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Generalised hessian, max function and weak convexity

Published online by Cambridge University Press:  17 April 2009

X. Q. Yang
X. Q. Yang
Affiliation:
Department of Mathematics, The University of Western Australia, Nedlands, QA 6009 Australia
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In this paper, a second-order characterisation of η-convex C1, 1 functions is derived in a Hilbert space using a generalised second-order directional derivative. Using this result it is then shown that every C1, 1 function is locally weakly convex, that is, every C1, 1 real-valued function f can be represented as f (x) = h (x) − η‖x‖2 on a neighbourhood of x where h is a convex function and η > 0. Moreover, a characterisation of the generalised second-order directional derivative for max functions is given.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 53 , Issue 1 , February 1996 , pp. 21 - 32
Copyright
Copyright © Australian Mathematical Society 1996

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