Generalised hessian, max function and weak convexity
1996 ◽
Vol 53
(1)
◽
pp. 21-32
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Keyword(s):
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.
1984 ◽
Vol 22
(3)
◽
pp. 381-404
◽
1996 ◽
Vol 34
(4)
◽
pp. 1220-1234
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Keyword(s):
2020 ◽
2009 ◽
Vol 142
(1)
◽
pp. 85-106
◽
1995 ◽
Vol 36
(3)
◽
pp. 274-285
◽
2013 ◽
Vol 34
(12)
◽
pp. 2992-2998