An unified scalarization and Ekeland’s variational principle for partial set order

This paper is devoted to the study of scalarization and Ekeland’s variational principle for a partial set order relation, which is defined by Minkowski difference. Firstly, a kind of scalarization function with the properties of order representing and order preserving is introduced, and it is used to establish an unified scheme for the minimal element of set-valued maps; As two special cases of the unified scheme, an Oriented distance-type function and a Gerstewitz-type function are provided to illustrate the effectiveness of the scheme. Secondly, an Ekeland’s variational principle related to partial set order relation is formulated under the condition of dynamical closeness. Finally, the obtained results are applied to examine the optimality and existence for an uncertain multi-objective optimization problem.