Limit Superior of Subdifferentials of Uniformly Convergent Functions

Limit Superior of Subdifferentials of Uniformly Convergent Functions In this paper we show that the $$G$$ – subdifferential of a lower semicontinuous function is contained in the limit superior of the $$G$$ – subdifferential of lower semicontinuous uniformly convergent family to this function. It happens that this result is equivalent to the corresponding normal cones formulas for family of sets which converges in the sense of the bounded Hausdorff distance. These results extend to the infinite dimensional case those of Ioffe for $$C^2$$ – functions and of Benoist for Clarke’s normal cone. As an application we characterize the subdifferential of any function which is bounded from below by a negative quadratic form in terms of its Moreau–Yosida proximal approximation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Limit Superior of Subdifferentials of Uniformly Convergent Functions

Positivity , Volume 3 (1) – Oct 22, 2004
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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 1999 by Kluwer Academic Publishers
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009740914637
Publisher site
See Article on Publisher Site

Abstract

In this paper we show that the $$G$$ – subdifferential of a lower semicontinuous function is contained in the limit superior of the $$G$$ – subdifferential of lower semicontinuous uniformly convergent family to this function. It happens that this result is equivalent to the corresponding normal cones formulas for family of sets which converges in the sense of the bounded Hausdorff distance. These results extend to the infinite dimensional case those of Ioffe for $$C^2$$ – functions and of Benoist for Clarke’s normal cone. As an application we characterize the subdifferential of any function which is bounded from below by a negative quadratic form in terms of its Moreau–Yosida proximal approximation.

Journal

PositivitySpringer Journals

Published: Oct 22, 2004

References

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