Approximate weakly efficient solutions of set-valued vector equilibrium problems
Tóm tắt
In this paper, we introduce a new kind of approximate weakly efficient solutions to the set-valued vector equilibrium problems with constraints in locally convex Hausdorff topological vector spaces; then we discuss a relationship between the weakly efficient solutions and approximate weakly efficient solutions. Under the assumption of near cone-subconvexlikeness, by using the separation theorem for convex sets we establish Kuhn–Tucker-type and Lagrange-type optimality conditions for set-valued vector equilibrium problems, respectively.
Tài liệu tham khảo
Anh, L.Q., Duy, T.Q., Khanh, P.Q.: Continuity properties of solution maps of parametric lexicographic equilibrium problems. Positivity 20, 1–20 (2015)
Gong, X.H.: Optimality conditions for vector equilibrium problems. J. Math. Anal. Appl. 342, 1455–1466 (2008)
Gong, X.H.: Scalarization and optimality conditions for vector equilibrium problems. Nonlinear Anal. TMA 73, 3598–3612 (2010)
Gong, X.H.: Efficiency and Henig efficiency for vector equilibrium problems. J. Optim. Theory Appl. 108, 139–154 (2001)
Long, X.J., Huang, Y.Q., Peng, Z.Y.: Optimality conditions for the Henig efficient solution of vector equilibrium problems with constraints. Optim. Lett. 5, 717–728 (2011)
Qiu, Q.S.: Optimality conditions for vector equilibrium problems with constraints. J. Ind. Manag. Optim. 5, 783–790 (2017)
Luu, D.V., Hang, D.D.: Efficient solutions and optimality conditions for vector equilibrium problems. Math. Methods Oper. Res. 79, 163–177 (2014)
Luu, D.V.: Optimality condition for local efficient solutions of vector equilibrium problems via convexificators and applications. J. Optim. Theory Appl. 171, 643–665 (2016)
Loridan, P.: Necessary conditions for ϵ-optimality. Math. Program. 19, 140–152 (1982)
Loridan, P.: ϵ-solutions in vector minimization problems. J. Optim. Theory Appl. 43, 265–276 (1984)
Li, B.X., Li, S.J.: Continuity of approximate solution mappings for parametric equilibrium problems. J. Glob. Optim. 51, 541–548 (2011)
Yang, X.M., Li, D., Wang, S.Y.: Near-subconvexlikeness in vector optimization with set-valued functions. J. Optim. Theory Appl. 110, 413–427 (2001)
Sach, P.H.: New generalized convexity notion for set-valued maps and application to vector optimization. J. Optim. Theory Appl. 125, 157–179 (2005)
Xu, Y.H., Song, X.S.: The relationship between ic-cone-convexness and nearly cone-subconvexlikeness. Appl. Math. Lett. 24, 1622–1624 (2011)
Rong, W.D., Wu, Y.N.: ϵ-weak minimal solutions of vector optimization problems with set-valued maps. J. Optim. Theory Appl. 106, 569–579 (2000)
Li, Z.F., Chen, G.Y.: Lagrangian multipliers, saddle points, and duality in vector optimization of set-valued maps. J. Math. Anal. Appl. 215, 279–316 (1997)
Yang, X.M., Yang, X.Q., Chen, G.Y.: Theorems of the alternative and optimization with set-valued maps. J. Optim. Theory Appl. 107, 627–640 (2000)
Sach, P.H.: Nearly subconvexlike set-valued maps and vector optimization problems. J. Optim. Theory Appl. 119, 335–356 (2003)