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Vietnam Journal of Mathematics

, Volume 43, Issue 2, pp 471–486 | Cite as

The Adaptive Parameter Control Method and Linear Vector Optimization

  • Nguyen Thi Thu Huong
  • Nguyen Dong YenEmail author
Article
  • 94 Downloads

Abstract

The question of constructing a set of equidistant points, with a given small approximate distance, in the efficient and weakly efficient frontiers of a linear vector optimization problem of a general form, is considered in this paper. It is shown that the question can be solved by combining Pascoletti–Serafini’s scalarization method (1984) and Eichfelder’s adaptive parameter control method (2009) with a sensitivity analysis formula in linear programming, which was obtained by J. Gauvin (2001). Our investigation shows that one can avoid the strong second-order sufficient condition used by G. Eichfelder, which cannot be imposed on linear vector optimization problems.

Keywords

Vector optimization Linear vector optimization Pascoletti–Serafini’s scalarization method Eichfelder’s adaptive parameter control method Sensitivity analysis in linear programming 

Mathematics Subject Classification (2010)

90C29 90C31 90C05 

Notes

Acknowledgments

The research of N. T. T. Huong is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.39 and supported in part by Department of Information Technology, Le Qui Don University. The research of N. D. Yen is funded by the Vietnam Institute for Advanced Study in Mathematics (VIASM) and supported in part by Institute of Mathematics, Vietnam Academy of Science and Technology. The authors would like to thank the two anonymous referees for their very careful reading and helpful suggestions.

References

  1. 1.
    Benson, H.P.: An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J. Global. Optim. 13, 1–24 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Benson, H. P.: Hybrid approach for solving multiple-objective linear programs in outcome space. J. Optim. Theory. Appl. 98, 17–35 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Eichfelder, G.: Adaptive Scalarization Methods in Multiobjective Optimization. Springer-Verlag, Berlin (2008)CrossRefzbMATHGoogle Scholar
  4. 4.
    Eichfelder, G.: An adaptive scalarization method in multiobjective optimization. SIAM J. Optim. 19, 1694–1718 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Eichfelder, G.: Scalarizations for adaptively solving multi-objective optimization problems. Comput. Optim. Appl. 44, 249–273 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Eichfelder, G.: Multiobjective bilevel optimization. Math. Program., Ser. A 123, 419–449 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Gauvin, J.: Formulae for the sensitivity analysis of linear programming problems. In: Lassonde, M (ed.) Approximation, Optimization, Economics, Mathematical, pp 117–120. Physica-Verlag, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Gerstewitz, Chr. [Tammer, Chr.]: Nichtkonvexe Dualitat in der Vektoroptimierung. (German) [Nonconvex duality in vector optimization] Wiss. Z. Tech. Hochsch. Leuna-Merseburg 25, 357–364 (1983)Google Scholar
  9. 9.
    Hamel, A.H.: Translative sets and functions and their applications to risk measure theory and nonlinear separation. http://carma.newcastle.edu.au/jon/Preprints/Books/CUP/CUPold/acrm.pdf
  10. 10.
    Helbig, S.: An interactive algorithm for nonlinear vector optimization. Appl. Math. Optim. 22, 147–151 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Helbig, S.: An algorithm for quadratic vector optimization problems. Z. Angew. Math. Mech. 70, T751—T753 (1990)MathSciNetGoogle Scholar
  12. 12.
    Helbig, S.: On a constructive approximation of the efficient outcomes in bicriterion vector optimization. J. Global Optim. 5, 35–48 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    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. Global Optim. 37, 1–10 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Huong, N.T.T., Yen, N.D.: The Pascoletti–Serafini scalarization scheme and linear vector optimization. J. Optim. Theory Appl. 162, 559–576 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland Publishing Company, Amsterdam–New York–Oxford (1979)zbMATHGoogle Scholar
  16. 16.
    Luc, D.T.: Theory of Vector Optimization. Springer-Verlag, Berlin–Heidelberg (1989)CrossRefGoogle Scholar
  17. 17.
    Pascoletti, A., Serafini, P.: Scalarizing vector optimization problems. J. Optim. Theory Appl. 42, 499–524 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton, NJ (1970)zbMATHGoogle Scholar
  19. 19.
    Sterna-Karwat, A.: Continuous dependence of solutions on a parameter in a scalarization method. J. Optim. Theory Appl. 55, 417–434 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Sterna-Karwat, A.: Lipschitz and differentiable dependence of solutions on a parameter in a scalarization method. J. Aust. Math. Soc. Ser. A 42, 353–364 (1987)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  1. 1.Department of Information TechnologyLe Qui Don UniversityHanoiVietnam
  2. 2.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

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