https://nova.newcastle.edu.au/vital/access/ /manager/Index en-au 5 https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:6464 Wed 11 Apr 2018 14:48:32 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13022 Wed 11 Apr 2018 14:18:53 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13023 m → R is arcwise essentially smooth on R^m and each function f_j : R^n → R, 1 ≤ j ≤ m, is strictly differentiable almost everywhere in Rⁿ, then g ○ f is strictly differentiable almost everywhere in Rⁿ, where f ≡ (f₁,f₂,...,f_m). We also show that all the semismooth and all the pseudoregular functions are arcwise essentially smooth. Thus, we provide a large and robust lattice algebra of Lipschitz functions whose generalized derivatives are well behaved.]]> Wed 11 Apr 2018 12:51:09 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13012 Wed 11 Apr 2018 11:34:45 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:12980 Wed 11 Apr 2018 09:29:42 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:6988 Sat 24 Mar 2018 08:37:48 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:13130 Sat 24 Mar 2018 08:15:42 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13055 0 they are s-Hölder, and so proximally, subdifferentiable only on dyadic rationals and nowhere else. As applications we construct Lipschitz functions with prescribed Hölder and approximate subderivatives.]]> Sat 24 Mar 2018 08:15:39 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 ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13099 Sat 24 Mar 2018 08:15:13 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:13100 Sat 24 Mar 2018 08:15:12 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:9931 Sat 24 Mar 2018 08:14:18 AEDT ]]>