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乐观型二层随机规划逼近问题最优解集的上半收敛性

周婉娜1, 霍永亮2, 吴凡3   

  1. 1. 西安翻译学院, 西安 710105;
    2. 重庆文理学院数学与大数据学院, 重庆 402160;
    3. 重庆文理学院, 重庆 402160
  • 收稿日期:2020-06-17 修回日期:2022-04-22 发布日期:2022-08-31
  • 通讯作者: 霍永亮,Email:yongliang-huo@126.com.
  • 基金资助:
    陕西省科技厅自然科学基础研究项目(2022JQ-712),陕西省教育厅专项项目(20JK0641)资助课题.

周婉娜, 霍永亮, 吴凡. 乐观型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, 2022, 42(7): 1805-1819.

ZHOU Wanna, HUO Yongliang, WU Fan. The Upper Semi-Convergence of Optimal Solution Set of Approximation Problem for Optimistic Bi-Level Stochastic Programming[J]. Journal of Systems Science and Mathematical Sciences, 2022, 42(7): 1805-1819.

The Upper Semi-Convergence of Optimal Solution Set of Approximation Problem for Optimistic Bi-Level Stochastic Programming

ZHOU Wanna1, HUO Yongliang2, WU Fan3   

  1. 1. Xi'an Fanyi University, Xi'an 710105;
    2. College of Mathematics and Big Data, Chongqing University of Arts and Sciences, Chongqing 402160;
    3. Chongqing University of Arts and Sciences, Chongqing 402160
  • Received:2020-06-17 Revised:2022-04-22 Published:2022-08-31
文章针对下层随机规划反馈的最优解不唯一,上层为单目标约束随机规划的一类乐观型二层随机规划逼近问题,构建了求解乐观型二层随机规划逼近最优解集上半收敛的理论框架.首先将乐观型二层随机规划等价转化为单层随机规划问题,通过逼近方法建立了无界可积函数在有限区域上以及全空间上的一致逼近定理,应用此结果给出了目标函数的连续收敛性和约束集的K-收敛性.其次利用上图收敛理论,得到了乐观型二层随机规划逼近最优解集的上半收敛性.该结论提供了乐观型二层随机规划逼近最优解集可以近似替代精确的最优解集的理论依据,结果表明离散化逼近方法是可行的、有效的、合理的.
Our goal is to construct a theoretical framework of the upper semiconvergence of the approximately optimal solution set for the optimistic bi-level stochastic programming problem, where the optimal solution set of the lower level stochastic programming is not singleton and the upper level is single objective constrained stochastic programming. Firstly, the optimistic bi-level stochastic programming is equivalently transformed into single-level programming, and the uniform approximation theorems of unbounded integrable function in finite region and full space are established by approximation method, respectively. By the uniform approximation theorems, the continuous convergence of the objective function and the k-convergence of the constraint set are given. Secondly, the upper semi-convergence of approximately optimal solution set for the optimistic bi-level model is obtained by using epi-convergence theory. This conclusion provides a theoretical basis that the approximately optimal solution set of optimistic bi-level stochastic programming can approximate the accurate optimal solution set. The results show that the discrete approximation method is feasible, effective and reasonable.

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[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] 霍永亮;刘三阳. 非线性参数规划问题$\varepsilon$-最优解集集值映射的连续性[J]. 系统科学与数学, 2009, 29(6): 735-741.
[3] 霍永亮;刘三阳. 概率约束规划逼近最优解集的稳定性和最优值的连续性[J]. 系统科学与数学, 2007, 27(6): 908-914.
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