On the Intrinsic Core of Convex Cones in Real Linear Spaces

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

“…6.3.1 ], the topological dual space of E, namely (E, τ c ) * , is exactly the algebraic dual space E . In recent works (see, e.g., Khazayel et al [36], Novo and Zȃlinescu [40]), the convex core topology τ c is used to derive properties for algebraic interiority notions (such as core and intrinsic core).…”

“…Notice that the τ c -closedness assumption concerning K in Lemma 2.1 cannot be omitted, as the example by Khazayel et al [36,Ex. 4.3] shows.…”

“…It is known that in vector optimization (see Jahn [34]) as well as in Image Space Analysis (ISA) in infinite dimensional linear spaces (see Giannessi [19,20] and references therein) difficulties may arise because of the possible non-solidness of ordering cones (for instance in the fields of optimal control, approximation theory, duality theory). Thus, it is of increasing interest to derive optimality conditions and duality results for such vector optimization problems using generalized interiority conditions (see, e.g., Adán and Novo [1][2][3][4], Bagdasar and Popovici [6], Bao and Mordukhovich [7], Borwein and Goebel [10], Borwein and Lewis [11], Grad [23,24], Grad and Pop [25], Khazayel et al [36], Zȃlinescu [43,44], and Cuong et al [14]). Such conditions can be formulated using the well-established generalized interiority notions given by quasi-interior, quasi-relative interior, algebraic interior (also known as core), relative algebraic interior (also known as intrinsic core, pseudo-relative interior or intrinsic relative interior).…”

“…In recent works related to vector optimization in real linear spaces (see, e.g., Adán and Novo [1][2][3][4], Bao and Mordukhovich [7], Khazayel et al [36], Novo and Zȃlinescu [40], Popovici [41], and Zhou, Yang and Peng [45]), the intrinsic core notion is studied in more detail. Having two real linear spaces X and E, a vector-valued objective function f : X → E, a certain set of constraints Ω ⊆ X , a convex (ordering) cone K ⊆ E (with possibly empty algebraic interior), a vector optimization problem is defined by f (x) → min w.r.t.…”

“…2 we recall important algebraic properties of convex sets and convex cones in linear spaces. In our main results, we will deal with relatively solid, convex cones, and for proving them, we will use separation techniques in linear spaces that are based on the intrinsic core notion (see [36] and Proposition 2.2).…”