On the relation between u-Hessians and second-order epi-derivatives

We consider second-order objects for a convex function defined as the maximum of a finite number of C2-functions. Variational analysis yields explicit formulae for the second-order epi-derivatives of such max-functions. Another second-order object can be defined by means of a space decomposition tha...

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Bibliographic Details
Published in:European journal of operational research 2004-08, Vol.157 (1), p.28
Main Authors: Mifflin, Robert, Sagastizábal, Claudia
Format: Article
Language:English
Subjects:
Operations research
Studies
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Summary:We consider second-order objects for a convex function defined as the maximum of a finite number of C2-functions. Variational analysis yields explicit formulae for the second-order epi-derivatives of such max-functions. Another second-order object can be defined by means of a space decomposition that allows us to identify a subspace on which a Lagrangian related to a max-function is smooth. This decomposition yields an explicit expression for the so-called U-Hessian of the function, defined as the Hessian of the related Lagrangian. We show that the second-order epi-derivative relative to the U-subspace and the U-Hessian are equivalent second-order objects. Thus, the U-Lagrangian of a max-function captures the function's second-order epi-differential behavior with ordinary second derivatives. [PUBLICATION ABSTRACT]
ISSN:0377-2217
1872-6860
DOI:10.1016/j.ejor.2003.08.010