参数广义弱向量拟平衡问题解映射的H-连续性刻画
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摘要
研究了Hausdorff拓扑向量空间中的一类参数广义弱向量拟平衡问题(PGWVQEP)的稳定性.首先,给出了此问题的参数间隙函数,研究了参数间隙函数的连续性.然后,提出了一个与参数间隙函数相关的关键假设,讨论了它的连续性,并给出关键假设的等价刻画.最后,借助于假设,获得了PGWVQEP解映射Hausdorff半连续的充分必要条件.并举例验证了所得结果.
The stability of a class of parametric generalized weak vector quasi-equilibrium problems(PGWVQEP) in Hausdorff topological vector spaces, were studied. First, a parametric gap function for the problem was given, and the continuity property of the function was studied. Next, a key hypothesis related to the gap function for the considered problem was presented, the characterizations of this hypothesis were discussed, and an equivalence theorem for the key hypothesis was given. Finally, by means of the hypothesis, the sufficient and necessary conditions for the Hausdorff semicontinuity of the solution mapping to PGWVQEP were obtained. Examples were given to verify the obtained results.
The stability of a class of parametric generalized weak vector quasi-equilibrium problems(PGWVQEP) in Hausdorff topological vector spaces, were studied. First, a parametric gap function for the problem was given, and the continuity property of the function was studied. Next, a key hypothesis related to the gap function for the considered problem was presented, the characterizations of this hypothesis were discussed, and an equivalence theorem for the key hypothesis was given. Finally, by means of the hypothesis, the sufficient and necessary conditions for the Hausdorff semicontinuity of the solution mapping to PGWVQEP were obtained. Examples were given to verify the obtained results.
引文
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[14] LI S J, CHEN C R. Stability of weak vector variational inequality[J]. Nonlinear Analysis: Theory, Methods & Applications, 2009, 70(4): 1528-1535.
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[17] ZHONG R Y, HUANG N J. Lower semicontinuity for parametric weak vector variational inequalities in reflexive Banach spaces[J]. Journal of Optimization Theory and Applications, 2011, 149(3): 564-579.
[18] ANH L Q, HUNG N V. Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems[J]. Journal of Industrial & Management Optimization, 2018, 14(1): 65-79.
[19] 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.
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[2] BIANCHI M, HADJISAVVAS N, SCHAIBLE S. Vector equilibrium problems with generalized monotone bifunctions[J]. Journal of Optimization Theory and Applications, 1997, 92(3): 527-542.
[3] ANSARI Q H, OETTLI W, SCHLAGER D. A generalization of vectorial equilibria[J]. Mathematical Methods of Operations Research, 1997, 46(2): 147-152.
[4] LONG X J, HUANG N J, TEO K L. Existence and stability of solutions for generalized strong vector quasi-equilibrium problem[J]. Mathematical and Computer Modelling, 2008, 47(3/4): 445-451.
[5] GONG X H. Continuity of the solution set to parametric weak vector equilibrium problems[J]. Journal of Optimization Theory and Applications, 2008, 139(1): 35-46.
[6] GONG X H, YAO J C. Lower semicontinuity of the set of efficient solutions for generalized systems[J]. Journal of Optimization Theory and Applications, 2008, 138(2): 197-205.
[7] ANH L Q, KHANH P Q. Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks ii: lower semicontinuities applications[J]. Set-Valued Analysis, 2008, 16(7/8): 943-960.
[8] ANH L Q, KHANH P Q. Continuity of solution maps of parametric quasiequilibrium problems[J]. Journal of Global Optimization, 2010, 46(2): 247-259.
[9] KIMURA K, YAO J C. Semicontinuity of solution mappings of parametric generalized vector equilibrium problems[J]. Journal of Optimization Theory and Applications, 2008, 138(3): 429-443.
[10] KIMURA K, YAO J C. Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems[J]. Journal of Global Optimization, 2008, 41(2): 187-202.
[11] 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))
[12] 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(1): 55-69.
[13] PENG Z Y, WANG X F, YANG X M. Connectedness of approximate efficient solutions for generalized semi-infinite vector optimization problems[J]. Set-Valued and Variational Analysis, 2019, 27(1): 103-118.
[14] LI S J, CHEN C R. Stability of weak vector variational inequality[J]. Nonlinear Analysis: Theory, Methods & Applications, 2009, 70(4): 1528-1535.
[15] CHEN C R, LI S J. Semicontinuity of the solution set map to a set-valued weak vector variational inequality[J]. Journal of Industrial and Management Optimization, 2007, 3(3): 519-528.
[16] CHEN C R, LI S J, FANG Z M. On the solution semicontinuity to a parametric generalized vector quasi- variational inequality[J]. Computers & Mathematics With Applications, 2010, 60(8): 2417-2425.
[17] ZHONG R Y, HUANG N J. Lower semicontinuity for parametric weak vector variational inequalities in reflexive Banach spaces[J]. Journal of Optimization Theory and Applications, 2011, 149(3): 564-579.
[18] ANH L Q, HUNG N V. Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems[J]. Journal of Industrial & Management Optimization, 2018, 14(1): 65-79.
[19] 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.
[20] AUBIN J P, EKELAND I. Applied Nonlinear Analysis[M]. New York: John Wiley and Sons, 1984.
[21] BERGE C. Topological Spaces[M]. London: Oliver and Boyd, 1963.
[22] Gerstewitz C. Nichtkonvexe dualitat in der vektaroptimierung[J]. Wissenschafliche Zeitschift der Technischen Hochschule Leuna-Mensehung, 1983, 25: 357-364.
[23] LUC D T. Theory of Vector Optimization, Lecture Notes in Economic and Mathematical Systems[M]. Berlin: Springer-Verlag, 1989.