July  2012, 8(3): 691-703. doi: 10.3934/jimo.2012.8.691

Upper Hölder estimates of solutions to parametric primal and dual vector quasi-equilibria

1. 

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

Received  March 2011 Revised  January 2012 Published  June 2012

In this paper, on one hand, we discuss upper Hölder type estimates of solutions to parametric vector quasi-equilibria with general settings, which generalize and extend the results of Chen et al. (Optim. Lett. 5: 85-98, 2011). On the other hand, combining the technique used for primal problems with suitable modifications, we also study upper Hölder type estimates of solutions to Minty-type parametric dual vector quasi-equilibria. The consequences obtained for dual problems are new in the literature.
Citation: Chunrong Chen, Shengji Li. Upper Hölder estimates of solutions to parametric primal and dual vector quasi-equilibria. Journal of Industrial & Management Optimization, 2012, 8 (3) : 691-703. doi: 10.3934/jimo.2012.8.691
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. 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. doi: 10.1007/s10957-007-9250-9. Google Scholar

[3]

L. Q. Anh and P. Q. Khanh, On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems,, J. Math. Anal. Appl., 321 (2006), 308. doi: 10.1016/j.jmaa.2005.08.018. Google Scholar

[4]

L. Q. Anh and P. Q. Khanh, Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces,, J. Glob. Optim., 37 (2007), 449. doi: 10.1007/s10898-006-9062-8. Google Scholar

[5]

L. Q. Anh and P. Q. Khanh, Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions,, J. Glob. Optim., 42 (2008), 515. doi: 10.1007/s10898-007-9268-4. Google Scholar

[6]

L. Q. Anh and P. Q. Khanh, Sensitivity anlysis for weak and strong vector quasiequilibrium problems,, Vietnam J. Math., 37 (2009), 237. Google Scholar

[7]

Q. H. Ansari, A. H. Siddiqi and S. Y. Wu, Existence and duality of generalized vector equilibrium problems,, J. Math. Anal. Appl., 259 (2001), 115. doi: 10.1006/jmaa.2000.7397. Google Scholar

[8]

E. M. Bednarczuk, Strong pseudomonotonicity, sharp efficiency and stability for parametric vector equilibria,, in, 17 (2007), 9. doi: 10.1051/proc:071702. Google Scholar

[9]

E. M. Bednarczuk, Weak sharp efficiency and growth condition for vector-valued functions with applications,, Optimization, 53 (2004), 455. doi: 10.1080/02331930412331330478. Google Scholar

[10]

M. Bianchi and R. Pini, Sensitivity for parametric vector equilibria,, Optimization, 55 (2006), 221. doi: 10.1080/02331930600662732. Google Scholar

[11]

J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems,", Springer Series in Operations Research, (2000). Google Scholar

[12]

G.-Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization: Set-Valued and Variational Analysis,", Lecture Notes in Economics and Mathematical Systems, 541 (2005). Google Scholar

[13]

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. Google Scholar

[14]

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. Google Scholar

[15]

C. R. Chen, S. J. Li, J. Zeng and X. B. Li, Error analysis of approximate solutions to parametric vector quasiequilibrium problems,, Optim. Lett., 5 (2011), 85. Google Scholar

[16]

F. Giannessi, ed., "Vector Variational Inequalities and Vector Equilibria: Mathematical Theories,", Nonconvex Optimization and its Applications, 38 (2000). Google Scholar

[17]

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/s10957-008-9378-2. Google Scholar

[18]

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. Google Scholar

[19]

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

[20]

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. Google Scholar

[21]

K. Kimura and J. C. Yao, Semicontinuity of solution mappings of parametric generalized vector equilibrium problems,, J. Optim. Theory Appl., 138 (2008), 429. doi: 10.1007/s10957-008-9386-2. Google Scholar

[22]

K. Kimura and J.-C. Yao, Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems,, J. Ind. Manag. Optim., 4 (2008), 167. Google Scholar

[23]

I. V. Konnov and S. Schaible, Duality for equilibrium problems under generalized monotonicity,, J. Optim. Theory Appl., 104 (2000), 395. doi: 10.1023/A:1004665830923. Google Scholar

[24]

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), 283. doi: 10.1023/A:1014830925232. Google Scholar

[25]

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. Google Scholar

[26]

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. Google Scholar

[27]

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

[28]

S. J. Li, X .B. Li and K. L. Teo, The Hölder continuity of solutions to generalized vector equilibrium problems,, European J. Oper. Res., 199 (2009), 334. doi: 10.1016/j.ejor.2008.12.024. Google Scholar

[29]

S. J. Li, C. R. Chen, X. B. Li and K. L. Teo, Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems,, European J. Oper. Res., 210 (2011), 148. doi: 10.1016/j.ejor.2010.10.005. Google Scholar

[30]

P. H. Sach, L. A. Tuan and G. M. Lee, Sensitivity results for a general class of generalized vector quasi-equilibrium problems with set-valued maps,, Nonlinear Anal., 71 (2009), 571. doi: 10.1016/j.na.2008.10.098. Google Scholar

