# American Institute of Mathematical Sciences

April  2019, 15(2): 791-815. doi: 10.3934/jimo.2018071

## Henig proper efficiency in vector optimization with variable ordering structure

 1 Faculty of Mathematics, "Alexandru Ioan Cuza" University, Bd. Carol Ⅰ, nr. 11, Iaşi, 700506, Romania 2 "Octav Mayer" Institute of Mathematics of the Romanian Academy, Bd. Carol Ⅰ, nr. 8, Iaşi, 700505, Romania 3 Department of Mathematics, "Gh. Asachi" Technical University, Bd. Carol Ⅰ, nr. 11, Iaşi, 700506, Romania

* Corresponding author: Marius Durea

Received  October 2016 Revised  March 2018 Published  June 2018

Fund Project: The work of the authors was supported by a grant of Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-Ⅲ-P4-ID-PCE-2016-0188, within PNCDI Ⅲ

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.

Citation: Marius Durea, Elena-Andreea Florea, Radu Strugariu. Henig proper efficiency in vector optimization with variable ordering structure. Journal of Industrial & Management Optimization, 2019, 15 (2) : 791-815. doi: 10.3934/jimo.2018071
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