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March  2020, 16(2): 511-529. doi: 10.3934/jimo.2018165

Strong vector equilibrium problems with LSC approximate solution mappings

Nguyen Ba Minh 1, and  Pham Huu Sach 2,,

1. 

Thuongmai University, Hanoi, Vietnam
2. 

Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, 10307 Hanoi, Vietnam

* Corresponding author: Pham Huu Sach

Received  December 2017 Revised  June 2018 Published  March 2020 Early access  October 2018

Fund Project: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2016.11

This paper introduces two classes of parametric strong vector equilibrium problems whose approximate solution mappings are lower semicontinuous. In the first class, the objective set-valued maps satisfy some cone-convexity/cone-concavity assumptions, and in the second one, they satisfy some strongly proper cone-quasiconvexconcavity assumptions. All these mentioned concepts of generalized cone-convexity/cone-concavity/ strongly proper cone-quasiconvexconcavity are new and different from the traditional ones. Some upper semicontinuity/continuity results are also obtained. Applications to parametric weak u-set and l-set optimization problems and weak vector multivalued equilibrium problems are given.

Keywords: Vector equilibrium problem, set optimization problem, approximate solution, semicontinuity, set-valued map.
Mathematics Subject Classification: Primary: 49K40, 90C29, 90C31.
Citation: Nguyen Ba Minh, Pham Huu Sach. Strong vector equilibrium problems with LSC approximate solution mappings. Journal of Industrial & Management Optimization, 2020, 16 (2) : 511-529. doi: 10.3934/jimo.2018165
References:
[1]

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

[2]

Q.H. Ansari, E. Kobis and J.-C. Yao, Vector Variational Inequalities and Vector Optimization: Theory and Applications, Springer, Berlin, 2018. doi: 10.1007/978-3-319-63049-6.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

F. Ferro, A minimax theorem for vector valued functions, part 2, J. Optim. Theory Appl., 68 (1991), 35-48.  doi: 10.1007/BF00939934.  Google Scholar

[7]

Chr. Gerth and P. Weidner, Nonconvex separation theorems and some applications in vector optimization, J. Optim. Theory Appl., 67 (1990), 297-320.  doi: 10.1007/BF00940478.  Google Scholar

[8]

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

[9]

E. Hernandez and L. Rodriguez-Marin, Nonconvex scalarization in set optimization with set-valued maps, J. Math. Anal. Appl., 325 (2007), 1-18.  doi: 10.1016/j.jmaa.2006.01.033.  Google Scholar

[10]

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

[11]

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

[12]

D. Kuroiwa, On set-valued optimization, Nonlinear Anal., 47 (2001), 1395-1400.  doi: 10.1016/S0362-546X(01)00274-7.  Google Scholar

[13]

S. J. Li and Z. M. Fang, Lower semicontinuity of the solution mappings to parametric generalized Ky Fan inequality, J. Optim. Theory Appl., 147 (2010), 507-515.  doi: 10.1007/s10957-010-9736-8.  Google Scholar

[14]

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

[15]

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

[16]

D. T. Luc, Theory of Vector Optimization, Springer, Berlin, 1989. doi: 10.1007/978-3-642-50280-4.  Google Scholar

[17]

Z. Y. Peng, X. M. Yang and J. W. Peng, On the lower semicontinuity of the solution mappings to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 152 (2012), 256-264.  doi: 10.1007/s10957-011-9883-6.  Google Scholar

[18]

Z. Y. Peng, Y. Zhao and X. Q. Yang, Semicontinuity of approximate solution mappings to parametric set-valued weak vector equilibrium problems, Numer. Funct. Anal. Optim., 36 (2015), 481-500.  doi: 10.1080/01630563.2015.1013551.  Google Scholar

[19]

P. H. Sach, Stability property in bifunction-set optimization, J. Optim. Theory Appl., 177 (2018), 376-398.  doi: 10.1007/s10957-018-1280-y.  Google Scholar

[20]

