Characterization of properly optimal elements with variable ordering structures

Extensions of proper efficiency in complex space

“…Proof. Since z 0 is properly efficient solution of (2) in the complex sense of Kuhn-Tucker, there is no h ∈ C n satisfying (6) and (7). Then from theorem 1.7.22 in Duca, 17 there exist t > 0 and v ∈ [ S(g(z 0 )) ] * such that…”

“…Proof. Assume that z 0 is efficient but not properly efficient of (2) in the complex sense of Kuhn-Tucker, then there exists h ∈ C n such that (6) and (7) are satisfied. Without loss of generality of (6), we can assume that…”

“…Since z 0 is a qualified point, then (7) implies the existence of a differentiable arc ( ) ⊂ C n and > 0 such that (0) = z 0 , ∇ (0) = h for some > 0, and g( ( )) ∈ S for 0 ≦ ≦ . Consider a sequence of positive numbers { k } → 0.…”

“…Generalized treatments with applications, in this topic, exist in previous works. [6][7][8][9][10] On the other hand, complex mathematical programming problems were initially studied by Levinson,11 in a particular linear form. Henceforward, several topics of the optimization theory were extended to complex space (see Elbrolosy 12 for example).…”

Henig proper efficiency in vector optimization with variable ordering structure

“…Proper efficiency in variable ordering structure (VOS, for short) setting was studied in [9] and [8] in the framework defined in the book [7]. In this paper we are interested in Henig proper efficiency in VOS setting, but using the paradigm introduced in [6], that is the case where the order is expressed by means of a set-valued map acting between the same spaces as the objective mapping.…”