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March  2021, 17(2): 869-887. doi: 10.3934/jimo.2020002

On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set

Jingjing Wang , Zaiyun Peng , , Zhi Lin and  Daqiong Zhou

College of Mathematics and Statistics, Chongqing JiaoTong University, Chongqing 400074, China

* Corresponding author: Zaiyun Peng

Received  June 2018 Revised  March 2019 Published  January 2020

Fund Project: The first and the fourth authors are supported by the Graduate Innovation Foundation of Chongqing Jiaotong University (2019S0123). The second author is supported by the National Natural Science Foundation of China (11301571), the Basic and Advanced Research Project of Chongqing (cstc2018jcyjAX0337), the Program for University Innovation Team of Chongqing (CXTDX201601022), the Innovation Project for Returned Overseas Scholars in Chongqing (cx2019148), the open project funded by the Chongqing Key Lab on ORSE (CSSXKFKTZ201801) and the Education Committee Project Foundation of Bayu Scholar. The third author is supported by the National Natural Science Foundation of China (11271389)

In this paper, we mainly discuss the stability of generalized vector quasi-equilibrium problems (GVQEPs) where the ordering relations are defined by free-disposal set. Firstly, by virtue of the oriented distance function $ (\triangle) $, gap functions for (GVQEPs) are given and some properties of them are studied. Then, under some types of continuity assumption, the sufficient conditions of the upper semicontinuity and the upper Painlevé-Kuratowski convergence of solutions for (GVQEPs) are talked about. Moreover, sufficient and necessary conditions of the lower semicontinuity and the lower Painlevé-Kuratowski convergence of solutions for (GVQEPs) are obtained in normed linear spaces. Some examples are given to illustrate the results, and our results are new and extend some known results in the literature.

Keywords: Generalized vector quasi-equilibrium problems, free-disposal set, the oriented distance function, semicontinuity, Painlevé-Kuratowski convergence.
Mathematics Subject Classification: 49K40, 90C31, 91B50.
Citation: Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002
References:
[1]

L. Q. Anh and N. V. Hung, Gap function and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems, Journal of Industrial and Management Optimization, 14 (2018), 65-79.  doi: 10.3934/jimo.2017037.  Google Scholar

[2]

L. Q. Anh, T. Bantaojai and N. V. Hung, Painlevé-Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems, Computational and Applied Mathematics, 37 (2018), 3832-3845.  doi: 10.1007/s40314-017-0548-4.  Google Scholar

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

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C. R. Chen, X. Zuo, F. Lu and S. J. Li, Vector equilibrium problems under improvement sets and linear scalarization with stability applications, Optimization Methods and Software, 31 (2016), 1240-1257.  doi: 10.1080/10556788.2016.1200043.  Google Scholar

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J. C. Chen and X. H. Gong, The stability of set of solutions for symmetric vector quasi-equilibrium problems, Journal of Optimization Theory and Applications, 136 (2008), 359-374.  doi: 10.1007/s10957-007-9309-7.  Google Scholar

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X. H. Gong, Lower semicontinuity of the efficient solution mapping in semi-infinite vector optimization, Journal of Systems Science and Complexity, 28 (2015), 1312-1325.  doi: 10.1007/s11424-015-3058-8.  Google Scholar

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S. H. Hou, X. H. Gong and X. M. Yang, Existence and stability of solutions for generalized Ky Fan inequality problems with trifunctions, Journal of Optimization Theory and Applications, 146 (2010), 387-398.  doi: 10.1007/s10957-010-9656-7.  Google Scholar

[11]

C. S. Lalitha and P. Chatterjee, Stability and scalarization in vector optimization using improvement sets, Journal of Optimization Theory and Applications, 166 (2015), 825-843.  doi: 10.1007/s10957-014-0686-4.  Google Scholar

[12]

S. J. Li and C. R. Chen, Stability of weak vector variational inequality, Nonlinear Analysis Theory Methods Applications, 70 (2009), 1528-1535.  doi: 10.1016/j.na.2008.02.032.  Google Scholar

[13]

X. J. Long, N. J. Huang and K. L. Teo, Existence and stability of solutions for generalized strong vector quasi-equilibrium problem, Mathematical and Computer Modelling, 47 (2008), 445-451.  doi: 10.1016/j.mcm.2007.04.013.  Google Scholar

[14]

J. W. Peng, S. Y. Wu and Y. Wang, Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems with functional constraints, Journal of Global Optimization, 52 (2012), 779-795.  doi: 10.1007/s10898-011-9711-4.  Google Scholar

[15]

