Stability of Vector Optimization Problems Under Approximate Equilibrium Constraints via Free-Disposal Sets
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摘要: 在自由支配集下,对一类近似平衡约束向量优化问题(AOPVF)的稳定性进行研究.首先,在较弱的凸性假设下获得了约束集映射的Berge-半连续性和约束集的闭性、凸性和紧性结果.然后,在目标函数列Gamma-收敛的假设下,分别得到了AOPVF弱有效解映射Berge半连续和弱有效解集下Painlevé-Kuratowski收敛的充分条件,并给出例子说明结论是新颖和有意义的.
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关键词:
- 自由支配集 /
- Gamma-收敛 /
- Berge-半连续 /
- 下Painlevé-Kuratowski收敛
Abstract: The stability of vector optimization problems under approximate equilibrium constraints (AOPVF) via free-disposal sets was discussed. Firstly, the Berge-semicontinuity of the constraint set mapping and the closedness, the convexity and the compactness of the constraint set were obtained with the weaker convexity assumption. Moreover, under the assumption of Gamma-convergence for the objective functional sequences, the lower Painlevé-Kuratowski convergence of the weak efficient solution set and the Berge-semicontinuity of weak efficient solution mappings for AOPVF were obtained respectively. Some examples illustrate that the results are new and meaningful. -
LUO Z Q, PANG J S, RALPH D.Mathematical Programs With Equilibrium Constraints[M]. Cambridge: Cambridge University Press, 1996. [2]HUANG X X, YANG X Q, ZHU D L. Levitin-Polyak well-posedness of variational inequality problems with functional constraints[J].Journal of Global Optimization,2009,44(2): 159-174. [3]LIGNOLA M B, MORGAN J.α-well-posedness for Nash equilibria and for optimization problems with Nash equilibrium constraints[J].Journal of Global Optimization,2006,36(3): 439-459. [4]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. [5]PENG Z Y, YANG X M. Painlevé-Kuratowski convergences of the solution sets for perturbed vector equilibrium problems without monotonicity[J].Acta Mathematicae Applicatae Sinica(English Series),2014,30(4): 845-858. [6]PENG Z Y, WANG Z Y, YANG X M. Connectedness of solution sets for weak generalized symmetric Ky Fan inequality problems via addition-invariant sets[J].Journal of Optimization Theory and Applications,2020,185(1): 188-206. [7]MISHRA S K, JAISWAL M, LE THI H A. Nonsmooth semi-infinite programming problem using limiting subdifferentials[J].Journal of Global Optimization,2012,53(2): 285-296. [8]CHEN G Y, CRAVEN B D. Existence and continuity of solutions for vector optimization[J].Journal of Optimization Theory and Applications,1994,81(3): 459-468. [9]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. [10]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)) [11]CHUONG T D, HUY N Q, YAO J C. Stability of semi-infinite vector optimization problems under functional perturbations[J].Journal of Global Optimization,2009,45(4): 583-595. [12]FAN X, CHENG C, WANG H. Stability of semi-infinite vector optimization problems without compact constraints[J].Nonlinear Analysis: Theory, Methods & Applications,2011,74(6): 2087-2093. [13]ZHAO Y, PENG Z Y, YANG X M. Painlevé-Kuratowski convergences of the solution sets for perturbed generalized systems[J].Journal of Nonlinear and Convex Analysis,2014,15(6): 1249-1259. [14]彭再云, 熊勤, 王泾晶, 等. 近似平衡约束向量优化问题解集的上Painlevé-Kuratowski收敛性[J]. 系统科学与数学, 2018,38(8): 960-970.(PENG Zaiyun, XIONG Qin, WANG Jingjing, et al. On upper Painlevé-Kuratowski convergence of the solutions set to vector optimization problems under approximate equilibrium constraints[J].Journal of Systems Science and Mathematical Sciences,2018,38(8): 960-970.(in Chinese)) [15]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. [16]邵重阳, 彭再云, 刘芙萍, 等. 改进集映射下参数广义向量拟平衡问题解映射的Berge下半连续性[J]. 应用数学和力学, 2020,41(8): 912-920.(SHAO Chongyang, PENG Zaiyun, LIU Fuping, et al. 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.(in Chinese)) [17]WANG J J, SHAO C Y, PENG Z Y. Stability and scalarization for perturbed set-valued optimization problems with constraints via general ordering sets[J].Pacific Journal of Optimization,2019,15(4): 529-549. [18]LUC D T.Theory of Vector Optimization[M]. Berlin: Springer-Verlag, 1989. [19]AUBIN J P, EKELAND I.Applied Nonlinear Analysis[M]. New York: John Wiley and Sons, 1984. [20]BERGE C.Topological Spaces[M]. London: Oliver and Boyd, 1963. [21]ROCKAFELLAR R T, WETS R J B.Variational Analysis[M]. Berlin: Springer Science & Business Media, 2009. [22]OPPEZZI P, ROSSI A M. A convergence for vector valued functions[J].Optimization,2008,57(3): 435-448.
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