Coderivative necessary optimality conditions for sharp and robust efficiencies in vector optimization with variable ordering structure

Henig proper efficiency in vector optimization with variable ordering structure

“…This newly introduced concept clearly covers the case of Henig proper efficiency described before provided that the base of K(x) with x close to x is a.c., and, moreover, this is in the same line with other notions of efficiency developed in [6] and [10]. We give a short account on these concepts, since we investigate in the sequel some interesting links between all these efficiencies.…”

“…Remark that the previous theorem can be formulated for (x, z) ∈ Gr G and (x, w) ∈ Gr K instead of (x, 0) ∈ Gr G ∩ Gr K, but we prefer the present form because of the later use in the paper. Observe that if in Theorem 4.7 we take K (x) := 0 for every x ∈ X, then we obtain Theorem 4.2 from [14], which is an openness result for F + G. Furthermore, in case G (x) := 0 for every x ∈ X, then we obtain Theorem 5.2 from [10].…”

“…Then, the extended notion of Henig efficiency is investigated in detail from the point of view of necessary optimality conditions: we show that using a Clarke-type penalization technique (see [2,Proposition 2.4.3]) we can reduce a Henig proper efficient point in VOS case of constrained problems to a notion of robust efficiency explored in [10]. In turn, as shown in [10], this is not compatible with the weak openness of the sum between new (penalized) objective mapping and the ordering multifunction. Therefore, in order to get necessary optimality conditions for VOS Henig efficiency following this argument, we firstly have to get sufficient conditions for the weak openness involving three applications: the initial objective set-valued map, the penalization term and the ordering set-valued map.…”

“…As a natural generalization of the above notions, the following concepts of robustness were introduced in [10].…”

“…Recall (see [10]) that the pair (F, K) is weakly open at (x, y) , or F + K is weakly open at (x, y) ∈ Gr(F + K) if for every neighborhood U of x, there exists a neighborhood V of y such that V ⊂ F (U )+K (U ). Motivated by the incompatibility proved in [10] between weak openness and the robust efficiency, and by the fact that the penalty results presented before involve three mappings, in this section we obtain sufficient conditions for the weak openness of pairs of the form (F + G, K) , where G : X ⇒ Y is a multifunction.…”