• Previous Article
    Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps
  • JIMO Home
  • This Issue
  • Next Article
    Neural network smoothing approximation method for stochastic variational inequality problems
April  2015, 11(2): 661-671. doi: 10.3934/jimo.2015.11.661

Stability of solution mapping for parametric symmetric vector equilibrium problems

1. 

College of Sciences, Chongqing Jiaotong University, Chongqing, 400074, China

2. 

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067

Received  October 2013 Revised  May 2014 Published  September 2014

This paper is concerned with the stability for a parametric symmetric vector equilibrium problem. A parametric gap function for the parametric symmetric vector equilibrium problem is introduced and investigated. By virtue of this function, we establish the sufficient and necessary conditions for the Hausdorff lower semicontinuity of solution mapping to a parametric symmetric vector equilibrium problem. The results presented in this paper generalize and improve the corresponding results in the recent literature.
Citation: Xiao-Bing Li, Xian-Jun Long, Zhi Lin. Stability of solution mapping for parametric symmetric vector equilibrium problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 661-671. doi: 10.3934/jimo.2015.11.661
References:
[1]

Q. H. Ansari, I. V. Konnov and J. C. Yao, Existence of a solution and variational principles for vector equilibrium problems,, J. Optim. Theory Appl., 110 (2001), 481. doi: 10.1023/A:1017581009670. Google Scholar

[2]

Q. H. Ansari, I. V. Konnov and J. C. Yao, Characterizations for vector equilibrium problems,, J. Optim. Theory Appl., 113 (2002), 435. doi: 10.1023/A:1015366419163. Google Scholar

[3]

J. P. Aubin and H. Frankowska, Set-Valued Analysis,, Systems & Control: Foundations & Applications, 2 (1990). Google Scholar

[4]

B. Bank, J. Guddat, D. Klattle, B. Kummer and K. Tammar, Non-Linear Parametric Optimization,, Akademie-Verlag, (1982). doi: 10.1007/978-3-0348-6328-5. Google Scholar

[5]

C. Berge, Topological Spaces., Oliver and Boyd, (1963). Google Scholar

[6]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems,, Math. Stud., 63 (1994), 123. Google Scholar

[7]

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

[8]

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

[9]

C. R. Chen, S. J. Li and Z. M. Fang, On the solution semicontinuity to a parametric generalize vector quasivariational inequality,, Comput. Math. Appl., 60 (2010), 2417. doi: 10.1016/j.camwa.2010.08.036. Google Scholar

[10]

G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-valued and Variational Anyasis,, in: Lecture Notes in Econonics and Mathematical Systems, (2005). Google Scholar

[11]

G. Y. Chen, X. Q. Yang and H. Yu, A nonlinear scalarization function and generalized quai-vector equilibrium problem,, J. Global Optim., 32 (2005), 451. doi: 10.1007/s10898-003-2683-2. Google Scholar

[12]

J. C. Chen and X. H. Gong, The stability of set of solutions for symmetric quasi-equilibrium problems,, J. Optim. Theory Appl., 136 (2008), 359. doi: 10.1007/s10957-007-9309-7. Google Scholar

[13]

A. P. Farajzadeh, On the symmetric vector quasi-equilibrium problems,, J. Math. Anal. Appl., 322 (2006), 1099. doi: 10.1016/j.jmaa.2005.09.079. Google Scholar

[14]

J. Y. Fu, Symmetric vector quasi-equilibrium problems,, J. Math. Anal. Appl., 285 (2003), 708. doi: 10.1016/S0022-247X(03)00479-7. Google Scholar

[15]

F. Giannessi, Theorem of the alternative, quadratic programs, and comlementarity problems,, Variational Inequalities and Complementarity, (1980), 151. Google Scholar

[16]

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

[17]

N. J. Huang, J. Li and H. B.Thompson, Implicit vector equilibrium problems with applications,, Math. Comput. Modelling., 37 (2003), 1343. doi: 10.1016/S0895-7177(03)90045-8. Google Scholar

[18]

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. doi: 10.1007/s10957-007-9190-4. Google Scholar

[19]

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

[20]

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

[21]

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

[22]

M. A. Noor and W. Oettli, On general nonlinear complementary problems and quasi-equilibria,, Le Matematiche, 49 (1994), 313. Google Scholar

[23]

W. Y. Zhang, Well-posedness for convex symmetric vector quasi-equilibrium problems,, J. Math. Anal. Appl., 387 (2012), 909. doi: 10.1016/j.jmaa.2011.09.052. Google Scholar

[24]

