Abstract
We show that for a set-valued mapping F: X → Y between Banach spaces the property of metric regularity near a point of its graph is separably determined in the sense that it holds, provided for any separable subspaces L 0 ⊂ X and M ⊂ Y, containing the corresponding components of the point, there is a separable subspace L ⊂ X containing L 0 such that the mapping whose graph is the intersection of the graph of F with L × M (restriction of F to L × M) is metrically regular near the same point. Moreover, it is shown that the rates of regularity of the mapping near the point can be recovered from the rates of such restrictions.
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Notes
- 1.
WCG spaces are distinguished by the property that the limiting versions of all subdifferentials trusted on the space coincide for locally Lipschitz functions.
- 2.
I am indebted to one of the reviewers for pointing out that compact regularity coincides with “partial cone property up to a compact set” of the inverse map introduced in a just published Penot’s monograph [14].
- 3.
We have chosen a slightly different way for writing the inclusion to emphasize close relationship with the popular “compact epi-Lipschitz property” introduced by Borwein and Strojwas in 1986.
- 4.
It is appropriate to quote here a comment by the other reviewer who states that (6) is equivalent to compact regularity and the closure operation can be harmlessly omitted. This is certainly true in the “full” regularity case when P ={ 0}—we have explicitly used this fact in the proof of Proposition 2 So I believe that the reviewer’s statement is correct and wish to express my thanks for it.
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Acknowledgements
I wish to thank the reviewers for valuable comments and many helpful remarks.
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Ioffe, A.D. (2014). Separable Reduction of Metric Regularity Properties. In: Demyanov, V., Pardalos, P., Batsyn, M. (eds) Constructive Nonsmooth Analysis and Related Topics. Springer Optimization and Its Applications, vol 87. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8615-2_3
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