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2014, 10(4): 1225-1234. doi: 10.3934/jimo.2014.10.1225

Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem

1. 

College of Sciences, Chongqing Jiaotong University, Chongqing, 400074

2. 

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331

Received  March 2013 Revised  September 2013 Published  February 2014

This paper deals with the lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem. Under new assumptions, which do not contain any information about solution mappings, we establish the lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem by using a scalarization method. These results improve the corresponding ones in recent literature. Some examples are given to illustrate our results.
Citation: Qilin Wang, Shengji Li. Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1225-1234. doi: 10.3934/jimo.2014.10.1225
References:
[1]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems,, J. Math. Anal. Appl., 294 (2004), 699. doi: 10.1016/j.jmaa.2004.03.014.

[2]

L. Q. Anh and P. Q. Khanh, On the stability of the solution sets of general multivalued vector quasiequilibrium problems,, J. Optim. Theory Appl., 135 (2007), 271. doi: 10.1007/s10957-007-9250-9.

[3]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Wiley, (1984).

[4]

Berge, Topological Spaces,, Oliver and Boyd, (1963).

[5]

M. Bianchi, N. Hadjisavvas and S. Schaible, Vector equilibrium problems with generalized monotone bifunctions,, J. Optim. Theory Appl., 92 (1997), 527. doi: 10.1023/A:1022603406244.

[6]

C. R. Chen and S. J. Li, Semicontinuity of the solution set map to a set-valued weak vector variational inequality,, J. Ind. Manag. Optim., 3 (2007), 519. doi: 10.3934/jimo.2007.3.519.

[7]

C. R. Chen, S. J. Li and K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems,, J. Glob. Optim., 45 (2009), 309. doi: 10.1007/s10898-008-9376-9.

[8]

C. R. Chen and S. J. Li, On the solution continuity of parametric generalized systems,, Pac. J. Optim., 6 (2010), 141.

[9]

Y. H. Cheng and D. L. Zhu, Global stability results for the weak vector variational inequality,, J. Glob. Optim., 32 (2005), 543. doi: 10.1007/s10898-004-2692-9.

[10]

F. Ferro, A minimax theorem for vector-valued functions,, J. Optim. Theory Appl., 60 (1989), 19. doi: 10.1007/BF00938796.

[11]

J. F. Fu, Vector equilibrium problems, existence theorems and convexity of solution set,, J. Glob. Optim., 31 (2005), 109. doi: 10.1007/s10898-004-4274-2.

[12]

F. Giannessi, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories,, Kluwer Academic Publishers, (2000). doi: 10.1007/978-1-4613-0299-5.

[13]

X. H. Gong, Connectedness of the solution sets and scalarization for vector equilibrium problems,, J. Optim. Theory Appl., 133 (2007), 151. doi: 10.1007/s10957-007-9196-y.

[14]

X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems,, J. Optim. Theory Appl., 138 (2008), 197. doi: 10.1007/s10957-008-9379-1.

[15]

X. H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems,, J. Optim. Theory Appl., 139 (2008), 35. doi: 10.1007/s10957-008-9429-8.

[16]

X. H. Gong and J. C. Yao, Connectedness of the set of efficient solutions for generalized systems,, J. Optim. Theory Appl., 138 (2008), 189. doi: 10.1007/s10957-008-9378-2.

[17]

N. J. Huang, J. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems,, Math. Comput. Model., 43 (2006), 1267. doi: 10.1016/j.mcm.2005.06.010.

[18]

B. T. Kien, On the lower semicontinuity of optimal solution sets,, Optimization, 54 (2005), 123. doi: 10.1080/02331930412331330379.

[19]

K. Kimura and J. C.Yao, Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems,, J. Glob. Optim., 41 (2008), 187. doi: 10.1007/s10898-007-9210-9.

[20]

K. Kimura and J. C.Yao, Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems,, J. Ind. Manag. Optim., 4 (2008), 167. doi: 10.3934/jimo.2008.4.167.

[21]

K. Kimura and J. C.Yao, Sensitivity analysis of solution mappings of parametric generalized quasivector equilibrium problems,, Taiwanese J. Math., 12 (2008), 2233.

[22]

K. Kimura and J. C.Yao, Sensitivity analysis of vector equilibrium problems,, Taiwanese J. Math., 12 (2008), 649.

[23]

P. Q. Khanh and L. M. Luu, Lower and upper semicontinuity of the solution sets and approximate solution sets to parametric multivalued quasivariational inequalities,, J. Optim. Theory Appl., 133 (2007), 329. doi: 10.1007/s10957-007-9190-4.

