Approximate Proper Solutions in Vector Equilibrium Problems

Limit Behavior and Linear Scalarization Results


The paper concerns with properties of approximate Benson and Henig proper efficient solutions of vector equilibrium problems. Relationships between both kinds of solutions and the approximate weak efficient solutions of the problem are stated. Roughly speaking, it is proved that they coincide provided that the ordering cone of the problem can be separated from another closed cone by an approximating sequence of dilating cones. As a result, the limit behavior of approximate Benson and Henig proper efficient solutions when the error tends to zero is studied. It is shown that efficient and proper efficient solutions of the problem can be obtained as the limit of approximate Benson and Henig proper efficient solutions whenever a suitable domination set is considered. Finally, optimality conditions for approximate Benson and Henig proper efficient solutions are derived by approximate solutions of scalar equilibrium problems. They are formulated via linear scalarization and the necessary conditions hold true in problems satisfying a kind of nearly subconvexlikeness condition. The main results of the paper generalize some recent ones because they involve weaker assumptions.

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  1. 1.

    Ansari, Q.H.: Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and fixed point theory. J. Math. Anal. Appl. 334, 561–575 (2007)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ansari, Q.H., Köbis, E., Yao, J.-C.: Vector Variational Inequalities and Vector Optimization. Theory and Applications. Springer, Cham (2018)

    Google Scholar 

  3. 3.

    Ansari, Q.H., Oettli, W., Schläger, D.: A generalization of vectorial equilibria. Math. Methods Oper. Res. 46, 147–152 (1997)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Ansari, Q.H., Yao, J.-C. (eds.): Recent Developments in Vector Optimization. Springer, Berlin (2012)

  5. 5.

    Bianchi, M., Hadjisavvas, N., Schaible, S.: Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theory Appl. 92, 527–542 (1997)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bianchi, M., Kassay, G., Pini, R.: Ekeland’s principle for vector equilibrium problems. Nonlinear Anal. 66, 1454–1464 (2007)

    MathSciNet  Article  Google Scholar 

  7. 7.

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

    MathSciNet  Article  Google Scholar 

  8. 8.

    Borwein, J.M., Zhuang, D.: Super efficiency in vector optimization. Trans. Amer. Math. Soc. 338, 105–122 (1993)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Chen, G.-Y., Huang, X.X., Yang, X.Q.: Vector Optimization. Set-Valued and Variational Analysis. Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005)

    Google Scholar 

  10. 10.

    Chen, B., Liu, Q.-Y., Liu, Z.-B., Huang, N.-J.: Connectedness of approximate solutions set for vector equilibrium problems in Hausdorff topological vector spaces. Fixed Point Theory Appl. 2011, 36 (2011)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Chen, C.-R., Zuo, X., Lu, F., Li, S.-J.: Vector equilibrium problems under improvement sets and linear scalarization with stability applications. Optim. Methods Softw. 31, 1240–1257 (2016)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Durea, M.: On the existence and stability of approximate solutions of perturbed vector equilibrium problems. J. Math. Anal. Appl. 333, 1165–1179 (2007)

    MathSciNet  Article  Google Scholar 

  13. 13.

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

  14. 14.

    Gong, X.H.: Efficiency and Henig efficiency for vector equilibrium problem. J. Optim. Theory Appl. 108, 139–154 (2001)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Gong, X.H.: Connectedness of the solution sets and scalarization for vector equilibrium problems. J. Optim. Theory Appl. 133, 151–161 (2007)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Gong, X.-H.: Scalarization and optimality conditions for vector equilibrium problems. Nonlinear Anal. 73, 3598–3612 (2010)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Gutiérrez, C., Huerga, L., Jiménez, B., Novo, V.: Henig approximate proper efficiency and optimization problems with difference of vector mappings. J. Convex Anal. 23, 661–690 (2016)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Gutiérrez, C., Huerga, L., Novo, V.: Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems. J. Math. Anal. Appl. 389, 1046–1058 (2012)

    MathSciNet  Article  Google Scholar 

  19. 19.

    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 

  20. 20.

    Hai, L.P., Huerga, L., Khanh, P.Q., Novo, V.: Variants of the Ekeland variational principle for approximate proper solutions of vector equilibrium problems. J. Global Optim. 74, 361–382 (2019)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Han, Y., Huang, N.-J.: Some characterizations of the approximate solutions to generalized vector equilibrium problems. J. Ind. Manag. Optim. 12, 1135–1151 (2016)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Henig, M.I.: A cone separation theorem. J. Optim. Theory Appl. 36, 451–455 (1982)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Jahn, J.: Vector optimization. Theory, Applications and Extensions. Springer, Berlin (2011)

    Google Scholar 

  24. 24.

    Kaliszewski, I.: Quantitative Pareto Analysis by Cone Separation Technique. Kluwer Academic Publishers, Boston (1994)

    Google Scholar 

  25. 25.

    Kassay, G., Rădulescu, V.: Equilibrium Problems and Applications. Academic Press, London (2019)

    Google Scholar 

  26. 26.

    Nieuwenhuis, J.W.: Properly efficient and efficient solutions for vector maximization problems in Euclidean space. J. Math. Anal. Appl. 84, 311–317 (1981)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Oettli, W.: A remark on vector-valued equilibria and generalized monotonicity. Acta. Math. Vietnam. 22, 213–221 (1997)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Papageorgiou, N.S.: Pareto efficiency in locally convex spaces. J. Numer. Funct. Anal. Optim. 8, 83–116 (1985)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Qiu, J.H.: On solidness of polar cones. J. Optim. Theory Appl. 109, 199–214 (2001)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Qiu, Q., Yang, X.: Scalarization of approximate solution for vector equilibrium problems. J. Ind. Manag. Optim. 9, 143–151 (2013)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Ródenas-Pedregosa, J.L.: Caracterización de Soluciones de Problemas de Equilibrio Vectoriales. PhD Thesis, Universidad Nacional de educación a Distancia, Madrid Spain (2018)

  32. 32.

    Sterna-Karwat, A.: Approximating families of cones and proper efficiency in vector optimization. Optimization 20, 809–817 (1989)

    MathSciNet  Article  Google Scholar 

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The author is very grateful to the referees for their valuable comments and suggestions. This research was partially supported by Ministerio de Economía y Competitividad (Spain) under project MTM2015-68103-P (MINECO/FEDER), by Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) (Spain) and Fondo Europeo de Desarrollo Regional (FEDER) under project PGC2018-096899-B-I00 (MCIU/AEI/FEDER, UE).

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This work is dedicated to Professor Marco Antonio López Cerdá on the occasion of his 70th birthday.

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Gutiérrez, C. Approximate Proper Solutions in Vector Equilibrium Problems. Vietnam J. Math. (2020).

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  • Vector equilibrium problem
  • Proper efficient solution
  • Approximate solution
  • Dilating cone
  • Approximating sequence of cones
  • Generalized convexity
  • Linear scalarization

Mathematics Subject Classification (2010)

  • 90C33
  • 49J40
  • 90C29
  • 90C48