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A Simple Formula for the Second-Order Subdifferential of Maximum Functions

Abstract

We derive a simple formula for the second-order subdifferential of the maximum of coordinates which allows us to construct this set immediately from its argument and the direction to which it is applied. This formula can be combined with a chain rule recently proved by Mordukhovich and Rockafellar (SIAM J. Optim. 22:953–986, 2012) in order to derive a similarly simple formula for the extended partial second-order subdifferential of finite maxima of smooth functions. Analogous formulas can be derived immediately for the full and conventional partial second-order subdifferentials.

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Acknowledgements

This work was supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.

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Correspondence to René Henrion.

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This paper is dedicated to Prof. Boris Mordukhovich on the occasion of his 65th birthday.

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Emich, K., Henrion, R. A Simple Formula for the Second-Order Subdifferential of Maximum Functions. Vietnam J. Math. 42, 467–478 (2014). https://doi.org/10.1007/s10013-013-0052-0

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  • DOI: https://doi.org/10.1007/s10013-013-0052-0

Keywords

  • Second-order subdifferential
  • Extended partial second-order subdifferential
  • Maximum function
  • Calculus rules

Mathematics Subject Classification (2010)

  • 49J52
  • 49J53