*(English)*Zbl 1201.90179

The authors extend the notion of $\varphi $-strict minimizer to set-valued maps, in such a way that the concept of a $\varphi $-strict minimizer, presented in *E. M. Bednarczuk* [Optimization 53, No. 5–6, 455–474 (2004; Zbl 1153.90529)] for vector functions, as well as the notion of a minimizer of order one introduced for set-valued maps in *G. P. Crespi, I. Ginchev* and *M. Rocca* [Math. Methods Oper. Res. 63, No. 1, 87–106 (2006; Zbl 1103.90089)] are generalized in a unified manner.

A structure theorem is proved for a vector-valued function which states that a point is a $\varphi $-strict minimizer for a vector valued function $f$ over a set if and only if the point is a $\varphi $-strict minimizer for a family of scalar functions and sets, each of these functions being the composition of $f$ with a positive, continuous and linear functional, and the family of sets, a covering of the initial set.

Different kinds of strict minimizers for the set-valued problem are characterized through different kinds of strict minimizers for an associated scalarized problem. Finally, several optimality conditions are established for strict minimizers of order one. A characterization is given for a global minimizer through the radial derivative. Comparisons with other results are made, and some illustrative examples are provide.

##### MSC:

90C29 | Multi-objective programming; goal programming |

90C46 | Optimality conditions, duality |

90C31 | Sensitivity, stability, parametric optimization |