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Regularities and their relations to error bounds

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Abstract.

In this paper, we mainly study various notions of regularity for a finite collection {C 1 ,⋯,C m } of closed convex subsets of a Banach space X and their relations with other fundamental concepts. We show that a proper lower semicontinuous function f on X has a Lipschitz error bound (resp., ϒ-error bound) if and only if the pair {epi(f),X×{0}} of sets in the product space X×ℝ is linearly regular (resp., regular). Similar results for multifunctions are also established. Next, we prove that {C 1 ,⋯,C m } is linearly regular if and only if it has the strong CHIP and the collection {N C 1(z),⋯,N C m (z)} of normal cones at z has property (G) for each z∈C:=∩ i=1 m C i . Provided that C 1 is a closed convex cone and that C 2 =Y is a closed vector subspace of X, we show that {C 1 ,Y} is linearly regular if and only if there exists α>0 such that each positive (relative to the order induced by C 1 ) linear functional on Y of norm one can be extended to a positive linear functional on X with norm bounded by α. Similar characterization is given in terms of normal cones.

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Authors and Affiliations

  1. Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territory, Hong Kong

    Kung Fu Ng

  2. Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territory, Hong Kong

    Wei Hong Yang

Authors
  1. Kung Fu Ng
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  2. Wei Hong Yang
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Correspondence to Kung Fu Ng.

Additional information

Mathematics Subject Classifications: 90C25, 90C31, 49J52, 46A40

This research was supported by an Earmarked grant from the Research Grant Council of Hong Kong

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Ng, K., Yang, W. Regularities and their relations to error bounds. Math. Program., Ser. A 99, 521–538 (2004). https://doi.org/10.1007/s10107-003-0464-9

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  • DOI: https://doi.org/10.1007/s10107-003-0464-9

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