J. W. Peng and S. Y. Wu, The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems, Optimization Letters, 4 (2010), 501-512.  doi: 10.1007/s11590-010-0179-9.  Google Scholar

[16]

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, Journal of Optimization Theory and Applications, 152 (2012), 256-264.  doi: 10.1007/s10957-011-9883-6.  Google Scholar

[17]

Z. Y. Peng, J. W. Peng, X. J. Long and J. C. Yao, On the stability of solutions for semi-infinite vector optimization problems, Journal of Global Optimization, 70 (2018), 55-69.  doi: 10.1007/s10898-017-0553-6.  Google Scholar

[18]

R. T. Rockafellar and R. J. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[19]

W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Analysis, 70 (2009), 45-57.  doi: 10.1016/j.na.2007.11.031.  Google Scholar

[20]

H. Yang and J. Yu, Essential components of the set of weakly Pareto-Nash equilibrium points, Applied Mathematics Letters, 15 (2002), 553-560.  doi: 10.1016/S0893-9659(02)80006-4.  Google Scholar

[21]

J. Yu, Essential weak efficient solution in multiobjective optimization problems, Journal of Mathematical Analysis and Applications, 166 (1992), 230-235.  doi: 10.1016/0022-247X(92)90338-E.  Google Scholar

[22]

J. Yu, Z. X. Liu, D. T. Peng, D. Y. Xu and Y. H. Zhou, Existence and stability analysis of optimal control, Optimal Control Applications and Methods, 35 (2014), 721-729.  doi: 10.1002/oca.2096.  Google Scholar

[23]

A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM Journal on Control and Optimization, 42 (2003), 1071-1086.  doi: 10.1137/S0363012902411532.  Google Scholar

[24]

J. Zeng, Z. Y. Peng, X. K. Sun and X. J. Long, Existence of solutions and Hadamard well-posedness for generalized strong vector quasi-equilibrium problems, Journal of Nonlinear Science and Applications, 9 (2016), 4104-4113.   Google Scholar

[25]

Y. Zhao, Z. Y. Peng and X. M. Yang, Semicontinuity and convergence for vector optimization problems with approximate equilibrium constraints, Optimization, 65 (2016), 1397-1415.  doi: 10.1080/02331934.2016.1149711.  Google Scholar

[26]

R. Y. Zhong and N. J. Huang, Lower semicontinuity for parametric weak vector variational inequalities in reflexive banach spaces, Journal of Optimization Theory and Applications, 150 (2011), 317-326.  doi: 10.1007/s10957-011-9843-1.  Google Scholar

[27]

R. Y. Zhong and N. J. Huang, On the stability of solution mapping for parametric generalized vector quasiequilibrium problems, Computers and Mathematics with Applications, 63 (2012), 807-815.  doi: 10.1016/j.camwa.2011.11.046.  Google Scholar

[28]

Y. H. Zhong, J. Yu and S. W. Xiang, Essential stability in games with infinitely many 1pure strategies, International Journal of Game Theory, 35 (2007), 493-503.  doi: 10.1007/s00182-006-0063-0.  Google Scholar

show all references

References:
[1]

L. Q. Anh and N. V. Hung, Gap function and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems, Journal of Industrial and Management Optimization, 14 (2018), 65-79.  doi: 10.3934/jimo.2017037.  Google Scholar

[2]

L. Q. Anh, T. Bantaojai and N. V. Hung, Painlevé-Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems, Computational and Applied Mathematics, 37 (2018), 3832-3845.  doi: 10.1007/s40314-017-0548-4.  Google Scholar

[3]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984.  Google Scholar

[4]

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

[5]

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

[6]

C. R. Chen, X. Zuo, F. Lu and S. J. Li, Vector equilibrium problems under improvement sets and linear scalarization with stability applications, Optimization Methods and Software, 31 (2016), 1240-1257.  doi: 10.1080/10556788.2016.1200043.  Google Scholar

[7]

J. C. Chen and X. H. Gong, The stability of set of solutions for symmetric vector quasi-equilibrium problems, Journal of Optimization Theory and Applications, 136 (2008), 359-374.  doi: 10.1007/s10957-007-9309-7.  Google Scholar

[8]

X. H. Gong, Lower semicontinuity of the efficient solution mapping in semi-infinite vector optimization, Journal of Systems Science and Complexity, 28 (2015), 1312-1325.  doi: 10.1007/s11424-015-3058-8.  Google Scholar

[9]

J. B. Hiriart-Urruty, New concepts in nondifferentiable programming, Mémoires de la Société Mathématique de France, 60 (1979), 57-85.   Google Scholar

