet  Google Scholar 

  2. Benoist, J.: Contractibility of the efficient set in strictly quasiconcave vector maximization. J. Optim. Theory Appl. 110, 325–336 (2001)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Bochnak, R., Coste, M., Roy, M.F.: Real Algebraic Geometry. Spinger, Berlin (1998)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Chew, K.L., Choo, E.U.: Pseudolinearity and efficiency. Math. Program. 28, 226–239 (1984)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Crespi, G.P.: Proper efficiency and vector variational inequalities. J. Inform. Optim. Sci. 23, 49–62 (2002)

    MathSciNet  MATH  Google Scholar 

  11. Crespi, G.P., Ginchev, I., Rocca, M.: Existence of solutions and star-shapedness in Minty variational inequalities. J. Global Optim. 32, 485–494 (2005)

    Article  MathSciNet  Google Scholar 

  12. Crespi, G.P., Ginchev, I., Rocca, M.: Increasing-along-rays property for vector functions. J. Nonlinear Convex Anal. 7, 39–50 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 613–630 (1968)

    Article  MathSciNet  Google Scholar 

  14. Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria. Kluwer Academic Publishers, Dordrecht (2000)

    MATH  Google Scholar 

  15. Guerraggio, A., Molho, E., Zaffaroni, A.: On the notion of proper efficiency in vector optimization. J. Optim. Theory Appl. 82, 1–21 (1994)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. 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 

  18. Hoa, T.N., Phuong, T.D., Yen, N.D.: On the parametric affine variational inequality approach to linear fractional vector optimization problems. Vietnam J. Math. 33, 477–489 (2005)

    MathSciNet  MATH  Google Scholar 

  19. Hoa, T.N., Huy, N.Q., Phuong, T.D., Yen, N.D.: Unbounded components in the solution sets of strictly quasiconcave vector maximization problems. J. Glob. Optim. 37, 1–10 (2007)

    Article  MathSciNet  Google Scholar 

  20. Huong, N.T.T., Hoa, T.N., Phuong, T.D., Yen, N.D.: A property of bicriteria affine vector variational inequalities. Appl. Anal. 10, 1867–1879 (2012)

    Article  MathSciNet  Google Scholar 

  21. Huong, N.T.T., Yao, J.-C., Yen, N.D.: Connectedness structure of the solution sets of vector variational inequalities. Optimization 66, 889–901 (2017)

    Article  MathSciNet  Google Scholar 

  22. Huy, N.Q., Yen, N.D.: Remarks on a conjecture of. J. Benoist. Nonlinear Anal. Forum 9, 109–117 (2004)

    MATH  Google Scholar 

  23. Lee, G.M., Tam, N.N., Yen, N.D.: Quadratic Programming and Affine Variational Inequalities: A Qualitative Study Series: Nonconvex Optimization and its Applications, vol. 78. Springer, New York (2005)

    Google Scholar 

  24. Luc, D.T.: Multiobjective Linear Programming: An Introduction. Springer, Berlin (2016)

    Book  Google Scholar 

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

  26. Robinson, S.M.: Generalized equations and their solutions. Part I: Basic theory. Math. Program. Study 10, 128–141 (1979)

    Article  Google Scholar 

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

    Book  Google Scholar 

  28. Rubinov, A.M., Glover, B.M.: Increasing convex-along-rays functions with applications to global optimization. J. Optim. Theory Appl. 102, 615–642 (1999)

    Article  MathSciNet  Google Scholar 

  29. Stancu-Minasian, I.M.: A ninth bibliography of fractional programming. Optimization 68, 2123–2167 (2019)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  31. 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 

  32. Yen N.D., Phuong T.D.: Connectedness and stability of the solution set in linear fractional vector optimization problems. 38, 479–489. https://link.springer.com/chapter/10.1007/978-1-4613-0299-5_29. (2000)

  33. Yen, N.D., Yang, X.Q.: Affine variational inequalities on normed spaces. J. Optim. Theory Appl. 178, 36–55 (2018)

    Article  MathSciNet  Google Scholar 

  34. Yen, N.D., Yao, J.-C.: Monotone affine vector variational inequalities. Optimization 60, 53–68 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are greatly indebted to the referees for making us familiar with the significant works of Choo  [7], Chew and Choo  [6], Stancu-Minasian  [29], and for valuable suggestions. We thank Prof. Phan Quoc Khanh and Prof. Christiane Tammer for useful discussions on the first draft of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Information Technology, Le Quy Don Technical University, 236 Hoang Quoc Viet, Hanoi, Vietnam

    N. T. T. Huong

  2. Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung, Taiwan

    J.-C. Yao

  3. Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, 10307, Vietnam

    N. D. Yen

Authors
  1. N. T. T. Huong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J.-C. Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. D. Yen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. D. Yen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors were supported by National Foundation for Science and Technology Development (Vietnam) under Grant No. 101.01-2018.306, Le Quy Don Technical University (Vietnam), Grant MOST 105-2115-M-039-002-MY3 (Taiwan), and the Vietnam Institute for Advanced Study in Mathematics (VIASM, Vietnam).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huong, N.T.T., Yao, JC. & Yen, N.D. Geoffrion’s proper efficiency in linear fractional vector optimization with unbounded constraint sets. J Glob Optim 78, 545–562 (2020). https://doi.org/10.1007/s10898-020-00927-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-020-00927-7

Keywords

Mathematics Subject Classification

Search

Navigation