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运筹学学报 >

2016 , Vol. 20 >Issue 1: 75 - 83

DOI: https://doi.org/10.15960/j.cnki.issn.1007-6093.2016.01.007

运筹学

极大极小随机规划逼近问题最优解集和最优值的稳定性

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  • 1. 重庆文理学院数学研究所，重庆市群与图的理论及其应用重点实验室，重庆永川 402160

收稿日期: 2014-12-29

  网络出版日期: 2016-03-15

基金资助

重庆高校创新团队建设计划项目(No. KJTD201321), 中国博士后科学基金资助项目(No. 2015M57016), 重庆市教委科学技术研究项目(No. KJ1500334)

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Stability of optimal solution set and optimal value for minimax stochastic programming approximation problems

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  • 1. Institute of Mathematics, Chongqing University of Arts and Sciences，Chongqing Key Lab. of Group & Graph Theories and Applications，Yongchuan  Chongqing  402160，China

Received date: 2014-12-29

  Online published: 2016-03-15

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摘要

研究了特殊的二层极大极小随机规划逼近收敛问题. 首先将下层初始随机规划最优解集拓展到非单点集情形, 且可行集正则的条件下, 讨论了下层随机规划逼近问题最优解集关于上层决策变量参数的上半收敛性和最优值函数的连续性. 然后把下层随机规划的epsilon-最优解向量函数反馈到上层随机规划的目标函数中, 得到了上层随机规划逼近问题的最优解集关于最小信息概率度量收敛的上半收敛性和最优值的连续性.

关键词： 极大极小随机规划; 正则条件; 最小信息概率度量; 最优解集; 上半收敛性

本文引用格式

霍永亮 . 极大极小随机规划逼近问题最优解集和最优值的稳定性[J]. 运筹学学报, 2016 , 20(1) : 75 -83 . DOI: 10.15960/j.cnki.issn.1007-6093.2016.01.007

Abstract

In this paper, we research convergence of minimax approximation problems  of special class of bilevel stochastic programming. First, under regularity conditions of feasible set, we expand optimal solution set of lower level original stochastic programming to into non-singleton set. And we give continuity of optimal value and upper semi-convergence of the optimal solution  set on the upper level decision variables for lower level stochastic programming approximation problem. Furthermore, we feedback $\varepsilon$-optimal solution vector function provided by the lower level stochastic programming into the objective function of the upper level stochastic programming problems, and obtain the continuity of optimal value and the upper semi-convergence of optimal solution set with respect to the minimal information (m.i.) probability metric for upper level programming.

Key words： minimax stochastic programming; regularity condition; minimal information (m.i.) probability metric; optimal solution set, upper semi-convergence

参考文献

[1] 霍永亮. 二层随机规划逼近问题最优解集的上半收敛性 [J]. 系统科学与数学, 2014, 34(6): 674-681.
[2] 王广民, 万仲平, 王先甲．二(双)层规划综述 [J]. 数学进展, 2007, 36(5): 513-524.
[3] 郑跃, 万仲平, 吕一兵. 非线性二层规划问题的全局优化方法 [J]. 系统科学与数学, 2012, 32(5): 513-521.
[4] Svetlozar T, Rachev, Romisch W. Quantitative stability in stochastic programming: the method of probability metrics [J]. Mathematics of Operations Research, 2002, 27(4): 792-818.
[5] Romisch W. Stability of stochastic programming problems [M]//Stochastic Programming, Amsterdam： Elsevier Science, 2003: 483-554.
[6] 霍永亮, 刘三阳. 随机规划逼近问题最优解集的下半收敛性 [J]. 数学进展, 2012, 41(6): 747-754.
[7] Pennanen T. Epi-convergent discretizations of multistage stochastic programs via integration quadratures [J].  Mathematical Programming, 2009, 116: 461-479.
[8] Shapiro A, Ahmed S. On a class of minimax stochastic programs [J]. SIAM Journal on Optimization,  2004, 14(4): 1237-1249.
[9] Riis M, Andersen K A. Applying the minimax criterion in stochastic recourse programs [J]. European Journal of Operational Research, 2005, 165(3): 569-584.
[10] Shapiro A. Asymptotics of minimax stochastic programs [J]. Statistics and Probability Letters, 2008, 78(2): 150-157.
[11] Dupacova J.  Uncertainties in minimax stochastic programs [J]. Optimization, 2011, 60(10): 1235-1250.
[12] 霍永亮, 刘三阳. 可测函数序列关于弱收敛概率测度序列积分的极限定理 [J]. 系统科学与数学, 2008, 28(6): 709-717.
[13] Rockafellar R T, Wets R J-B. Variational Analysis [M]. Berlin: Springer-Verlag, 1998.
[14] Koivu M. Variance reduction in sample approximations of stochastic programs [J]. Mathematical Programming, 2005, 103: 463-485.
[15] Pennanen T, Koivu M. Epi-convergent discretizations of stochastic programs via integration quadratures [J]. Numerische Mathematik, 2005, 100(2): 141-163.
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