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);云南省教育厅科学研究基金(2022J0478);云南财经大学科学研究基金(2021D08);云南财经大学科学研究基金(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);Scientific Research Fund of Yunnan Province Education Department(2022J0478);Scientific Research Fund of Yunnan University of Finance and Economics(2021D08);Scientific Research Fund of Yunnan University of Finance and Economics(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.

参考文献

[1]	Aubin J P. Set-Valued Analysis. Boston: Birkh?user, 1999
[2]	Auslender A, Teboulle M. Asymptotic Cones and Functions in Optimization and Variational Inequalities. New York: Springer-Verlag, 2003
[3]	Aussel D, Daniillids A, Thibault L. Subsmooth sets: functional characterizations and related concepts. Trans Amer Math Soc, 2005, 357(4): 1275-1301
[4]	Azé D, Corvellec J N. Characterizations of error bounds for lower semicontinuous functions on metric spaces. ESAIM Control Optim Calc Var, 2004, 10(3): 409-425
[5]	Azé D, Corvellec J N. On the sensitivity analysis of Hoffman constants for systems of linear inequalities. SIAM J Optim, 2002, 12(4): 913-927
[6]	Beck A, Teboulle M. Convergence rate analysis and error bounds for projection algorithms in convex feasibility problems. Optim Methods Softw, 2003, 18(4): 377-394
[7]	Bolte J, Nguyen T P, Peypouquet J, Suter B. From error bounds to the complexity of first-order descent methods for convex functions. Math Program, 2017, 165(2): 471-507
[8]	Burke J V, Deng S. Weak sharp minima revisited, part II: application to linear regularity and error bounds. Math Program, 2005, 104: 235-261
[9]	Clarke F H. Optimization and Nonsmooth Analysis. New York: Wiley, 1990
[10]	Dontchev A L, Rockafellar R T. Regularity and conditioning of solution mappings in variational analysis. Set-Valued Analysis, 2004, 12(1/2): 79-109
[11]	Drusvyatskiy D, Ioffe A D, Lewis A S. Nonsmooth optimization using Taylor-like models: error bounds, convergence, and termination criteria. Math Program, 2021, 185(2): 357-383
[12]	Drusvyatskiy D, Ioffe A D, Lewis A S. Transversality and alternating projections for nonconvex sets. Found Comput Math, 2015, 15(6): 1637-1651
[13]	Ekeland I. On the variational principle. J Math Anal Appl, 1974, 47(2): 324-353
[14]	Hoffman A. On approximate solutions of systems of linear inequalities. Journal of Research of the National Bureau of Standards, Section B, Mathematical Sciences, 1952, 49(4): 263-265
[15]	Ioffe A D. Metric regularity and subdifferential calculus. Russ Math Surv, 2000, 55(3): 501-558
[16]	Kruger A Y, López M A, Théra M A. Perturbation of error bounds. Math Program, 2018, 168(2): 533-554
[17]	Li M H, Meng K W, Yang X Q. On error bound moduli for locally Lipschitz and regular functions. Math Program, 2018, 171(1/2): 463-487
[18]	Li G, Mordukhovich B S, Nghia T T A, Pham T S. Error bounds for parametric polynomial systems with applications to higher-order stability analysis and convergence rates. Math Program, 2018, 168(1/2): 313-346
[19]	Luo X D, Luo Z Q. Extension of Hoffman's error bound to polynomial systems. SIAM J Optim, 1994, 4(2): 383-392
[20]	Luo Z Q, Tseng P. Perturbation analysis of a condition number for linear systems. SIAM J Matrix Anal Appl, 1994, 15(2): 636-660
[21]	Mordukhovich B S. Variational Analysis and Generalized Differentiation I. Berlin: Springer-Verlag, 2006
[22]	Ng K F, Zheng X Y. Characterization of error bounds for convex multifunctions on Banach spaces. Math Oper Res, 2004, 29(1): 45-63
[23]	Ng K F, Zheng X Y. Error bounds for lower semicontinuous functions in normed spaces. SIAM J Optim, 2001, 12(1): 1-17
[24]	Ngai H V, Kruger A, Théra M. Stability of error bounds for semi-infinite convex constraint systems. SIAM J Optim, 2010, 20(4): 2080-2096
[25]	Ngai H V, Théra M. Error bounds for convex differentiable inequality systems in Banach spaces. Math Program, 2005, 104(2/3): 465-482
[26]	Ngai H V, Théra M. Error bounds for systems of lower semicontinuous functions in Asplund spaces. Math Program, 2009, 116(1/2): 397-427
[27]	Pang J S. Error bounds in mathematical programming. Math Program, 1997, 79(1): 299-332
[28]	Shapiro A. Existence and differentiability of metric projections in Hilbert spaces. SIAM J Optim, 1994, 4(1): 130-141
[29]	Shapiro A, Al-Khayyal F. First-order conditions for isolated locally optimal solutions. J Optim Theory Appl, 1993, 77(1): 189-196
[30]	Shen Z S, Yao J C, Zheng X Y. Calmness and the Abadie CQ for multifunctions and linear regularity for a collection of closed sets. SIAM J Optim, 2019, 29(3): 2291-2319
[31]	Wu Z, Ye J J. On error bounds for lower semicontinuous functions. Math Program, 2002, 92(2): 301-314
[32]	Zhang B B, Zheng X Y. Well-posedness and generalized metric subregularity with respect to an admissible function. Sci China Math, 2019, 62(4): 809-822
[33]	Zheng X Y. Slater condition for tangent derivatives. Math Oper Res, 2022, 47(4): 3282-3303
[34]	Zheng X Y, Ng K F. Calmness for L-subsmooth multifunctions in Banach spaces. SIAM J Optim, 2009, 19(4): 1648-1673
[35]	Zheng X Y, Ng K F. Perturbation analysis of error bounds for systems of conic linear inequalities in Banach spaces. SIAM J Optim, 2005, 15(4): 1026-1041
[36]	Zheng X Y, Ng K F. Stability of error bounds for conic subsmooth inequalities. ESAIM Control Optim Calc Var, 2019, 25: Article 55
[37]	Zheng X Y, Wei Z. Perturbation analysis of error bounds for quasi-subsmooth inequalities and semi-infinite constraint systems. SIAM J Optim, 2012, 22(1): 41-65
[38]	Zheng X Y, Zhu J X. Generalized metric subregularity and regularity with respect to an admissible function. SIAM J Optim, 2016, 26(1): 535-563