# Approximate Proper Solutions in Vector Equilibrium Problems

Limit Behavior and Linear Scalarization Results

## Abstract

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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## References

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)

2. 2.

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

3. 3.

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

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)

6. 6.

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

7. 7.

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

8. 8.

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

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)

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)

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)

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)

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)

15. 15.

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

16. 16.

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

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)

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)

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)

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)

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)

22. 22.

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

23. 23.

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

24. 24.

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

25. 25.

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

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)

27. 27.

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

28. 28.

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

29. 29.

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

30. 30.

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

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)

## Acknowledgements

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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Correspondence to C. Gutiérrez.