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Separable reduction of metric regularity properties. (English) Zbl 1282.49012

Demyanov, Vladimir F. (ed.) et al., Constructive nonsmooth analysis and related topics. New York, NY: Springer (ISBN 978-1-4614-8614-5/hbk; 978-1-4614-8615-2/ebook). Springer Optimization and Its Applications 87, 25-37 (2014).
The paper presented is concerned with the separable reduction for the regularity properties of set-valued mappings \(F:X\rightrightarrows Y,\, X,Y\) Banach spaces (see the survey paper of the author, [A. D. Ioffe, Russ. Math. Surv. 55, No. 3, 501–558 (2000); translation from Usp. Mat. Nauk 55, No. 3, 103–162 (2000; Zbl 0979.49017)], or the book [A. L. Dontchev andR. T. Rockafellar, Implicit functions and solution mappings. A view from variational analysis. Springer Monographs in Mathematics. New York, NY: Springer (2009; Zbl 1178.26001)]). The author works with the essentially equivalent notions of metric regularity and linear openness and considers also a new notion, called compact regularity, meaning the existence of a compact subset \(P\) of \(Y\), and of two numbers \(\varepsilon,r>0\) such that \[ \left(F(x)\cap B(\bar y,\varepsilon)\right)+trB_Y\subset f\left(B(x,t)\right)+tP, \] for all \(x\in X\) with \(\|x-\bar x\|<\varepsilon\) and \(t\in(0,\varepsilon),\) which agrees with the standard linear openness for \(P=\{0\}.\) The main result (Theorem 2) asserts that \(F:X\rightrightarrows Y\) is regular at \((\bar x,\bar y)\in\) graph \(F\) if and only for all separable subspaces \(L_0\subset X,M\subset Y\) such that \((\bar x,\bar y)\in L_0\times M\) there exists a bigger separable subspace \(L\supset L_0\) of \(X\) such that the set-valued mapping \(F_{L\times M}\) with graph \((L\times M)\cap\,\)graph \(F\) is regular at \((\bar x,\bar y).\)
The last part of the paper contains a discussion on compact regularity, a notion which appears also in the recent book by [J.-P. Penot, Calculus without derivatives. Graduate Texts in Mathematics 266. New York, NY: Springer (2013; Zbl 1264.49014)], under the name “partial cone property up to a compact set”.
For the entire collection see [Zbl 1278.49002].

MSC:

49J53 Set-valued and variational analysis
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
47H04 Set-valued operators
49J52 Nonsmooth analysis
49N60 Regularity of solutions in optimal control
54C60 Set-valued maps in general topology
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