P. H. Sach and N. B. Minh, Continuity of solution mappings in some non-weak vector Ky Fan inequalities, J. Glob. Optim., 57 (2013), 1401-1418.  doi: 10.1007/s10898-012-0015-0.  Google Scholar

[21]

P. H. Sach and L. A. Tuan, New scalarizing approach to the stability analysis in parametric generalized Ky Fan inequality problems, J. Optim. Theory Appl., 157 (2013), 347-364.  doi: 10.1007/s10957-012-0105-7.  Google Scholar

[22]

Y. D. Xu and S. J. Li, Continuity of the solution mappings to parametric generalized non-weak vector Ky Fan inequalities, J. Ind. Manag. Optim., 13 (2017), 967-975.  doi: 10.3934/jimo.2016056.  Google Scholar

show all references

References:
[1]

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

[2]

Q.H. Ansari, E. Kobis and J.-C. Yao, Vector Variational Inequalities and Vector Optimization: Theory and Applications, Springer, Berlin, 2018. doi: 10.1007/978-3-319-63049-6.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

F. Ferro, A minimax theorem for vector valued functions, part 2, J. Optim. Theory Appl., 68 (1991), 35-48.  doi: 10.1007/BF00939934.  Google Scholar

[7]

Chr. Gerth and P. Weidner, Nonconvex separation theorems and some applications in vector optimization, J. Optim. Theory Appl., 67 (1990), 297-320.  doi: 10.1007/BF00940478.  Google Scholar

[8]

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

[9]

E. Hernandez and L. Rodriguez-Marin, Nonconvex scalarization in set optimization with set-valued maps, J. Math. Anal. Appl., 325 (2007), 1-18.  doi: 10.1016/j.jmaa.2006.01.033.  Google Scholar

[10]

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

[11]

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

[12]

D. Kuroiwa, On set-valued optimization, Nonlinear Anal., 47 (2001), 1395-1400.  doi: 10.1016/S0362-546X(01)00274-7.  Google Scholar

[13]

S. J. Li and Z. M. Fang, Lower semicontinuity of the solution mappings to parametric generalized Ky Fan inequality, J. Optim. Theory Appl., 147 (2010), 507-515.  doi: 10.1007/s10957-010-9736-8.  Google Scholar

[14]

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

[15]

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

[16]

D. T. Luc, Theory of Vector Optimization, Springer, Berlin, 1989. doi: 10.1007/978-3-642-50280-4.  Google Scholar

[17]

Z. Y. Peng, X. M. Yang and J. W. Peng, On the lower semicontinuity of the solution mappings to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 152 (2012), 256-264.  doi: 10.1007/s10957-011-9883-6.  Google Scholar

[18]

Z. Y. Peng, Y. Zhao and X. Q. Yang, Semicontinuity of approximate solution mappings to parametric set-valued weak vector equilibrium problems, Numer. Funct. Anal. Optim., 36 (2015), 481-500.  doi: 10.1080/01630563.2015.1013551.  Google Scholar

[19]

P. H. Sach, Stability property in bifunction-set optimization, J. Optim. Theory Appl., 177 (2018), 376-398.  doi: 10.1007/s10957-018-1280-y.  Google Scholar

[20]

P. H. Sach and N. B. Minh, Continuity of solution mappings in some non-weak vector Ky Fan inequalities, J. Glob. Optim., 57 (2013), 1401-1418.  doi: 10.1007/s10898-012-0015-0.  Google Scholar

[21]

P. H. Sach and L. A. Tuan, New scalarizing approach to the stability analysis in parametric generalized Ky Fan inequality problems, J. Optim. Theory Appl., 157 (2013), 347-364.  doi: 10.1007/s10957-012-0105-7.  Google Scholar

[22]

Y. D. Xu and S. J. Li, Continuity of the solution mappings to parametric generalized non-weak vector Ky Fan inequalities, J. Ind. Manag. Optim., 13 (2017), 967-975.  doi: 10.3934/jimo.2016056.  Google Scholar

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