On cone convexity of set-valued maps

A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory

“…Therefore, will work with even more specialized subspaces of P (Z ). The starting point is a popular definition, compare for example [12,20,27,32,39].…”

“…The construction has been described by Kuroiwa, Tanaka and Truong [32] in the context of optimization problems, but it has been well-known before, for example in Algebra and Computer Science [8] and Economics [29]. As in [34] and [35], we will use these "set orders" to define appropriate image spaces for set-valued functions which will turn out to be complete lattices without any special assumptions to the underlying vector order.…”

On solutions of set-valued optimization problems

“…The set relation ≤ l was presented by the first time in the framework of linear spaces in [13] and the weak set relation l was introduced in [14].…”

A derivative-free descent method in set optimization

“…Although most of the papers on set optimization work with the notion of a minimizer and variants of it, nowadays one works with a more realistic order relation for the comparison of sets which has been introduced to optimization by Kuroiwa (e.g., see [12]; a first publication has been given by Kuroiwa et al. [13]). Outside the optimization community this notion has been already used by Young [27] in algebra, by Nishnianidze [20] in fixed point theory and by Chiriaev and Walster [2] in computer science and interval analysis.…”