Generalised Hessian, max function and weak convexity

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Abstract

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(cursive greek chi) = h(cursive greek chi) - η ∥cursive greek chi∥2 on a neighbourhood of cursive greek chi where h is a convex function and η > 0. Moreover, a characterisation of the generalised second-order directional derivative for max functions is given.
Original languageEnglish
Pages (from-to)21-32
Number of pages12
JournalBulletin of the Australian Mathematical Society
Volume53
Issue number1
Publication statusPublished - 1 Feb 1996
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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