Henig proper efficiency in vector optimization with variable ordering structure

Abstract

In this paper we introduce a notion of Henig proper efficiency for constrained vector optimization problems in the setting of variable ordering structure. In order to get an appropriate concept, we have to explore firstly the case of fixed ordering structure and to observe that, in certain situations, the well-known Henig proper efficiency can be expressed in a simpler way. Then, we observe that the newly introduced notion can be reduced, by a Clarke-type penalization result, to the notion of unconstrained robust efficiency. We show that this penalization technique, coupled with sufficient conditions for weak openness, serves as a basis for developing necessary optimality conditions for our Henig proper efficiency in terms of generalized differentiation objects lying in both primal and dual spaces.

Approximate efficiency in set-valued optimization with variable order

“…Minimality in the context of variable order structures was studied in [19] with respect to the third type of cone enlargement. The Lipschitz property from above was used to penalize the Henig-type nondominated efficiency and some necessary optimality conditions were given for it by using the incompatibility between minimality and openness.…”

“…The Lipschitz property from above was used to penalize the Henig-type nondominated efficiency and some necessary optimality conditions were given for it by using the incompatibility between minimality and openness. This will be for us also one of the purposes in the next section, that is obtaining necessary optimality conditions for the Henigtype minimality, but we will not follow the line from [19], but instead, using the conclusion above, we will use a cone separation result and the properties of Liminf It is interesting to record the following result that gives the lower semicontinuity of the lower limit set-valued map.…”

New Notions of Proper Efficiency in Set Optimization with the Set Criterion

“…Moreover, since this notion is based on cones with nonempty interior, these solutions can be characterized through nonlinear scalarization as well, for which no convexity conditions are required (see, for example, [12]). Because of that, Henig proper efficiency has resulted to be a fruitful notion in vector optimization, as it is proved by the numerous papers dealing with it (see, for instance, [6,9,[11][12][13]).…”

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

“…Some well-known generalizations of the mentioned proper efficiency concepts are given by Benson [8], Borwein [9], Borwein and Zhuang [12], Hartley [29], Henig [30], and Hurwicz [33]. These concepts and corresponding generalizations are discussed, among others, by Durea, Florea and Strugariu [15], Eichfelder and Kasimbeyli [16], Gutiérrez et al [28], Hernández, Jiménez and Novo [31], Jahn [34,Ch. 4], and Luc [38,Def.…”

“…) for all k ∈ K , c ∈ C and x ∈ Ω. Finally, letting k = 0 and c = 0 in(15), it followsx ( f (x)) ≤ x ( f (x)) for all x ∈ Ω, which means that x ∈ argmin x∈Ω (x • f )(x). The proof of 1 • is complete.…”