• Previous Article
    Optimal ordering policy for a two-warehouse inventory model use of two-level trade credit
  • JIMO Home
  • This Issue
  • Next Article
    Artificial intelligence combined with nonlinear optimization techniques and their application for yield curve optimization
October  2017, 13(4): 1685-1699. doi: 10.3934/jimo.2017013

Continuity of approximate solution maps to vector equilibrium problems

1. 

Department of Mathematics, Teacher College, Can Tho University, Can Tho, 900000, Viet Nam

2. 

Department of Mathematics, Vo Truong Toan University, Hau Giang, Viet Nam

3. 

Department of Mathematics, Nam Can Tho University, Can Tho, 900000, Viet Nam

* Corresponding author

Received  October 2015 Revised  October 2016 Published  December 2016

Fund Project: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.44

This paper considers the parametric primal and dual vector equilibrium problems in locally convex Hausdorff topological vector spaces. Based on linear scalarization technique, we establish sufficient conditions for the continuity of approximate solution maps to these problems. As applications, some new results for vector optimization problem and vector variational inequality are derived. Our results are new and improve the existing ones in the literature.

Citation: 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
References:
[1]

M. Ait Mansour and H. Riahi, Sensitivity analysis for abstract equilibrium problems, J. Math. Anal. Appl., 306 (2005), 684-691. doi: 10.1016/j.jmaa.2004.10.011. Google Scholar

[2]

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

[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, Various kinds of semicontinuity and solution sets of parametric multivalued symmetric vector quasiequilibrium problems, J. Glob. Optim., 41 (2008), 539-558. doi: 10.1007/s10898-007-9264-8. Google Scholar

[5]

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

[6]

L. Q. AnhP. Q. Khanh and T. N. Tam, On Hölder continuity of approximate solutions to parametric equilibrium problems, Nonlinear Anal., 75 (2012), 2293-2303. doi: 10.1016/j.na.2011.10.029. Google Scholar

[7]

L. Q. AnhP. Q. Khanh and T. N. Tam, Hausdorff continuity of approximate solution maps to parametric primal and dual equilibrium problems, TOP, 24 (2016), 242-258. doi: 10.1007/s11750-015-0390-z. Google Scholar

[8]

Q. H. AnsariI. V. Konnov and J. C. Yao, Existence of a solution and variational principles for vector equilibrium problems, J. Optim. Theory Appl., 110 (2001), 481-492. doi: 10.1023/A:1017581009670. Google Scholar

[9]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990. Google Scholar

[10]

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

[11]

M. Bianchi and S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl., 90 (1996), 31-43. doi: 10.1007/BF02192244. Google Scholar

[12]

M. BianchiN. Hadjisavas and S. Schaible, Equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl., 92 (1997), 527-542. doi: 10.1023/A:1022603406244. Google Scholar

[13]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student., 63 (1994), 123-145. Google Scholar

[14]

C. R. ChenS. L. 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

[15]

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

[16]

K. Fan, A minimax inequality and applications, In: Shisha O (ed) Inequality III Academic Press, New York, (1972), 103–113. Google Scholar

[17] A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, London, 1983.
[18]

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

[19]

N. X. HaiP. Q. Khanh and N. H. Quan, On the existence of solutions to quasivariational inclusion problems, J. Glob. Optim., 45 (2009), 565-581. doi: 10.1007/s10898-008-9390-y. Google Scholar

[20]

J. Jahn, Mathematical Vector Optimization in Partially Ordered Linear Spaces, Peter Lang, Frankfurt, 1986. Google Scholar

[21]

P. Q. Khanh and V. S. T. Long, Invariant-point theorems and existence of solutions to optimization-related problems, J. Global. Optim., 58 (2014), 545-564. doi: 10.1007/s10898-013-0065-y. Google Scholar

[22]

W. K. KimS. Kum and K. H. Lee, Semicontinuity of the solution multifunctions of the parametric generalized operator equilibrium problems, Nonlinear Anal., 71 (2009), 2182-2187. doi: 10.1016/j.na.2009.04.036. Google Scholar

[23]

K. Kimura and J. C. Yao, Semicontinuity of solution mappings of parametric generalized vector equilibrium problems, J. Optim. Theory Appl., 138 (2008), 429-443. doi: 10.1007/s10957-008-9386-2. Google Scholar

[24]

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

[25]

S. J. LiH. M. Liu and C. R. Chen, Lower semicontinuity of parametric generalized weak vector equilibrium problems, Bull. Aust. Math. Soc., 81 (2010), 85-95. doi: 10.1017/S0004972709000628. Google Scholar

[26]

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

[27]

H. Nikaido and K. Isoda, Note on non-copperative convex games, Pacific J. Math., 5 (1955), 807-815. doi: 10.2140/pjm.1955.5.807. Google Scholar

show all references

References:
[1]

M. Ait Mansour and H. Riahi, Sensitivity analysis for abstract equilibrium problems, J. Math. Anal. Appl., 306 (2005), 684-691. doi: 10.1016/j.jmaa.2004.10.011. Google Scholar

[2]

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

[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, Various kinds of semicontinuity and solution sets of parametric multivalued symmetric vector quasiequilibrium problems, J. Glob. Optim., 41 (2008), 539-558. doi: 10.1007/s10898-007-9264-8. Google Scholar

[5]

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

[6]

