Regularities and their relations to error bounds
Regularities and their relations to error bounds
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DOI：
10.1007/s10107-003-0464-9
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发表时间：
2004-04
影响因子：
2.7
通讯作者：
K. Ng;W. Yang
中科院分区：
文献类型：
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作者：
K. Ng;W. Yang
在本文中，我们主要研究了Banach空间的闭凸子集的有限集合{C1，⋯，Cm}的各种正则性概念，以及它们与其他基本概念的关系。我们证明了一个合适的下半连续函数x具有一个Lipschitz误差界。， ϒ-error bound)当且仅当积空间exx上集合的对{epi(f)， xx{0}}是线性正则的(resp。、定期)。对于多函数也建立了类似的结果。接下来，我们证明{C1，⋯，Cm}是线性正则的当且仅当它具有强CHIP和正锥的集合{NC1(z),⋯,NCm(z)}对于每个z∈C:=∩i=1mCi具有atzhas性质(G)。假设C1是闭凸锥，c2 =Y是x的闭向量子空间，我们证明了{C1，Y}是线性正则的当且仅当存在α>，使得范数1的每一个正（相对于C1诱导的阶数）线性泛函Y可以推广为范数以α为界的x上的正线性泛函。对于正锥体也给出了类似的描述。
In this paper, we mainly study various notions of regularity for a finite collection {C1,⋯,Cm} of closed convex subsets of a Banach spaceXand their relations with other fundamental concepts. We show that a proper lower semicontinuous functionfonXhas a Lipschitz error bound (resp., ϒ-error bound) if and only if the pair {epi(f),X×{0}} of sets in the product spaceX×ℝ is linearly regular (resp., regular). Similar results for multifunctions are also established. Next, we prove that {C1,⋯,Cm} is linearly regular if and only if it has the strong CHIP and the collection {NC1(z),⋯,NCm(z)} of normal cones atzhas property (G) for eachz∈C:=∩i=1mCi. Provided thatC1is a closed convex cone and thatC2=Yis a closed vector subspace ofX, we show that {C1,Y} is linearly regular if and only if there exists α>0 such that each positive (relative to the order induced byC1) linear functional onYof norm one can be extended to a positive linear functional onXwith norm bounded by α. Similar characterization is given in terms of normal cones.