Scalarization approaches for set-valued vector optimization problems and vector variational inequalities

The scalarization approaches of Giannessi, Mastroeni and Pellegrini are extended to the study of set-valued vector optimization problems and set-valued weak vector optimization problems. Some equivalence results among set-valued (scalar) optimization problems, set-valued (scalar) quasi-optimization problems, set-valued vector optimization problems and set-valued weak vector optimization problems are established under the convexity assumption of objective functions. Some examples are provided to illustrate these results. The approaches are furthermore exploited to investigate the set-valued vector variational inequalities and set-valued weak vector variational inequalities, which are different from that suggested by Konnov. Some equivalence relations among set-valued (scalar) variational inequalities, set-valued (scalar) quasi-variational inequalities, set-valued vector variational inequalities and set-valued weak vector variational inequalities are also derived under some suitable conditions.