On the cone minima and maxima of directed convex free disposal subsets and applications

In this paper, we first present new existence theorems of cone-supremum/infimum for directed convex and/or free disposal subsets in their closure. Then, we provide various conditions through which this kind of subsets admits a cone-maximum/minimum point, the so-called strongly maximal/minimal or ideal efficient points with respect to a cone. Next, we present a unifying result on the existence of these remarkable points, which we apply to extend, improve and unify the existence of an ideal efficient point for hypo/epi-graphical level sets of a given vector-valued function recently considered in [2–4]. A global set-valued analysis on the hypo/epi-profile mappings for general vector-valued maps is also presented. As a consequence, we extend the regularizations and radial epi-derivatives of [2,23] and, henceforth, obtain optimality conditions for global strong Pareto optimums of non-convex nondifferentiable extended vector-valued maps under different assumptions on the ordering cone and the topology of the target space, improving and generalizing the classic global optimality conditions of quasi-convex differentiable extended real-valued functions.