https://nova.newcastle.edu.au/vital/access/ /manager/Index en-au 5 https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:11705 Wed 11 Apr 2018 15:39:28 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:6475 Sat 24 Mar 2018 10:23:55 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:1421 Sat 24 Mar 2018 08:28:15 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:2253 Sat 24 Mar 2018 08:27:15 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:14234 Sat 24 Mar 2018 08:24:44 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:14070 Sat 24 Mar 2018 08:22:33 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:14689 Sat 24 Mar 2018 08:19:10 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13141 ’(x), h) is continuous. Recall that a convex Gateaux differentiable function is strictly Gateaux differentiable. In the case of a locally Lipschitz function our definition coincides with more standard ones: it requires that f^’ be norm to weak-star continuous.]]> Sat 24 Mar 2018 08:18:08 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13044 Sat 24 Mar 2018 08:16:32 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13045 Sat 24 Mar 2018 08:16:32 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13172 Sat 24 Mar 2018 08:16:07 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13165 n or for Lipschitz functions. To obtain more delicate results it is necessary to restrict either the spaces or the functions. Many examples are available in Borwein and Strojwas which illustrate how badly wrong things may go outside of a Baire metrizable or Banach space setting. In this paper we restrict our attention primarily to a Banach space X and consider what properties a set C in X should have for the Clarke tangent cone T_c(x) and normal cone N_c(x) to adequately measure boundary behaviour of x in C.]]> Sat 24 Mar 2018 08:16:06 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13052 Sat 24 Mar 2018 08:15:41 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13074 Sat 24 Mar 2018 08:15:37 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13076 o(x;v):=[formula cannot be replicated], and the Clarke subdifferential is defined by ∂_cf(x) = {⏀∈X*:⏀(v) ≤ f^o(x;v) for all v∈X}. This subdifferential has been widely used as a powerful tool in nonsmooth analysis with applications in diverse areas of optimization. Recently, substantial progress has been made on understanding the limitations of the Clarke derivative. Among other things, it is shown that on any Banach space X, the 1-Lipschitz functions for which ∂_cf(x)=B_x* for all x∈X, is a residual set among all the 1-Lipschitz functions on X (where B_x* denotes the dual unit ball). That is, even though the Clarke derivative is an effective tool in a wide variety of both theoretical and applied optimization problems, just like the classical derivative, the class of pathological Lipschitz functions for which it provides no additional information is larger in the category sense. In this note, we begin by considering the following related question, which asks how profuse (from the point of view of extensions) the functions in the aforementioned result are.]]> Sat 24 Mar 2018 08:15:36 AEDT ]]>