Set Order Relations, Set Optimization, and Ekeland’s Variational Principle

Strong subdifferential calculus for convex set-valued mappings and applications to set optimization

“…In this section, we give some basic definitions and results. In what follows, let X and Y be two Hausdorff locally convex topological vector spaces and Y + ⊂ Y be a pointed (Y + ∩ −Y + = {0 Y }) closed and convex cone with nonempty topological interior inducing a partial order in Y , which is defined by y 1 ≤ Y + y 2 ⇐⇒ y 2 − y 1 ∈ Y + , for y 1 , y 2 ∈ Y (see [2,6]). We adjoin to Y an abstract maximal element +∞ Y such that, for any y ∈ Y , y ≤ Y + +∞ Y .…”

Some topological properties of solution sets in partially ordered set optimization

“…Set optimization problem is based on comparison of values of the set-valued objective map by means of set order relations. Several kinds of set order relations are available in the literature; see, e.g., [2,3,4]. Recently, Karaman et al [5] introduced set order relations on the family of sets involving the Minkowski difference.…”

Levitin-Polyak Well-Posedness for Parametric Set Optimization Problem

“…Kuroiwa [20,21] proposed various set order relations for comparison of sets to define notions of minimal solution of a set optimization problem. For details, we refer the reader to the survey paper [2]. The study of the solution sets of a perturbed set optimization problem, perturbed with respect to the feasible set or the objective set-valued map is a fast growing topic and is studied under stability theory.…”