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Some characterizations of the approximate solutions to generalized vector equilibrium problems
1. | Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China |
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. |
[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. |
[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. |
[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. |
[5] |
J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984. |
[6] | |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer, Berlin Heidelberg, New York, 2003. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[5] |
J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984. |
[6] | |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer, Berlin Heidelberg, New York, 2003. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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