[24]

Z. F. Li and G. Y. Chen, Lagrangian Multipliers, saddle points and duality in vector optimization of set-valued maps,, J. Math. Anal. Appl., 215 (1997), 297. doi: 10.1006/jmaa.1997.5568.

[25]

Z. F. Li and S. Y. Wang, Lagrange Multipliers and saddle points in multiobjective programming,, J. Optim. Theory Appl., 83 (1994), 63. doi: 10.1007/BF02191762.

[26]

S. J. Li, G. Y. Chen and K. L. Teo, On the stability of generalized vector quasivariational inequality problems,, J. Optim. Theory Appl., 113 (2002), 297. doi: 10.1023/A:1014830925232.

[27]

S. J. Li and Z. M. Fang, On the stability of a dual weak vector variational inequality problem,, J. Ind. Manag. Optim., 4 (2008), 155. doi: 10.3934/jimo.2008.4.155.

[28]

S. J. Li and C. R. Chen, Stability of weak vector variational inequality,, Nonlinear Anal., 70 (2009), 1528. doi: 10.1016/j.na.2008.02.032.

[29]

S. J. Li and Z. M.Fang, Lower semicontinuity of the solution mappings to a parametric generalized Ky Fan inequality,, J. Optim. Theory Appl., 147 (2010), 507. doi: 10.1007/s10957-010-9736-8.

[30]

S. J. Li, H. M. Liu and C. R. Chen, Lower semicomtinuity of parametric generalized weak vector equilibrium problems,, Bull. Aust. Math. Soc., 81 (2010), 85. doi: 10.1017/S0004972709000628.

[31]

S. J. Li, H. M. Liu, Y. Zhang and Z. M. Fang, Continuity of solution mappings to parametric generalized strong vector equilibrium problems,, J. Glob. Optim., 55 (2013), 597. doi: 10.1007/s10898-012-9985-1.

[32]

Y. D. Xu and S. J. Li, On the lower semicontinuity of the solution mappings to a para- metric generalized strong vector equilibrium problem,, Positivity, 17 (2013), 341. doi: 10.1007/s11117-012-0170-z.

[33]

X. M. Yang, D. Li and S. Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions,, J. Optim. Theory Appl., 110 (2001), 413. doi: 10.1023/A:1017535631418.

show all references

References:
[1]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems,, J. Math. Anal. Appl., 294 (2004), 699. doi: 10.1016/j.jmaa.2004.03.014.

[2]

L. Q. Anh and P. Q. Khanh, On the stability of the solution sets of general multivalued vector quasiequilibrium problems,, J. Optim. Theory Appl., 135 (2007), 271. doi: 10.1007/s10957-007-9250-9.

[3]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Wiley, (1984).

[4]

Berge, Topological Spaces,, Oliver and Boyd, (1963).

[5]

M. Bianchi, N. Hadjisavvas and S. Schaible, Vector equilibrium problems with generalized monotone bifunctions,, J. Optim. Theory Appl., 92 (1997), 527. doi: 10.1023/A:1022603406244.

[6]

C. R. Chen and S. J. Li, Semicontinuity of the solution set map to a set-valued weak vector variational inequality,, J. Ind. Manag. Optim., 3 (2007), 519. doi: 10.3934/jimo.2007.3.519.

[7]

C. R. Chen, S. J. Li and K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems,, J. Glob. Optim., 45 (2009), 309. doi: 10.1007/s10898-008-9376-9.

[8]

C. R. Chen and S. J. Li, On the solution continuity of parametric generalized systems,, Pac. J. Optim., 6 (2010), 141.

[9]

Y. H. Cheng and D. L. Zhu, Global stability results for the weak vector variational inequality,, J. Glob. Optim., 32 (2005), 543. doi: 10.1007/s10898-004-2692-9.

[10]

F. Ferro, A minimax theorem for vector-valued functions,, J. Optim. Theory Appl., 60 (1989), 19. doi: 10.1007/BF00938796.

[11]

J. F. Fu, Vector equilibrium problems, existence theorems and convexity of solution set,, J. Glob. Optim., 31 (2005), 109. doi: 10.1007/s10898-004-4274-2.