[31]

A. Shapiro, Perturbation analysis of optimization problems in Banach spaces,, Numer. Funct. Anal. Optim., 13 (1992), 97. doi: 10.1080/01630569208816463. Google Scholar

[32]

M. M. Wong, Lower semicontinuity of the solution map to a parametric vector variational inequality,, J. Glob. Optim., 46 (2010), 435. doi: 10.1007/s10898-009-9447-6. 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. 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. doi: 10.1007/s10957-007-9250-9. Google Scholar

[3]

L. Q. Anh and P. Q. Khanh, On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems,, J. Math. Anal. Appl., 321 (2006), 308. doi: 10.1016/j.jmaa.2005.08.018. Google Scholar

[4]

L. Q. Anh and P. Q. Khanh, Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces,, J. Glob. Optim., 37 (2007), 449. doi: 10.1007/s10898-006-9062-8. Google Scholar

[5]

L. Q. Anh and P. Q. Khanh, Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions,, J. Glob. Optim., 42 (2008), 515. doi: 10.1007/s10898-007-9268-4. Google Scholar

[6]

L. Q. Anh and P. Q. Khanh, Sensitivity anlysis for weak and strong vector quasiequilibrium problems,, Vietnam J. Math., 37 (2009), 237. Google Scholar

[7]

Q. H. Ansari, A. H. Siddiqi and S. Y. Wu, Existence and duality of generalized vector equilibrium problems,, J. Math. Anal. Appl., 259 (2001), 115. doi: 10.1006/jmaa.2000.7397. Google Scholar

[8]

E. M. Bednarczuk, Strong pseudomonotonicity, sharp efficiency and stability for parametric vector equilibria,, in, 17 (2007), 9. doi: 10.1051/proc:071702. Google Scholar

[9]

E. M. Bednarczuk, Weak sharp efficiency and growth condition for vector-valued functions with applications,, Optimization, 53 (2004), 455. doi: 10.1080/02331930412331330478. Google Scholar

[10]

M. Bianchi and R. Pini, Sensitivity for parametric vector equilibria,, Optimization, 55 (2006), 221. doi: 10.1080/02331930600662732. Google Scholar

[11]

J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems,", Springer Series in Operations Research, (2000). Google Scholar

[12]

G.-Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization: Set-Valued and Variational Analysis,", Lecture Notes in Economics and Mathematical Systems, 541 (2005). Google Scholar

[13]

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. Google Scholar

[14]

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. Google Scholar

[15]

C. R. Chen, S. J. Li, J. Zeng and X. B. Li, Error analysis of approximate solutions to parametric vector quasiequilibrium problems,, Optim. Lett., 5 (2011), 85. Google Scholar

[16]

F. Giannessi, ed., "Vector Variational Inequalities and Vector Equilibria: Mathematical Theories,", Nonconvex Optimization and its Applications, 38 (2000). Google Scholar

[17]

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/s10957-008-9378-2. Google Scholar

[18]

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. Google Scholar

[19]

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

[20]

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. Google Scholar

[21]

K. Kimura and J. C. Yao, Semicontinuity of solution mappings of parametric generalized vector equilibrium problems,, J. Optim. Theory Appl., 138 (2008), 429. doi: 10.1007/s10957-008-9386-2. Google Scholar

[22]

K. Kimura and J.-C. Yao, Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems,, J. Ind. Manag. Optim., 4 (2008), 167. Google Scholar

[23]

I. V. Konnov and S. Schaible, Duality for equilibrium problems under generalized monotonicity,, J. Optim. Theory Appl., 104 (2000), 395. doi: 10.1023/A:1004665830923. Google Scholar

[24]

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), 283. doi: 10.1023/A:1014830925232. Google Scholar

[25]

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. Google Scholar

[26]

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. Google Scholar

[27]

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

[28]

S. J. Li, X .B. Li and K. L. Teo, The Hölder continuity of solutions to generalized vector equilibrium problems,, European J. Oper. Res., 199 (2009), 334. doi: 10.1016/j.ejor.2008.12.024. Google Scholar

[29]

S. J. Li, C. R. Chen, X. B. Li and K. L. Teo, Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems,, European J. Oper. Res., 210 (2011), 148. doi: 10.1016/j.ejor.2010.10.005. Google Scholar

[30]

P. H. Sach, L. A. Tuan and G. M. Lee, Sensitivity results for a general class of generalized vector quasi-equilibrium problems with set-valued maps,, Nonlinear Anal., 71 (2009), 571. doi: 10.1016/j.na.2008.10.098. Google Scholar

[31]

A. Shapiro, Perturbation analysis of optimization problems in Banach spaces,, Numer. Funct. Anal. Optim., 13 (1992), 97. doi: 10.1080/01630569208816463. Google Scholar

[32]

M. M. Wong, Lower semicontinuity of the solution map to a parametric vector variational inequality,, J. Glob. Optim., 46 (2010), 435. doi: 10.1007/s10898-009-9447-6. Google Scholar

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