两阶段全随机优化问题的定量统计鲁棒性

doi:10.15960/j.cnki.issn.1007-6093.2026.03.011

运筹学学报（中英文） ›› 2026, Vol. 30 ›› Issue (3): 183-203.doi: 10.15960/j.cnki.issn.1007-6093.2026.03.011

所属专题：运筹与管理前沿：模型、算法及其应用专辑

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两阶段全随机优化问题的定量统计鲁棒性

张锦立¹, 韦思瑶^1†, 童小娇², 孙海琳¹

1. 南京师范大学数学科学学院, 江苏南京 210023;
2. 湖南第一师范学院数学与统计学院, 湖南长沙 410205

收稿日期:2024-11-14 发布日期:2026-09-12
通讯作者: 韦思瑶 E-mail:weisiyao2025@163.com
基金资助:
国家重点研发计划(No.2023YFA1009300),国家自然科学基金(Nos.12331011,12171145)

Quantitative statistical robustness for full random two-stage stochastic optimization problems

ZHANG Jinli¹, WEI Siyao^1†, TONG Xiaojiao², SUN Hailin¹

1. School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, Jiangsu, China;
2. School of Mathematics and Statistics, Hunan First Normal University, Changsha 410205, Hunan, China

Received:2024-11-14 Published:2026-09-12

摘要/Abstract

摘要： 两阶段随机优化模型广泛应用于投资组合、交通运输管理等现代运营问题。在两阶段随机优化中常用样本均值逼近方法。已有的研究一般假设样本是由真实分布产生的,然而实际情况得到的样本有可能受到污染,此时需要知道两阶段随机优化问题的最优值是否可靠。针对第二阶段目标函数是线性函数、分片线性函数和一类凸函数的两阶段全随机优化问题,本文研究了其样本均值逼近问题最优值的定量统计鲁棒性。当目标函数是线性函数时,在全随机优化问题稳定性分析的基础上,研究了其样本均值逼近问题最优值的定量统计鲁棒性。当目标函数是分片线性函数时,通过证明对偶可行解的一致上界,推导出了第二阶段最优值函数的一致Lipschitz连续性,从而得到两阶段全随机优化问题的定量统计鲁棒性结果。进一步推广该研究方法到目标函数是一类凸函数的两阶段全随机优化问题,分析样本均值逼近问题最优值的定量统计鲁棒性。

关键词: 全随机, 统计鲁棒性, 两阶段随机优化, 样本均值逼近

Abstract: Two-stage stochastic optimization models are widely used in modern operational problems such as investment portfolios and transportation management. In two-stage stochastic optimization, the sample average approximation method is commonly used. The existing research generally assumes that the sample is generated by the real distribution, but the actual situation of the sample may be contaminated, then we need to know whether the optimal value of the two-stage stochastic optimization problem is reliable. This paper investigates the quantitative statistical robustness of the sample average approximation of the optimal value for full random two-stage stochastic optimization problems where the second-stage objective function is a linear function, a piecewise linear function, or a class of convex functions. When the objective function is a linear function, based on the stability analysis of the full random stochastic optimization problem, the quantitative statistical robustness of the sample average approximation to the optimal value is studied. When the objective function is a piecewise linear function, the uniform Lipschitz continuity of the optimal value function in the second stage is derived by proving a uniform bound on feasible dual solutions, thus obtaining the quantitative statistical robustness results of the full random two-stage stochastic optimization problem. The method is extended to a class of full random two-stage stochastic optimization problems where the objective function is a convex function, and the quantitative statistical robustness of the sample average approximation to the optimum of the problem is analyzed.

Key words: full random, statistical robustness, two-stage stochastic optimization, sample average approximation

中图分类号:

O221.5

张锦立, 韦思瑶, 童小娇, 孙海琳. 两阶段全随机优化问题的定量统计鲁棒性[J]. 运筹学学报（中英文）, 2026, 30(3): 183-203.

ZHANG Jinli, WEI Siyao, TONG Xiaojiao, SUN Hailin. Quantitative statistical robustness for full random two-stage stochastic optimization problems[J]. Operations Research Transactions, 2026, 30(3): 183-203.

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两阶段全随机优化问题的定量统计鲁棒性

Quantitative statistical robustness for full random two-stage stochastic optimization problems

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