Berge Lower Semi-Continuity of Parametric Generalized Vector Quasi-Equilibrium Problems Under Improvement Set Mappings

SHAO Chongyang; PENG Zaiyun; LIU Fuping; WANG Jingjing

doi:10.21656/1000-0887.400307

Volume 41 Issue 8

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SHAO Chongyang, PENG Zaiyun, LIU Fuping, WANG Jingjing. Berge Lower Semi-Continuity of Parametric Generalized Vector Quasi-Equilibrium Problems Under Improvement Set Mappings[J]. Applied Mathematics and Mechanics, 2020, 41(8): 912-920. doi: 10.21656/1000-0887.400307

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Berge Lower Semi-Continuity of Parametric Generalized Vector Quasi-Equilibrium Problems Under Improvement Set Mappings

doi: 10.21656/1000-0887.400307

¹College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, P.R.China;²School of Mathematical Sciences, Chongqing Normal University, Chongqing 400074, P.R.China

Funds: The National Natural Science Foundation of China(11301571)

Received Date: 2019-10-14
Rev Recd Date: 2019-12-17
Publish Date: 2020-08-01

Abstract

Abstract

The Berge lower semi-continuity of solution mapping for a new class of parametric generalized vector quasi-equilibrium problems was discussed. Firstly, the improvement set mapping was defined, based on which the order structure was generalized and applied to the study of vector quasi-equilibrium problems, to lead to parametric generalized vector quasi-equilibrium problems under improvement set mappings (IPGVQEP). Then, a nonlinear scalarization function Ψ associated with the improvement set mapping was introduced, the scalar problem (IPGVQEP)_Ψ corresponding to the above problem (IPGVQEP) was given, and the relation between solution sets of (IPGVQEP) and (IPGVQEP)_Ψ was obtained. Finally, by virtue of a key hypothesis H_Ψ and the relation between solution sets, the sufficient and necessary conditions for Berge lower semi-continuity of the solution mapping for (IPGVQEP) were established, and an example was given to verify the results.
- improvement set mapping,
- parametric generalized vector quasi-equilibrium problem,
- solution mapping,
- Berge lower semi-continuity,
- scalarization problem

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References(20)

References

[1]	BLUM E, OETTLI W. From optimization and variational inequalities to equilibrium problems[J]. The Mathmatics Student,1994,63: 123-145.
[2]	ANSARI Q H, OETTLI W, SCHLAGER D. A generalization of vectorial equilibria[J]. Mathematical Methods of Operations Research,1997,46(2): 147-152.
[3]	BIANCHI M, HADJISAVVAS N, SCHAIBLE S. Vector equilibrium problems with generalized monotone bifunctions[J]. Journal of Optimization Theory and Applications,1997,92: 527-542.
[4]	CHEN C R, LI S J, FANG Z M. On the solution semicontinuity to a parametric generalized vector quasivariational inequality[J]. Computers & Mathematics With Applications,2010,60(8): 2417-2425.
[5]	曾静, 彭再云, 张石生. 广义强向量拟平衡问题解的存在性和Hadamard适定性[J]. 应用数学和力学, 2015,36(6): 651-658.(ZENG Jing, PENG Zaiyun, ZHANG Shisheng. Existence and Hadamard well-posedness of solutions to generalized strong vector quasi-equilibrium problems[J]. Applied Mathematics and Mechanics,2015,36(6): 651-658.(in Chinese))
[6]	彭建文, 杨新民, 朱道立. 向量拟平衡问题系统及其应用[J]. 应用数学和力学, 2006,27(8): 963-970.(PENG Jianwen, YANG Xinmin, ZHU Daoli. System of vector quasi-equilibrium problems and its applications[J]. Applied Mathematics and Mechanics,2006,27(8): 963-970.(in Chinese))
[7]	GONG X H. Continuity of the solution set to parametric weak vector equilibrium problems[J]. Journal of Optimization Theory and Applications,2008,139: 35-46.
[8]	PENG Z Y, YANG X M. Semicontinuity of the solution mappings to weak generalized parametric Ky Fan inequality problems with trifunctions[J]. Optimization,2014,63(1): 447-457.
[9]	PENG Z Y, ZHAO Y, YANG X M. Semicontinuity of approximate solution mappings toparametric set-valued weak vector equilibrium problems[J]. Numerical Functional Analysis and Optimization,2015,36: 481-500.
[10]	PENG Z Y, PENG J W, LONG X J, et al. On the stability of solutions for semi-infinite vector optimization problems[J]. Journal of Global Optimization,2018,70: 55-69.
[11]	ZHAO J. The lower semicontinuity of optimal solution sets[J]. Journal of Mathematical Analysis and Applications,1997,207(1): 240-254.
[12]	ZHONG R Y, HUANG N J. On the stability of solution mapping for parametric generalized vector quasiequilibrium problems[J]. Computers & Mathematics With Applications,2012,63(4): 807-815.
[13]	ANH L Q, HUNG N V. Gap functions and Hausdorff continuity of solution mapping to parametric strong vector quasiequilibrium problem[J]. Journal of Industrial and Management Optimization,2018,14(1): 65-79.
[14]	邵重阳, 彭再云, 王泾晶, 等. 参数广义弱向量拟平衡问题解映射的H-连续性刻画[J]. 应用数学和力学, 2019,40(4): 452-462.(SHAO Chongyang, PENG Zaiyun, WANG Jingjing, et al. Characterizations of H-continuity for solution mapping to parametric generalized weak vector quasi-equilibrium problems[J]. Applied Mathematics and Mechanics,2019,40(4): 452-462.(in Chinese))
[15]	ANH L Q, TRAN Q D, DINH V H. Stability for parametric vector quasi-equilibrium problems with variable cones[J]. Numerical Functional Analysis and Optimization,2019,40(4): 461-483.
[16]	CHEN C R, ZHOU X, LI F, et al. Vector equilibrium problems under improvement sets and linear scalarization with stability applications[J]. Optimization Methods and Software,2016,31(6): 1240-1257.
[17]	WANG J J, PENG Z Y, LIN Z, et al. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set[J]. Journal of Industrial and Management Optimization,2019,40(4): 461-483.
[18]	CHICCO M, MIGNANEGO F, PUSILLO L, et al. Vector optimization problem via improvement sets[J]. Journal of Optimization Theory and Applications,2011,150: 516-529.
[19]	AUBIN J P, EKELAND I. Applied Nonlinear Analysis [M]. New York: John Wiley and Sons, 1984.
[20]	GPFERT A, RIAHI H, TAMMER C, et al. Variational Methods in Partially Ordered Spaces [M]. Berlin: Springer, 2003.