Monotonic analysis over ordered topological vector spaces. IV (Q1037365)

The authors deal with increasing and co-radiant (ICR) functions defined on a topological vector space \(X\) equipped with a closed pointed convex cone. They characterize non-negative ICR functions on \(X\) as those functions that are abstract convex with respect to a suitable family \(L\) of (elementary) ICR functions \(\ell_{y,\alpha}:X\to [0,+\infty]\) defined as follows: \[ \ell_{y,\alpha}(x):= \max\{0\leq \lambda\leq \alpha: \lambda y\leq x\},\qquad \forall y\in X,\;\alpha\in\mathbb R_{++}. \] Moreover, they provide a description of the support set \[ \text{supp}(f,L)=\{\ell_{y,\alpha}\in L: \ell_{y,\alpha}(x)\leq f(x),\;\forall x\in X\}, \] and they prove a nonemptyness result and a characterization of the \(L\)-subdifferential of a non-negative ICR function. Subsequently they introduce \(X\times\mathbb R_{++}\)-support sets for an ICR function, in order to characterize polar sets; furthermore, they give a kind of separation property for a certain class of co-radiant sets by elementary ICR functions. In the last section, they provide a characterization of the subdifferential of ICR functions by means of increasing and positively-homogeneous (IPH) functions, which are simpler. [For Parts I and II, see \textit{H. Mohebi} and \textit{H. Sadeghi}, Optimization 56, No. 3, 305--321 (2007; Zbl 1144.26012); ibid. 58, No. 2, 241--249 (2009; Zbl 1158.26305).]