Solution semicontinuity of parametric generalized vector equilibrium problems.(English)Zbl 1213.54028

The objective of a parametric generalized vector equilibrium problem (PGVEP) is defined as to find $$x\in A(\mu)$$ such that
$F(x,y,\mu)\subset Y\backslash(-\text{int}\,C)\quad \forall y\in A(\mu)$
Here $$F:\;B\times B\times\Lambda \rightarrow 2^Y\backslash\{\varnothing\}$$ and $$A:\;\Lambda\rightarrow 2^X\backslash\{\varnothing\}$$ are set-valued mappings, $$X$$ and $$Y$$ are Hausdorff topological vector spaces, $$Z$$ is a real topological space, $$B\subset X$$, $$\mu \in\Lambda\subset Z$$, and $$C\subset Y$$ is a convex cone with a nonempty interior. Let $$S(\mu)$$ denote the set of all PGVEP solutions for a given $$\mu$$. It is obtained that for $$S(\cdot)$$ to be continuous on $$\Lambda$$, it is sufficient that (1) the mapping $$A$$ is compact-valued and continuous, (2) $$F$$ is compact-valued and upper-semicontinuous, (3) $$F(\cdot,\cdot,\mu)$$ is $$C$$-strictly monotone for each $$\mu\in\Lambda$$, and (4) $$F(x,\cdot,\mu)$$ is $$C$$-convexlike, that is for an arbitrary $$\rho\in[0,1]$$ the inclusion
$\rho F(x,x_1,\mu)+(1-\rho) F(x,x_2,\mu)\subset F(x,x_3,\mu)+C$
holds true for some $$x_3\in\Lambda$$.

MSC:

 54C60 Set-valued maps in general topology 26E25 Set-valued functions 47H04 Set-valued operators 58C06 Set-valued and function-space-valued mappings on manifolds 90C31 Sensitivity, stability, parametric optimization
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References:

 [1] Anh L.Q., Khanh P.Q.: Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems. J. Math. Anal. Appl. 294, 699–711 (2004) · Zbl 1048.49004 · doi:10.1016/j.jmaa.2004.03.014 [2] Anh L.Q., Khanh P.Q.: On the stability of the solution sets of general multivalued vector quasiequilibrium problems. J. Optim. Theory Appl. 135, 271–284 (2007) · Zbl 1146.90516 · doi:10.1007/s10957-007-9250-9 [3] Aubin J.P., Ekeland I.: Applied Nonlinear Analysis. Wiley, New York (1984) · Zbl 0641.47066 [4] Berge C.: Topological Spaces. Oliver and Boyd, London (1963) · Zbl 0114.38602 [5] Chen C.R., Li S.J.: Semicontinuity of the solution set map to a set-valued weak vector variational inequality. J. Ind. Manag. Optim. 3, 519–528 (2007) · Zbl 1170.90496 [6] Chen, C.R., Li, S.J.: On the solution continuity of parametric generalized systems. (2008) (submitted) [7] Chen G.Y., Huang X.X., Yang X.Q.: Vector Optimization: Set-Valued and Variational Analysis. Springer, Berlin (2005) · Zbl 1104.90044 [8] Chen, C.R., Fang, Z.M., Li, S.J.: On the semicontinuity for a parametric generalized vector quasivariational inequality. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms. (2008) (to appear) [9] Cheng Y.H., Zhu D.L.: Global stability results for the weak vector variational inequality. J. Global Optim. 32, 543–550 (2005) · Zbl 1097.49006 · doi:10.1007/s10898-004-2692-9 [10] Ferro F.: A minimax theorem for vector-valued functions. J. Optim. Theory Appl. 60, 19–31 (1989) · Zbl 0631.90077 · doi:10.1007/BF00938796 [11] Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria: Mathematical Theories. Kluwer Academic Publishers, Dordrecht (2000) · Zbl 0952.00009 [12] Gong X.H.: Continuity of the solution set to parametric weak vector equilibrium problems. J. Optim. Theory Appl. 139, 35–46 (2008) · Zbl 1189.90195 · doi:10.1007/s10957-008-9429-8 [13] 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) · Zbl 1302.49018 · doi:10.1007/s10957-008-9379-1 [14] Huang N.J., Li J., Thompson H.B.: Stability for parametric implicit vector equilibrium problems. Math. Comput. Model. 43, 1267–1274 (2006) · Zbl 1187.90286 · doi:10.1016/j.mcm.2005.06.010 [15] Jahn J.: Vector Optimization-Theory, Applications and Extensions. Springer, Berlin (2004) · Zbl 1055.90065 [16] Khanh P.Q., Luu L.M.: Upper semicontinuity of the solution set to parametric vector quasivariational inequalities. J. Global Optim. 32, 569–580 (2005) · Zbl 1097.49013 · doi:10.1007/s10898-004-2694-7 [17] Kimura K., Yao J.C.: Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems. J. Global Optim. 41, 187–202 (2008) · Zbl 1165.49009 · doi:10.1007/s10898-007-9210-9 [18] Kimura K., Yao J.C.: Semicontinuity of solution mappings of parametric generalized vector equilibrium problems. J. Optim. Theory Appl. 138, 429–443 (2008) · Zbl 1162.47044 · doi:10.1007/s10957-008-9386-2 [19] Li S.J., Chen G.Y., Teo K.L.: On the stability of generalized vector quasivariational inequality problems. J. Optim. Theory Appl. 113, 283–295 (2002) · Zbl 1003.47049 · doi:10.1023/A:1014830925232 [20] Li, S.J., Chen, C.R.: Stability of weak vector variational inequality. Nonlinear Anal. doi: 10.1016/j.na.2008.02.032 (2008) [21] Li S.J., Fang Z.M.: On the stability of a dual weak vector variational inequality problem. J. Ind. Manag. Optim. 4, 155–165 (2008) · Zbl 1180.90339
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