[10]

S. H. Hou, X. H. Gong and X. M. Yang, Existence and stability of solutions for generalized Ky Fan inequality problems with trifunctions, Journal of Optimization Theory and Applications, 146 (2010), 387-398.  doi: 10.1007/s10957-010-9656-7.  Google Scholar

[11]

C. S. Lalitha and P. Chatterjee, Stability and scalarization in vector optimization using improvement sets, Journal of Optimization Theory and Applications, 166 (2015), 825-843.  doi: 10.1007/s10957-014-0686-4.  Google Scholar

[12]

S. J. Li and C. R. Chen, Stability of weak vector variational inequality, Nonlinear Analysis Theory Methods Applications, 70 (2009), 1528-1535.  doi: 10.1016/j.na.2008.02.032.  Google Scholar

[13]

X. J. Long, N. J. Huang and K. L. Teo, Existence and stability of solutions for generalized strong vector quasi-equilibrium problem, Mathematical and Computer Modelling, 47 (2008), 445-451.  doi: 10.1016/j.mcm.2007.04.013.  Google Scholar

[14]

J. W. Peng, S. Y. Wu and Y. Wang, Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems with functional constraints, Journal of Global Optimization, 52 (2012), 779-795.  doi: 10.1007/s10898-011-9711-4.  Google Scholar

[15]

J. W. Peng and S. Y. Wu, The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems, Optimization Letters, 4 (2010), 501-512.  doi: 10.1007/s11590-010-0179-9.  Google Scholar

[16]

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, Journal of Optimization Theory and Applications, 152 (2012), 256-264.  doi: 10.1007/s10957-011-9883-6.  Google Scholar

[17]

Z. Y. Peng, J. W. Peng, X. J. Long and J. C. Yao, On the stability of solutions for semi-infinite vector optimization problems, Journal of Global Optimization, 70 (2018), 55-69.  doi: 10.1007/s10898-017-0553-6.  Google Scholar

[18]

R. T. Rockafellar and R. J. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[19]

W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Analysis, 70 (2009), 45-57.  doi: 10.1016/j.na.2007.11.031.  Google Scholar

[20]

H. Yang and J. Yu, Essential components of the set of weakly Pareto-Nash equilibrium points, Applied Mathematics Letters, 15 (2002), 553-560.  doi: 10.1016/S0893-9659(02)80006-4.  Google Scholar

[21]

J. Yu, Essential weak efficient solution in multiobjective optimization problems, Journal of Mathematical Analysis and Applications, 166 (1992), 230-235.  doi: 10.1016/0022-247X(92)90338-E.  Google Scholar

[22]

J. Yu, Z. X. Liu, D. T. Peng, D. Y. Xu and Y. H. Zhou, Existence and stability analysis of optimal control, Optimal Control Applications and Methods, 35 (2014), 721-729.  doi: 10.1002/oca.2096.  Google Scholar

[23]

A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM Journal on Control and Optimization, 42 (2003), 1071-1086.  doi: 10.1137/S0363012902411532.  Google Scholar

[24]

J. Zeng, Z. Y. Peng, X. K. Sun and X. J. Long, Existence of solutions and Hadamard well-posedness for generalized strong vector quasi-equilibrium problems, Journal of Nonlinear Science and Applications, 9 (2016), 4104-4113.   Google Scholar

[25]

Y. Zhao, Z. Y. Peng and X. M. Yang, Semicontinuity and convergence for vector optimization problems with approximate equilibrium constraints, Optimization, 65 (2016), 1397-1415.  doi: 10.1080/02331934.2016.1149711.  Google Scholar

[26]

R. Y. Zhong and N. J. Huang, Lower semicontinuity for parametric weak vector variational inequalities in reflexive banach spaces, Journal of Optimization Theory and Applications, 150 (2011), 317-326.  doi: 10.1007/s10957-011-9843-1.  Google Scholar

[27]

R. Y. Zhong and N. J. Huang, On the stability of solution mapping for parametric generalized vector quasiequilibrium problems, Computers and Mathematics with Applications, 63 (2012), 807-815.  doi: 10.1016/j.camwa.2011.11.046.  Google Scholar

[28]

Y. H. Zhong, J. Yu and S. W. Xiang, Essential stability in games with infinitely many 1pure strategies, International Journal of Game Theory, 35 (2007), 493-503.  doi: 10.1007/s00182-006-0063-0.  Google Scholar

Figure 1.  the improvement set 1
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Figure 2.  the improvement set 2
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Figure 3.  the improvement set 3
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