J. Zhao, The lower semicontinuity of optimal solution sets,, J. Math. Anal. Appl., 207 (1997), 240. doi: 10.1006/jmaa.1997.5288. Google Scholar

[25]

R. Y. Zhong and N. J. Huang, On the stability of solution mapping for parametric generalized vector quasiequilibrium problems,, Comput. Math. Appl., 63 (2012), 807. doi: 10.1016/j.camwa.2011.11.046. Google Scholar

[26]

R. Y. Zhong and N. J. Huang, Lower semicontinuity for parametric weak vector variational inequalities in Reflexive Banach Spaces,, J. Optim. Theory Appl., 150 (2011), 317. doi: 10.1007/s10957-011-9843-1. Google Scholar

[27]

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

show all references

References:
[1]

Q. H. Ansari, I. V. Konnov and J. C. Yao, Existence of a solution and variational principles for vector equilibrium problems,, J. Optim. Theory Appl., 110 (2001), 481. doi: 10.1023/A:1017581009670. Google Scholar

[2]

Q. H. Ansari, I. V. Konnov and J. C. Yao, Characterizations for vector equilibrium problems,, J. Optim. Theory Appl., 113 (2002), 435. doi: 10.1023/A:1015366419163. Google Scholar

[3]

J. P. Aubin and H. Frankowska, Set-Valued Analysis,, Systems & Control: Foundations & Applications, 2 (1990). Google Scholar

[4]

B. Bank, J. Guddat, D. Klattle, B. Kummer and K. Tammar, Non-Linear Parametric Optimization,, Akademie-Verlag, (1982). doi: 10.1007/978-3-0348-6328-5. Google Scholar

[5]

C. Berge, Topological Spaces., Oliver and Boyd, (1963). Google Scholar

[6]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems,, Math. Stud., 63 (1994), 123. Google Scholar

[7]

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

[8]

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

[9]

C. R. Chen, S. J. Li and Z. M. Fang, On the solution semicontinuity to a parametric generalize vector quasivariational inequality,, Comput. Math. Appl., 60 (2010), 2417. doi: 10.1016/j.camwa.2010.08.036. Google Scholar

[10]

G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-valued and Variational Anyasis,, in: Lecture Notes in Econonics and Mathematical Systems, (2005). Google Scholar

[11]

G. Y. Chen, X. Q. Yang and H. Yu, A nonlinear scalarization function and generalized quai-vector equilibrium problem,, J. Global Optim., 32 (2005), 451. doi: 10.1007/s10898-003-2683-2. Google Scholar

[12]

J. C. Chen and X. H. Gong, The stability of set of solutions for symmetric quasi-equilibrium problems,, J. Optim. Theory Appl., 136 (2008), 359. doi: 10.1007/s10957-007-9309-7. Google Scholar

[13]

A. P. Farajzadeh, On the symmetric vector quasi-equilibrium problems,, J. Math. Anal. Appl., 322 (2006), 1099. doi: 10.1016/j.jmaa.2005.09.079. Google Scholar

[14]

J. Y. Fu, Symmetric vector quasi-equilibrium problems,, J. Math. Anal. Appl., 285 (2003), 708. doi: 10.1016/S0022-247X(03)00479-7. Google Scholar

[15]

F. Giannessi, Theorem of the alternative, quadratic programs, and comlementarity problems,, Variational Inequalities and Complementarity, (1980), 151. Google Scholar

[16]

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

[17]

N. J. Huang, J. Li and H. B.Thompson, Implicit vector equilibrium problems with applications,, Math. Comput. Modelling., 37 (2003), 1343. doi: 10.1016/S0895-7177(03)90045-8. Google Scholar

[18]

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. doi: 10.1007/s10957-007-9190-4. Google Scholar

[19]

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

[20]

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

[21]

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

[22]

M. A. Noor and W. Oettli, On general nonlinear complementary problems and quasi-equilibria,, Le Matematiche, 49 (1994), 313. Google Scholar

[23]

W. Y. Zhang, Well-posedness for convex symmetric vector quasi-equilibrium problems,, J. Math. Anal. Appl., 387 (2012), 909. doi: 10.1016/j.jmaa.2011.09.052. Google Scholar

[24]

J. Zhao, The lower semicontinuity of optimal solution sets,, J. Math. Anal. Appl., 207 (1997), 240. doi: 10.1006/jmaa.1997.5288. Google Scholar

[25]

R. Y. Zhong and N. J. Huang, On the stability of solution mapping for parametric generalized vector quasiequilibrium problems,, Comput. Math. Appl., 63 (2012), 807. doi: 10.1016/j.camwa.2011.11.046. Google Scholar