[12]

F. Giannessi, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories,, Kluwer Academic Publishers, (2000). doi: 10.1007/978-1-4613-0299-5.

[13]

X. H. Gong, Connectedness of the solution sets and scalarization for vector equilibrium problems,, J. Optim. Theory Appl., 133 (2007), 151. doi: 10.1007/s10957-007-9196-y.

[14]

X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems,, J. Optim. Theory Appl., 138 (2008), 197. doi: 10.1007/s10957-008-9379-1.

[15]

X. H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems,, J. Optim. Theory Appl., 139 (2008), 35. doi: 10.1007/s10957-008-9429-8.

[16]

X. H. Gong and J. C. Yao, Connectedness of the set of efficient solutions for generalized systems,, J. Optim. Theory Appl., 138 (2008), 189. doi: 10.1007/s10957-008-9378-2.

[17]

N. J. Huang, J. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems,, Math. Comput. Model., 43 (2006), 1267. doi: 10.1016/j.mcm.2005.06.010.

[18]

B. T. Kien, On the lower semicontinuity of optimal solution sets,, Optimization, 54 (2005), 123. doi: 10.1080/02331930412331330379.

[19]

K. Kimura and J. C.Yao, Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems,, J. Glob. Optim., 41 (2008), 187. doi: 10.1007/s10898-007-9210-9.

[20]

K. Kimura and J. C.Yao, Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems,, J. Ind. Manag. Optim., 4 (2008), 167. doi: 10.3934/jimo.2008.4.167.

[21]

K. Kimura and J. C.Yao, Sensitivity analysis of solution mappings of parametric generalized quasivector equilibrium problems,, Taiwanese J. Math., 12 (2008), 2233.

[22]

K. Kimura and J. C.Yao, Sensitivity analysis of vector equilibrium problems,, Taiwanese J. Math., 12 (2008), 649.

[23]

P. Q. Khanh and L. M. Luu, Lower and upper semicontinuity of the solution sets and approximate solution sets to parametric multivalued quasivariational inequalities,, J. Optim. Theory Appl., 133 (2007), 329. doi: 10.1007/s10957-007-9190-4.

[24]

Z. F. Li and G. Y. Chen, Lagrangian Multipliers, saddle points and duality in vector optimization of set-valued maps,, J. Math. Anal. Appl., 215 (1997), 297. doi: 10.1006/jmaa.1997.5568.

[25]

Z. F. Li and S. Y. Wang, Lagrange Multipliers and saddle points in multiobjective programming,, J. Optim. Theory Appl., 83 (1994), 63. doi: 10.1007/BF02191762.

[26]

S. J. Li, G. Y. Chen and K. L. Teo, On the stability of generalized vector quasivariational inequality problems,, J. Optim. Theory Appl., 113 (2002), 297. doi: 10.1023/A:1014830925232.

[27]

S. J. Li and Z. M. Fang, On the stability of a dual weak vector variational inequality problem,, J. Ind. Manag. Optim., 4 (2008), 155. doi: 10.3934/jimo.2008.4.155.

[28]

S. J. Li and C. R. Chen, Stability of weak vector variational inequality,, Nonlinear Anal., 70 (2009), 1528. doi: 10.1016/j.na.2008.02.032.

[29]

S. J. Li and Z. M.Fang, Lower semicontinuity of the solution mappings to a parametric generalized Ky Fan inequality,, J. Optim. Theory Appl., 147 (2010), 507. doi: 10.1007/s10957-010-9736-8.

[30]

S. J. Li, H. M. Liu and C. R. Chen, Lower semicomtinuity of parametric generalized weak vector equilibrium problems,, Bull. Aust. Math. Soc., 81 (2010), 85. doi: 10.1017/S0004972709000628.

[31]

S. J. Li, H. M. Liu, Y. Zhang and Z. M. Fang, Continuity of solution mappings to parametric generalized strong vector equilibrium problems,, J. Glob. Optim., 55 (2013), 597. doi: 10.1007/s10898-012-9985-1.

[32]

Y. D. Xu and S. J. Li, On the lower semicontinuity of the solution mappings to a para- metric generalized strong vector equilibrium problem,, Positivity, 17 (2013), 341. doi: 10.1007/s11117-012-0170-z.

[33]

X. M. Yang, D. Li and S. Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions,, J. Optim. Theory Appl., 110 (2001), 413. doi: 10.1023/A:1017535631418.

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