Abstract
In this paper, we give results on Chebyshev scalarization of weakly efficient solution, Henig efficient solution, globally efficient solution and superefficient solution to the vector equilibrium problems without convexity assumptions.
Similar content being viewed by others
References
Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Giannessi F.: Vector variational inequalities and vector equilibria. In: Giannessi, F. (eds) Mathematical Theories, Kluwer, Dordrecht (2000)
Isac G, Bulavsky V.A., Kalashnikov V.V.: Complementarity, Equilibrium, Efficiency and Economics. Kluwer, Dordrecht (2002)
Göpfert, A., Tammer, C., Riahi, H., Zălinescu, C.: Variational methods in partially ordered spaces. Springer, (2003)
Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L.: Pareto Optimality, game theory and equilibria. New York Series: Springer optimization and its applications. vol.17. Springer, New York (2008)
Lee G.M., Kim D.S., Lee B.S., Yun N.D.: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal. Theory Methods Appl. 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)
Gong X.H., Fu W.T., Liu W.: Super efficiency for a vector equilibrium in locally convex topological vector spaces. In: Giannessi, F. (eds) Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, pp. 233–252. Kluwer, Dordrecht (2000)
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)
Gong X.H., Yao J.C.: Lower semicontinuity of the set of efficient solutions for generalized systems. J. Optim. Theory Appl. 138, 197–205 (2008)
Gong X.H.: Continuity of the solution set to parametric weak vector equilibrium problems. J. Optim. Theory Appl. 139, 35–46 (2008)
Morgan J., Romaniello M.: Scalarization and Kuhn–Tucker-like conditions for weak vector generalized quasivarational inequalities. J. Optim. Theory Appl. 130, 309–316 (2006)
Chen G.Y., Yang X.Q., Yu H.: A nonlinear scalarization function and generalized quasi-vector equilibrium problems. J. Glob. Optim. 32, 451–466 (2005)
Yang X.Q., Zheng X.Y.: Approximate solutions and optimality conditions of vector variational inequalities in Banach spaces. J. Glob. Optim. 40, 455–462 (2008)
Borwein J.M., Zhuang D.M.: Super efficiency in vector optimization. Trans. Am. Math. Soc. 338, 105–122 (1993)
Zheng X.Y.: The domination property for efficiency in locally convex spaces. J. Math. Anal. Appl. 213, 455–467 (1997)
Jahn J.: Mathematical Vector Optimization in Partially Ordered Linear Spaces. Peter Lang, Frankfurt am Main (1986)
Yang F.M., Gong X.H.: \({C^\sharp}\) monotone norm and Chebyshev scalarization of Henig efficient point. J. Syst. Sci. Math. Sci. 22, 334–342 (2002)
Jameson, G.: Ordered linear space. In: Lecture Notes in Mathematics, vol. 141. Springer, Berlin (1970)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gong, X.H. Chebyshev scalarization of solutions to the vector equilibrium problems. J Glob Optim 49, 607–622 (2011). https://doi.org/10.1007/s10898-010-9553-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-010-9553-5
Keywords
- Vector equilibrium problem
- Chebyshev scalarization
- Weakly efficient solution
- Henig efficient solution
- Globally efficient solution
- Superefficient solution