##
**Metric subregularity for subsmooth generalized constraint equations in Banach spaces.**
*(English)*
Zbl 1268.49004

Let \(X\) and \(Y\) be Banach spaces, and let \(F:X \rightrightarrows Y\) be a closed multifunction. Next, let \(b\in Y\) be a given point , and \(A\subset X\) be a closed set. In the paper, the following constraint equation
\[
b\in F(x) \eqno (1)
\]
on the set \(A\), is considred.

The equation (1) is called metrically subregular at \(a\in S=\{x\in A:b\in F(x)\}\) if there exist \(\tau,\delta \in (0,\infty)\) such that \[ d(x,S)\leq \tau d(b,F(x))+d(x,A) \; \forall x\in B(a,\delta), \] where \(B(a,\delta)\) denotes the open ball of center \(a\) and radius \(\delta.\)

The equation (1) is called strongly subregular at \(a\in S\) if there exist \(\tau,\delta \in (0,\infty)\) such that \[ \|x-a\|\leq \tau d(b,F(x))+d(x,A) \; \forall x\in B(a,\delta). \] In this paper, necessary and sufficient conditions are obtained for subsmooth constraint equations to be metrically and strongly metrically subregular.

The equation (1) is called metrically subregular at \(a\in S=\{x\in A:b\in F(x)\}\) if there exist \(\tau,\delta \in (0,\infty)\) such that \[ d(x,S)\leq \tau d(b,F(x))+d(x,A) \; \forall x\in B(a,\delta), \] where \(B(a,\delta)\) denotes the open ball of center \(a\) and radius \(\delta.\)

The equation (1) is called strongly subregular at \(a\in S\) if there exist \(\tau,\delta \in (0,\infty)\) such that \[ \|x-a\|\leq \tau d(b,F(x))+d(x,A) \; \forall x\in B(a,\delta). \] In this paper, necessary and sufficient conditions are obtained for subsmooth constraint equations to be metrically and strongly metrically subregular.

Reviewer: Tamaz Tadumadze (Tbilisi)

### Keywords:

metric subregularity; subsmooth constraint equations; Banach spaces; Asplund spaces; coderivatives; normal cones
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\textit{Q. He} et al., J. Appl. Math. 2012, Article ID 185249, 16 p. (2012; Zbl 1268.49004)

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### References:

[1] | W. Li and I. Singer, “Global error bounds for convex multifunctions and applications,” Mathematics of Operations Research, vol. 23, no. 2, pp. 443-462, 1998. · Zbl 0977.90054 |

[2] | X. Y. Zheng and K. F. Ng, “Perturbation analysis of error bounds for systems of conic linear inequalities in Banach spaces,” SIAM Journal on Optimization, vol. 15, no. 4, pp. 1026-1041, 2005. · Zbl 1114.90134 |

[3] | W. Li, “Abadie’s constraint qualification, metric regularity, and error bounds for differentiable convex inequalities,” SIAM Journal on Optimization, vol. 7, no. 4, pp. 966-978, 1997. · Zbl 0891.90132 |

[4] | B. S. Mordukhovich and Y. Shao, “Nonconvex differential calculus for infinite-dimensional multifunctions,” Set-Valued Analysis, vol. 4, no. 3, pp. 205-236, 1996. · Zbl 0866.49024 |

[5] | X. Y. Zheng and K. F. Ng, “Metric subregularity and constraint qualifications for convex generalized equations in Banach spaces,” SIAM Journal on Optimization, vol. 18, no. 2, pp. 437-460, 2007. · Zbl 1190.90230 |

[6] | X. Y. Zheng and K. F. Ng, “Calmness for L-subsmooth multifunctions in Banach spaces,” SIAM Journal on Optimization, vol. 19, no. 4, pp. 1648-1673, 2008. · Zbl 1188.49019 |

[7] | D. Klatte and B. Kummer, Nonsmooth Equations in Optimization, vol. 60, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002. · Zbl 1173.49300 |

[8] | B. S. Mordukhovich, “Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions,” Transactions of the American Mathematical Society, vol. 340, no. 1, pp. 1-35, 1993. · Zbl 0791.49018 |

[9] | B. S. Mordukhovich, Variational Analysis and Generalized Differentiation. I, II, Springer, Berlin, Germany, 2006. |

[10] | R. Henrion, A. Jourani, and J. Outrata, “On the calmness of a class of multifunctions,” SIAM Journal on Optimization, vol. 13, no. 2, pp. 603-618, 2002. · Zbl 1028.49018 |

[11] | X. Y. Zheng and K. F. Ng, “Metric regularity and constraint qualifications for convex inequalities on Banach spaces,” SIAM Journal on Optimization, vol. 14, no. 3, pp. 757-772, 2003. · Zbl 1079.90103 |

[12] | X. Y. Zheng and K. F. Ng, “Linear regularity for a collection of subsmooth sets in Banach spaces,” SIAM Journal on Optimization, vol. 19, no. 1, pp. 62-76, 2008. · Zbl 1190.90231 |

[13] | F. H. Clarke, R. J. Stern, and P. R. Wolenski, “Proximal smoothness and the lower-C2 property,” Journal of Convex Analysis, vol. 2, no. 1-2, pp. 117-144, 1995. · Zbl 0881.49008 |

[14] | R. A. Poliquin, R. T. Rockafellar, and L. Thibault, “Local differentiability of distance functions,” Transactions of the American Mathematical Society, vol. 352, no. 11, pp. 5231-5249, 2000. · Zbl 0960.49018 |

[15] | D. Aussel, A. Daniilidis, and L. Thibault, “Subsmooth sets: functional characterizations and related concepts,” Transactions of the American Mathematical Society, vol. 357, no. 4, pp. 1275-1301, 2005. · Zbl 1094.49016 |

[16] | F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, NY, USA, 1983. · Zbl 0582.49001 |

[17] | R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, vol. 1364 of Lecture Notes in Mathematics, Springer, New York, NY, USA, 1989. · Zbl 0658.46035 |

[18] | R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, vol. 317, Springer, Berlin, Germany, 1998. · Zbl 0888.49001 |

[19] | B. S. Mordukhovich and Y. H. Shao, “Nonsmooth sequential analysis in Asplund spaces,” Transactions of the American Mathematical Society, vol. 348, no. 4, pp. 1235-1280, 1996. · Zbl 0881.49009 |

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