L. Q. AnhP. Q. Khanh and T. N. Tam, On Hölder continuity of approximate solutions to parametric equilibrium problems, Nonlinear Anal., 75 (2012), 2293-2303. doi: 10.1016/j.na.2011.10.029. Google Scholar

[7]

L. Q. AnhP. Q. Khanh and T. N. Tam, Hausdorff continuity of approximate solution maps to parametric primal and dual equilibrium problems, TOP, 24 (2016), 242-258. doi: 10.1007/s11750-015-0390-z. Google Scholar

[8]

Q. H. AnsariI. V. Konnov and J. C. Yao, Existence of a solution and variational principles for vector equilibrium problems, J. Optim. Theory Appl., 110 (2001), 481-492. doi: 10.1023/A:1017581009670. Google Scholar

[9]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990. Google Scholar

[10]

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

[11]

M. Bianchi and S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl., 90 (1996), 31-43. doi: 10.1007/BF02192244. Google Scholar

[12]

M. BianchiN. Hadjisavas and S. Schaible, Equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl., 92 (1997), 527-542. doi: 10.1023/A:1022603406244. Google Scholar

[13]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student., 63 (1994), 123-145. Google Scholar

[14]

C. R. ChenS. L. 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

[15]

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

[16]

K. Fan, A minimax inequality and applications, In: Shisha O (ed) Inequality III Academic Press, New York, (1972), 103–113. Google Scholar

[17] A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, London, 1983.
[18]

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

[19]

N. X. HaiP. Q. Khanh and N. H. Quan, On the existence of solutions to quasivariational inclusion problems, J. Glob. Optim., 45 (2009), 565-581. doi: 10.1007/s10898-008-9390-y. Google Scholar

[20]

J. Jahn, Mathematical Vector Optimization in Partially Ordered Linear Spaces, Peter Lang, Frankfurt, 1986. Google Scholar

[21]

P. Q. Khanh and V. S. T. Long, Invariant-point theorems and existence of solutions to optimization-related problems, J. Global. Optim., 58 (2014), 545-564. doi: 10.1007/s10898-013-0065-y. Google Scholar

[22]

W. K. KimS. Kum and K. H. Lee, Semicontinuity of the solution multifunctions of the parametric generalized operator equilibrium problems, Nonlinear Anal., 71 (2009), 2182-2187. doi: 10.1016/j.na.2009.04.036. Google Scholar

[23]

K. Kimura and J. C. Yao, Semicontinuity of solution mappings of parametric generalized vector equilibrium problems, J. Optim. Theory Appl., 138 (2008), 429-443. doi: 10.1007/s10957-008-9386-2. Google Scholar

[24]

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

[25]

S. J. LiH. M. Liu and C. R. Chen, Lower semicontinuity of parametric generalized weak vector equilibrium problems, Bull. Aust. Math. Soc., 81 (2010), 85-95. doi: 10.1017/S0004972709000628. Google Scholar

[26]

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

[27]

H. Nikaido and K. Isoda, Note on non-copperative convex games, Pacific J. Math., 5 (1955), 807-815. doi: 10.2140/pjm.1955.5.807. Google Scholar

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

Tran Ninh Hoa, Ta Duy Phuong, Nguyen Dong Yen. Linear fractional vector optimization problems with many components in the solution sets. Journal of Industrial & Management Optimization, 2005, 1 (4) : 477-486. doi: 10.3934/jimo.2005.1.477

[7]

Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483

[8]

Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial & Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673

[9]

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

[10]

O. Chadli, Z. Chbani, H. Riahi. Recession methods for equilibrium problems and applications to variational and hemivariational inequalities. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 185-196. doi: 10.3934/dcds.1999.5.185

[11]

Fengmin Wang, Dachuan Xu, Donglei Du, Chenchen Wu. Primal-dual approximation algorithms for submodular cost set cover problems with linear/submodular penalties. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 91-100. doi: 10.3934/naco.2015.5.91

[12]

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

[13]

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

[14]

Jen-Yen Lin, Hui-Ju Chen, Ruey-Lin Sheu. Augmented Lagrange primal-dual approach for generalized fractional programming problems. Journal of Industrial & Management Optimization, 2013, 9 (4) : 723-741. doi: 10.3934/jimo.2013.9.723

[15]

Kenji Kimura, Yeong-Cheng Liou, David S. Shyu, Jen-Chih Yao. Simultaneous system of vector equilibrium problems. Journal of Industrial & Management Optimization, 2009, 5 (1) : 161-174. doi: 10.3934/jimo.2009.5.161

[16]

Lili Li, Chunrong Chen. Nonlinear scalarization with applications to Hölder continuity of approximate solutions. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 295-307. doi: 10.3934/naco.2014.4.295

[17]

Rong Hu, Ya-Ping Fang, Nan-Jing Huang. Levitin-Polyak well-posedness for variational inequalities and for optimization problems with variational inequality constraints. Journal of Industrial & Management Optimization, 2010, 6 (3) : 465-481. doi: 10.3934/jimo.2010.6.465

[18]

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

[19]

Siqi Li, Weiyi Qian. Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 37-46. doi: 10.3934/naco.2015.5.37

[20]

Hong-Zhi Wei, Xin Zuo, Chun-Rong Chen. Unified vector quasiequilibrium problems via improvement sets and nonlinear scalarization with stability analysis. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019036

2018 Impact Factor: 1.025

Metrics

  • PDF downloads (38)
  • HTML views (211)
  • Cited by (0)

Other articles
by authors

[Back to Top]