First and second-order optimality conditions in a vector optimization problem with an uncertain preference relation

In this paper we consider a vector optimization problem with an uncertain preference relation. The uncertainty arises from the fact that only a family $\mathcal{P}$ of preference relations is known, one of which is actually realized, while the concrete realization is unknown. As an optimal solution of this problem, we propose to consider such admissible solutions that provide the objective mapping with the guaranteed (weak, strong) maximum relative to the family $\mathcal{P}$, i.e., those that provide the objective mapping with (weak, strong) maximum for any concrete realization of the preference relation from $\mathcal{P}$. Under some assumptions regarding the family $\mathcal{P}$, a theorem on the scalarization of this vector optimization problem by means of difference sublinear functions is proved. Then this theorem is used for deriving first- and second-order necessary and sufficient conditions for admissible solutions that provide the objective mapping with the guaranteed $\mathcal{P}$-maximum. The necessary conditions are established for solutions that provide the guaranteed weak $\mathcal{P}$-maximum, while the sufficient conditions are established for solutions that provide the guaranteed strong $\mathcal{P}$-maximum.