[26]

R. Y. Zhong and N. J. Huang, Lower semicontinuity for parametric weak vector variational inequalities in Reflexive Banach Spaces,, J. Optim. Theory Appl., 150 (2011), 317. doi: 10.1007/s10957-011-9843-1. Google Scholar

[27]

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

[1]

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

[2]

Kenji Kimura, Jen-Chih Yao. Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems. Journal of Industrial & Management Optimization, 2008, 4 (1) : 167-181. doi: 10.3934/jimo.2008.4.167

[3]

Xin Zuo, Chun-Rong Chen, Hong-Zhi Wei. Solution continuity of parametric generalized vector equilibrium problems with strictly pseudomonotone mappings. Journal of Industrial & Management Optimization, 2017, 13 (1) : 477-488. doi: 10.3934/jimo.2016027

[4]

Lam Quoc Anh, Nguyen Van Hung. Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems. Journal of Industrial & Management Optimization, 2018, 14 (1) : 65-79. doi: 10.3934/jimo.2017037

[5]

Lam Quoc Anh, Pham Thanh Duoc, Tran Ngoc Tam. Continuity of approximate solution maps to vector equilibrium problems. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1685-1699. doi: 10.3934/jimo.2017013

[6]

Qilin Wang, Shengji Li. Semicontinuity of approximate solution mappings to generalized vector equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1303-1309. doi: 10.3934/jimo.2016.12.1303

[7]

Yangdong Xu, Shengjie Li. Continuity of the solution mappings to parametric generalized non-weak vector Ky Fan inequalities. Journal of Industrial & Management Optimization, 2017, 13 (2) : 967-975. doi: 10.3934/jimo.2016056

[8]

Vítor Araújo. Semicontinuity of entropy, existence of equilibrium states and continuity of physical measures. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 371-386. doi: 10.3934/dcds.2007.17.371

[9]

Kenji Kimura, Yeong-Cheng Liou, Soon-Yi Wu, Jen-Chih Yao. Well-posedness for parametric vector equilibrium problems with applications. Journal of Industrial & Management Optimization, 2008, 4 (2) : 313-327. doi: 10.3934/jimo.2008.4.313

[10]

Qiusheng Qiu, Xinmin Yang. Scalarization of approximate solution for vector equilibrium problems. Journal of Industrial & Management Optimization, 2013, 9 (1) : 143-151. doi: 10.3934/jimo.2013.9.143

[11]

Zaiyun Peng, Xinmin Yang, Kok Lay Teo. On the Hölder continuity of approximate solution mappings to parametric weak generalized Ky Fan Inequality. Journal of Industrial & Management Optimization, 2015, 11 (2) : 549-562. doi: 10.3934/jimo.2015.11.549

[12]

Nguyen Huy Chieu, Jen-Chih Yao. Subgradients of the optimal value function in a parametric discrete optimal control problem. Journal of Industrial & Management Optimization, 2010, 6 (2) : 401-410. doi: 10.3934/jimo.2010.6.401

[13]

Yunan Wu, Guangya Chen, T. C. Edwin Cheng. A vector network equilibrium problem with a unilateral constraint. Journal of Industrial & Management Optimization, 2010, 6 (3) : 453-464. doi: 10.3934/jimo.2010.6.453

[14]

Liping Zhang, Soon-Yi Wu, Shu-Cherng Fang. Convergence and error bound of a D-gap function based Newton-type algorithm for equilibrium problems. Journal of Industrial & Management Optimization, 2010, 6 (2) : 333-346. doi: 10.3934/jimo.2010.6.333

[15]

Xinmin Yang. On symmetric and self duality in vector optimization problem. Journal of Industrial & Management Optimization, 2011, 7 (3) : 523-529. doi: 10.3934/jimo.2011.7.523

[16]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519

[17]

Nguyen Ba Minh, Pham Huu Sach. Strong vector equilibrium problems with LSC approximate solution mappings. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2018165

[18]

Chunrong Chen, Zhimiao Fang. A note on semicontinuity to a parametric generalized Ky Fan inequality. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 779-784. doi: 10.3934/naco.2012.2.779

[19]

Jiawei Chen, Guangmin Wang, Xiaoqing Ou, Wenyan Zhang. Continuity of solutions mappings of parametric set optimization problems. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2018138

[20]

Alexandre Caboussat, Roland Glowinski. Numerical solution of a variational problem arising in stress analysis: The vector case. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1447-1472. doi: 10.3934/dcds.2010.27.1447

2018 Impact Factor: 1.025

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]