Abstract
In this paper, the efficient solution set for generalized vector quasi-equilibrium problems is investigated. By means of the linear scalarization method, we establish the path connectedness of the efficient solution set for generalized vector quasi-equilibrium problems under some suitable conditions.
Similar content being viewed by others
Notes
This paper has been submitted to a journal.
References
Ansari, Q.H., Flores-Bazán, F.: Generalized vector quasi-equilibrium problems with applications. J. Math. Anal. Appl. 277, 246–256 (2003)
Ansari, Q.H., Köbis, E., Sharma, P.K.: Characterizations of set relations with respect to variable domination structures via oriented distance function. Optimization 67, 1389–1407 (2018)
Ansari, Q.H., Köbis, E., Sharma, P.K.: Characterizations of multiobjective robustness via oriented distance function and image space analysis. J. Optim. Theory Appl. 181, 817–839 (2019)
Ansari, Q.H., Hamel, A.H., Sharma, P.K.: Ekeland’s variational principle with weighted set order relations. Math. Methods Oper. Res. 91, 117–136 (2020)
Ansari, Q.H., Sharma, P.K., Qin, X.: Characterizations of robust optimality conditions via image space analysis. Optimization 69, 2063–2083 (2020)
Lee, G.M., Kim, D.S., Lee, B.S., Yen, N.D.: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal. 34, 745–765 (1998)
Cheng, Y.H.: On the connectedness of the solution set for the weak vector variational inequality. J. Math. Anal. Appl. 260, 1–5 (2001)
Gong, X.H.: Efficiency and henig efficiency for vector equilibrium problems. J. Optim. Theory Appl. 108, 139–154 (2001)
Liu, Q.Y., Long, X.J., Huang, N.J.: Connectedness of the sets of weak efficient solutions for generalized vector equilibrium problems. Math. Slovaca 62, 123–136 (2012)
Gong, X.H.: Connectedness of the solution sets and scalarization for vector equilibrium problems. J. Optim. Theory Appl. 133, 151–161 (2007)
Gong, X.H., Yao, J.C.: Connectedness of the set of efficient solutions for generalized systems. J. Optim. Theory Appl. 138, 189–196 (2008)
Han, Y., Huang, N.J.: The connectedness of the solutions set for generalized vector equilibrium problems. Optimization 65, 357–367 (2016)
Xu, Y.D., Zhang, P.P.: Connectedness of solution sets of strong vector equilibrium problems with an application. J. Optim. Theory Appl. 178, 131–152 (2018)
Han, Y., Huang, N.J.: Existence and connectedness of solutions for generalized vector quasi-equilibrium problems. J. Optim. Theory Appl. 179, 65–85 (2018)
Gong, X.H.: On the contractibility and connectedness of an efficient point set. J. Math. Anal. Appl. 264, 465–478 (2001)
Jahn, J.: Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin (2011)
Peng, Z.Y., Zhao, Y., Yang, X.M.: Semicontinuity of approximate solution mappings to parametric set-valued weak vector equilibrium problems. Numer. Funct. Anal. Optim. 36, 481–500 (2015)
Han, Y., Huang, N.J.: Existence and stability of solutions for a class of generalized vector equilibrium problems. Positivity 20, 829–846 (2016)
Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria: Mathematical Theories. Kluwer Academic Publishers, Dordrecht (2000)
Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)
Göpfert, A., Riahi, H., Tammer, C., Zǎlinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, Berlin (2003)
Fu, J.Y.: Generalized vector quasi-equilibrium problems. Math. Mathods Oper. Res. 52, 57–64 (2000)
Han, Y., Huang, N.J.: Some characterizations of the approximate solutions to generalized vector equilibrium problems. J. Ind. Manag. Optim. 12, 1135–1151 (2016)
Yen, N.D., Phuong, T.D.: Connectedness and stability of the solution set in linear fractional vector optimization problems. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria, pp. 479–489. Kluwer Academic Publishers, Dordrecht (2000)
Michael, E.: Continuous selections: I. Ann. Math. 63(2), 361–382 (1956)
Tan, N.X., Tinh, P.N.: On the existence of equilibrium points of vector functions. Numer. Funct. Anal. Optim. 19, 141–156 (1998)
Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)
Wei, H.Z., Chen, C.R., Li, S.J.: Characterizations for optimality conditions of general robust optimization problems. J. Optim. Theory Appl. 177, 835–856 (2018)
Acknowledgements
This research was supported by the National Natural Science Foundation of China (Grant No. 11971078).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Cui, C., Li, S. Path connectedness of the efficient solution set for generalized vector quasi-equilibrium problems. Optim Lett 16, 1881–1893 (2022). https://doi.org/10.1007/s11590-021-01809-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-021-01809-x
Keywords
- Generalized vector quasi-equilibrium problem
- Efficient solutions
- Path connectedness
- Linear scalarization