https://ogma.newcastle.edu.au/vital/access/ /manager/Index ${session.getAttribute("locale")} 5 https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:12997 Wed 11 Apr 2018 13:40:53 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:12999 n} is a family of maximal cyclically monotone operators defined on a Banach space X then there exists a real-valued locally Lipschitz function g such that ∂0g(x) = co{T₁(x), T₂(x),..., Tn(x)} for each x ∈ X; in a separable Banach space each non-empty weak compact convex subset in the dual space is identically equal to the approximate subdifferential mapping of some Lipschitz function and for locally Lipschitz functions defined on separable spaces the notions of strong and weak integrability coincide.]]> Wed 11 Apr 2018 11:35:39 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:13073 δ subset of the polar set, and (b) any nonsemicoercive proper convex lsc [weak*-lsc] function in a [dual] Banach space has a generic [dense G_δ] set of L^∞-perturbations which do not attain their infimum. We also characterize the proper convex functions that have inf-nonattaining L^∞-perturbations. This results also in a criterion for reflexivity.]]> Sat 24 Mar 2018 08:15:38 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:13079 X* is the Clarke subdifferential of some locally Lipschitz function on X. Related results for approximate subdifferentials are also given. Moreover, on smooth Banach spaces, for every locally Lipschitz function with minimal Clarke subdifferential, one can obtain a maximal Clarke subdifferential map via its ‘local Lipschitz-constant’ function. Finally, some results concerning the characterization and calculus of local Lipschitz-constant functions are developed.]]> Sat 24 Mar 2018 08:15:37 AEDT ]]> https://ogma.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 ]]>