Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces

Abstract

In vector optimization, it is of increasing interest to study problems where the image space (a real linear space) is preordered by a not necessarily solid (and not necessarily pointed) convex cone. It is well-known that there are many examples where the ordering cone of the image space has an empty (topological/algebraic) interior, for instance in optimal control, approximation theory, duality theory. Our aim is to consider Pareto-type solution concepts for such vector optimization problems based on the intrinsic core notion (a well-known generalized interiority notion). We propose a new Henig-type proper efficiency concept based on generalized dilating cones which are relatively solid (i.e., their intrinsic cores are nonempty). Using linear functionals from the dual cone of the ordering cone, we are able to characterize the sets of (weakly, properly) efficient solutions under certain generalized convexity assumptions. Toward this end, we employ separation theorems that are working in the considered setting.

Vectorial penalisation in vector optimisation in real linear-topological spaces

“…As far as we know, in the literature on vectorial penalisation techniques, no results are known for the set PEff( f , S, C ) (in particular, for Y = R m and the standard cone C = R m ≥ ). The topic of penalisation in vector optimisation in real linear spaces based on algebraic notions (such as algebraic/vectorial closure, algebraic interior, relative algebraic interior) could be interesting for further extensions of our results (see, e.g., Günther, Khazayel and Tammer [57], Novo and Zȃlinescu [58], and Schmölling [16]).…”

“…where F (C ) is a family of (nontrivial, pointed, solid, convex) Henig-type dilating cones (see, e.g., [42,43,57]). For the set WEff( f , S, D) (⊆ Eff( f , S, C )) with D ∈ F (C ) one can apply our derived penalisation results in the paper.…”

Nonlinear strict cone separation theorems in real normed spaces

“…Notice that cones from the set {C ⊆ E | C is a convex cone with Ω 1 \ {0} ⊆ intC} are known in the vector optimization / order theory literature as dilating cones for Ω 1 (Henig [23] used such dilating cones to define a proper efficiency solution concept in vector optimization; see also Grad [18,Ch. 3], Günther, Khazayel and Tammer [24], and Khan, Tammer and Zȃlinescu [25,Sec. 2.4]).…”