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July  2016, 12(3): 1135-1151. doi: 10.3934/jimo.2016.12.1135

Some characterizations of the approximate solutions to generalized vector equilibrium problems

Yu Han 1, and  Nan-Jing Huang 1,

1. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

Received  November 2014 Revised  May 2015 Published  September 2015

In this paper, a scalarization result and a density theorem concerned with the sets of weakly efficient and efficient approximate solutions to a generalized vector equilibrium problem are given, respectively. By using the scalarization result and the density theorem, the connectedness of the sets of weakly efficient and efficient approximate solutions to the generalized vector equilibrium problem are established without the assumptions of monotonicity and compactness. The lower semicontinuity of weakly efficient and efficient approximate solution mappings to the parametric generalized vector equilibrium problem with perturbing both the objective mapping and the feasible region are obtained without the assumptions of monotonicity and compactness. Furthermore, the upper semicontinuity of weakly efficient approximate solution mapping and the Hausdorff upper semicontinuity of efficient approximate solution mapping to the parametric generalized vector equilibrium problem with perturbing both the objective mapping and the feasible region are also given under some suitable conditions.
Keywords: upper semicontinuity., scalarization, Generalized vector equilibrium problem, approximate solution, lower semicontinuity, connectedness.
Mathematics Subject Classification: 90C31, 91B5.
Citation: Yu Han, Nan-Jing Huang. Some characterizations of the approximate solutions to generalized vector equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1135-1151. doi: 10.3934/jimo.2016.12.1135
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), 699-711. doi: 10.1016/j.jmaa.2004.03.014.  Google Scholar

[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), 271-284. doi: 10.1007/s10957-007-9250-9.  Google Scholar

[3]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems, Numer. Funct. Anal. Optim., 29 (2008), 24-42. doi: 10.1080/01630560701873068.  Google Scholar

[4]

L. Q. Anh and P. Q. Khanh, Continuity of solution maps of parametric quasiequilibrium problems, J. Glob. Optim., 46 (2010), 247-259. doi: 10.1007/s10898-009-9422-2.  Google Scholar

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J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984.  Google Scholar

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R. Brown, Topology, Ellis Horwood, New York, NY, 1988.  Google Scholar

[7]

B. Chen, Q. Y. Liu, Z. B. Liu and N. J. Huang, Connectedness of approximate solutions set for vector equilibrium problems in Hausdorff topological vector spaces, Fixed Point Theory and Applications, 2011 (2011), p36. doi: 10.1186/1687-1812-2011-36.  Google Scholar

[8]

B. Chen and N. J. Huang, Continuity of the solution mapping to parametric generalized vector equilibrium problems, J. Glob. Optim., 56 (2013), 1515-1528. doi: 10.1007/s10898-012-9904-5.  Google Scholar

[9]

C. R. Chen, S. J. Li and K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems, J. Glob. Optim., 45 (2009), 309-318. doi: 10.1007/s10898-008-9376-9.  Google Scholar

[10]

Y. H. Cheng, On the connectedness of the solution set for the weak vector variational inequality, J. Math. Anal. Appl., 260 (2001), 1-5. doi: 10.1006/jmaa.2000.7389.  Google Scholar

[11]

Y. H. Cheng and D. L. Zhu, Global stability results for the weak vector variational inequality, J. Glob. Optim., 32 (2005), 543-550. doi: 10.1007/s10898-004-2692-9.  Google Scholar

[12]

Y. Gao, X. M. Yang and K. L. Teo, Optimality conditions for approximate solutions of vector optimization problems, J. Ind. Manag. Optim., 7 (2011), 483-496. doi: 10.3934/jimo.2011.7.483.  Google Scholar

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Y. Gao, X. M. Yang, J. Yang and H. Yan, Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps, J. Ind. Manag. Optim., 11 (2015), 673-683. doi: 10.3934/jimo.2015.11.673.  Google Scholar

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X. H. Gong, Efficiency and Henig efficiency for vector equilibrium problems, J. Optim. Theory Appl., 108 (2001), 139-154. doi: 10.1023/A:1026418122905.  Google Scholar

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X. H. Gong, Connectedness of the solution sets and scalarization for vector equilibrium problems, J. Optim. Theory Appl., 133 (2007), 151-161. doi: 10.1007/s10957-007-9196-y.  Google Scholar

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X.H. Gong and J.C. Yao, Connectedness of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 189-196. doi: 10.1007/s10957-008-9378-2.  Google Scholar

[18]

X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 197-205. doi: 10.1007/s10957-008-9379-1.  Google Scholar

[19]

X. H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, J. Optim. Theory Appl., 139 (2008), 35-46. doi: 10.1007/s10957-008-9429-8.  Google Scholar

[20]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer, Berlin Heidelberg, New York, 2003.  Google Scholar

[21]

Y. Han and X. H. Gong, Lower semicontinuity of solution mapping to parametric generalized strong vector equilibrium problems, Appl. Math. Lett., 28 (2014), 38-41. doi: 10.1016/j.aml.2013.09.006.  Google Scholar

[22]

