集值映射误差界的稳定性

数学物理学报 ›› 2025, Vol. 45 ›› Issue (1): 214-235.

集值映射误差界的稳定性

沈宗山

云南财经大学统计与数学学院昆明 650221

收稿日期:2024-01-10 修回日期:2024-04-28 出版日期:2025-02-26 发布日期:2025-01-08
作者简介:沈宗山,E-mail:shenzongshan@126.com
基金资助:
云南省教育厅科学研究基金 (2022J0476, 2022J0478) 和云南财经大学科学研究基金 (2021D08, 2021D09)

Stability of Error Bounds for Multifunctions

Shen Zongshan

School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221

Received:2024-01-10 Revised:2024-04-28 Online:2025-02-26 Published:2025-01-08
Supported by:
Scientific Research Fund of Yunnan Province Education Department (2022J0476, 2022J0478) and the Scientific Research Fund of Yunnan University of Finance and Economics (2021D08, 2021D09)

摘要/Abstract

摘要： 该文主要研究集值映射关于序锥有局部误差界及其稳定性的原始刻画. 首先, 证明了一个集值映射 $\Psi$ 的 Bouligand 切导数关于序锥 $C$ 的 Slater 条件在 $\Psi$ 经 "小 calm" 扰动时总是稳定的. 基于此, 证明了若集值映射 $\Psi$ 的 Bouligand 切导数关于序锥 $C$ 满足 Slater 条件, 则 $\Psi$ 经小 calm 正则扰动时, 关于 $C$ 有稳定局部误差界. 这些结果把 Zheng [Math Oper Res, 2022, 47(4): 3282--3303] 建立的相应结果从向量值情形推广到集值情形. 作为应用, 该文给出了凸过程关于序锥有稳定全局误差界的充分条件.

关键词: 误差界, 切导数, Slater 条件, 凸过程

Abstract: In terms of the Slater condition of the Bouligand and Clarke tangent derivatives of the objective multifunction $\Psi$, this paper mainly studies the stability of error bound of $\Psi$ at a point $\bar{x}$ with respect to an ordering cone $C$. It is proved that the Slater condition of the Bouligand tangent derivative of $\Psi$ at $\bar{x}$ with respect to $C$ is always stable with respect to all small calm perturbations. Based on this result, we prove that the Slater condition of the Bouligand tangent derivative of $\Psi$ at $\bar{x}$ with respect to $C$ is a sufficient condition for $\Psi$ to have a stable error bound at $\bar{x}$ with respect to $C$ when $\Psi$ undergoes small calm and regular perturbations. These results extend the corresponding ones given by Zheng [Math Oper Res, 2022, 47(4): 3282--3303] from the vector-valued to the set-valued case. As applications, some sufficient conditions are provided for a convex progress to have a stable global error bound with respect to an ordering cone.

Key words: error bound, tangent derivative, Slater condition, convex progress

中图分类号:

O221.2

沈宗山. 集值映射误差界的稳定性[J]. 数学物理学报, 2025, 45(1): 214-235.

Shen Zongshan. Stability of Error Bounds for Multifunctions[J]. Acta mathematica scientia,Series A, 2025, 45(1): 214-235.

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