N. J. Huang, J. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Model., 43 (2006), 1267-1274. doi: 10.1016/j.mcm.2005.06.010.  Google Scholar

[23]

P. Q. Khanh and L. M. Luu, Lower and upper semicontinuity of the solution sets and the approxiamte solution sets to parametric multivalued quasivariational inequalities, J. Optim. Theory Appl., 133 (2007), 329-339. doi: 10.1007/s10957-007-9190-4.  Google Scholar

[24]

K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems, J. Glob. Optim., 41 (2008), 187-202. doi: 10.1007/s10898-007-9210-9.  Google Scholar

[25]

G. M. Lee, D. S. Kim, B. S. Lee and N. D. Yun, Vector variational inequalities as a tool for studing vector optimization problems, Nonlinear Anal., 34 (1998), 745-765. doi: 10.1016/S0362-546X(97)00578-6.  Google Scholar

[26]

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), 507-515. doi: 10.1007/s10957-010-9736-8.  Google Scholar

[27]

X. B. Li and S. J. Li, Continuity of approximate solution mappings for parametric equilibrium problems, J. Glob. Optim., 51 (2011), 541-548. doi: 10.1007/s10898-010-9641-6.  Google Scholar

[28]

S. J. Li, H. M. Liu, Y. Zhang and Z. M. Fang, Continuity of the solution mappings to parametric generalized strong vector equilibrium problems, J. Glob. Optim., 55 (2013), 597-610. doi: 10.1007/s10898-012-9985-1.  Google Scholar

[29]

D. T. Luc, Connectedness of the efficient point sets in quasiconcave vector maximization, J. Math. Anal. Appl., 122 (1987), 346-354. doi: 10.1016/0022-247X(87)90264-2.  Google Scholar

[30]

Q. S. Qiu and X. M. Yang, Some properties of approximate solutions for vector optimization problem with set-valued functions, J. Glob. Optim., 47 (2010), 1-12. doi: 10.1007/s10898-009-9452-9.  Google Scholar

[31]

Q. S. Qiu and X. M. Yang, Connectedness of Henig weakly efficient solution set for set-valued optimization problems, J. Optim. Theory Appl., 152 (2012), 439-449. doi: 10.1007/s10957-011-9906-3.  Google Scholar

[32]

Q. S. Qiu and X. M. Yang, Scalarization of approximate solution for vector equilibrium problems, J. Ind. Manag. Optim., 9 (2013), 143-151. doi: 10.3934/jimo.2013.9.143.  Google Scholar

[33]

E. J. Sun, On the connectedness of the efficient set for strictly quasiconvex vector minimization problems, J. Optim. Theory Appl., 89 (1996), 475-481. doi: 10.1007/BF02192541.  Google Scholar

[34]

Q. L. Wang and S. J. Li, Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem, J. Ind. Manag. Optim., 10 (2014), 1225-1234. doi: 10.3934/jimo.2014.10.1225.  Google Scholar

[35]

R. Y. Zhong, N. J. Huang and M. M. Wong, Connectedness and path-connectedness of solution sets to symmetric vector equilibrium problems, Taiwan. J. Math., 13 (2009), 821-836.  Google Scholar

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), 699-711. doi: 10.1016/j.jmaa.2004.03.014.  Google Scholar

[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), 271-284. doi: 10.1007/s10957-007-9250-9.  Google Scholar

[3]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems, Numer. Funct. Anal. Optim., 29 (2008), 24-42. doi: 10.1080/01630560701873068.  Google Scholar

[4]

L. Q. Anh and P. Q. Khanh, Continuity of solution maps of parametric quasiequilibrium problems, J. Glob. Optim., 46 (2010), 247-259. doi: 10.1007/s10898-009-9422-2.  Google Scholar

[5]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984.  Google Scholar

[6]

R. Brown, Topology, Ellis Horwood, New York, NY, 1988.  Google Scholar

[7]

B. Chen, Q. Y. Liu, Z. B. Liu and N. J. Huang, Connectedness of approximate solutions set for vector equilibrium problems in Hausdorff topological vector spaces, Fixed Point Theory and Applications, 2011 (2011), p36. doi: 10.1186/1687-1812-2011-36.  Google Scholar

[8]

B. Chen and N. J. Huang, Continuity of the solution mapping to parametric generalized vector equilibrium problems, J. Glob. Optim., 56 (2013), 1515-1528. doi: 10.1007/s10898-012-9904-5.  Google Scholar

[9]

C. R. Chen, S. J. Li and K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems, J. Glob. Optim., 45 (2009), 309-318. doi: 10.1007/s10898-008-9376-9.  Google Scholar

[10]

Y. H. Cheng, On the connectedness of the solution set for the weak vector variational inequality, J. Math. Anal. Appl., 260 (2001), 1-5. doi: 10.1006/jmaa.2000.7389.  Google Scholar

[11]

Y. H. Cheng and D. L. Zhu, Global stability results for the weak vector variational inequality, J. Glob. Optim., 32 (2005), 543-550. doi: 10.1007/s10898-004-2692-9.  Google Scholar

[12]

Y. Gao, X. M. Yang and K. L. Teo, Optimality conditions for approximate solutions of vector optimization problems, J. Ind. Manag. Optim., 7 (2011), 483-496. doi: 10.3934/jimo.2011.7.483.  Google Scholar

[13]

Y. Gao, X. M. Yang, J. Yang and H. Yan, Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps, J. Ind. Manag. Optim., 11 (2015), 673-683. doi: 10.3934/jimo.2015.11.673.  Google Scholar

[14]

X. H. Gong, Connectedness of efficiency solution sets for set-valued maps in normed spaces, J. Optim. Theory Appl., 83 (1994), 83-96. doi: 10.1007/BF02191763.  Google Scholar

[15]

X. H. Gong, Efficiency and Henig efficiency for vector equilibrium problems, J. Optim. Theory Appl., 108 (2001), 139-154. doi: 10.1023/A:1026418122905.  Google Scholar

[16]

X. H. Gong, Connectedness of the solution sets and scalarization for vector equilibrium problems, J. Optim. Theory Appl., 133 (2007), 151-161. doi: 10.1007/s10957-007-9196-y.  Google Scholar

[17]

X.H. Gong and J.C. Yao, Connectedness of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 189-196. doi: 10.1007/s10957-008-9378-2.  Google Scholar

[18]

X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 197-205. doi: 10.1007/s10957-008-9379-1.  Google Scholar

[19]

X. H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, J. Optim. Theory Appl., 139 (2008), 35-46. doi: 10.1007/s10957-008-9429-8.  Google Scholar

[20]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer, Berlin Heidelberg, New York, 2003.  Google Scholar

[21]

Y. Han and X. H. Gong, Lower semicontinuity of solution mapping to parametric generalized strong vector equilibrium problems, Appl. Math. Lett., 28 (2014), 38-41. doi: 10.1016/j.aml.2013.09.006.  Google Scholar

[22]

N. J. Huang, J. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Model., 43 (2006), 1267-1274. doi: 10.1016/j.mcm.2005.06.010.  Google Scholar

[23]

P. Q. Khanh and L. M. Luu, Lower and upper semicontinuity of the solution sets and the approxiamte solution sets to parametric multivalued quasivariational inequalities, J. Optim. Theory Appl., 133 (2007), 329-339. doi: 10.1007/s10957-007-9190-4.  Google Scholar

[24]

K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems, J. Glob. Optim., 41 (2008), 187-202. doi: 10.1007/s10898-007-9210-9.  Google Scholar

[25]

G. M. Lee, D. S. Kim, B. S. Lee and N. D. Yun, Vector variational inequalities as a tool for studing vector optimization problems, Nonlinear Anal., 34 (1998), 745-765. doi: 10.1016/S0362-546X(97)00578-6.  Google Scholar

[26]

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), 507-515. doi: 10.1007/s10957-010-9736-8.  Google Scholar

[27]

X. B. Li and S. J. Li, Continuity of approximate solution mappings for parametric equilibrium problems, J. Glob. Optim., 51 (2011), 541-548. doi: 10.1007/s10898-010-9641-6.  Google Scholar

[28]

S. J. Li, H. M. Liu, Y. Zhang and Z. M. Fang, Continuity of the solution mappings to parametric generalized strong vector equilibrium problems, J. Glob. Optim., 55 (2013), 597-610. doi: 10.1007/s10898-012-9985-1.  Google Scholar

[29]

D. T. Luc, Connectedness of the efficient point sets in quasiconcave vector maximization, J. Math. Anal. Appl., 122 (1987), 346-354. doi: 10.1016/0022-247X(87)90264-2.  Google Scholar

[30]

Q. S. Qiu and X. M. Yang, Some properties of approximate solutions for vector optimization problem with set-valued functions, J. Glob. Optim., 47 (2010), 1-12. doi: 10.1007/s10898-009-9452-9.  Google Scholar

[31]

Q. S. Qiu and X. M. Yang, Connectedness of Henig weakly efficient solution set for set-valued optimization problems, J. Optim. Theory Appl., 152 (2012), 439-449. doi: 10.1007/s10957-011-9906-3.  Google Scholar

[32]

Q. S. Qiu and X. M. Yang, Scalarization of approximate solution for vector equilibrium problems, J. Ind. Manag. Optim., 9 (2013), 143-151. doi: 10.3934/jimo.2013.9.143.  Google Scholar

[33]

E. J. Sun, On the connectedness of the efficient set for strictly quasiconvex vector minimization problems, J. Optim. Theory Appl., 89 (1996), 475-481. doi: 10.1007/BF02192541.  Google Scholar

[34]

Q. L. Wang and S. J. Li, Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem, J. Ind. Manag. Optim., 10 (2014), 1225-1234. doi: 10.3934/jimo.2014.10.1225.  Google Scholar

[35]

R. Y. Zhong, N. J. Huang and M. M. Wong, Connectedness and path-connectedness of solution sets to symmetric vector equilibrium problems, Taiwan. J. Math., 13 (2009), 821-836.  Google Scholar

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