Abstract:We have proposed firstly the vector variational inequality with variable partially ordering and the vector complementarity problem in the world. These mathematical models have become a new research direction in optimization theory field. We establish the Minimax inequalities for vector valued and set valued mappings. These work answer partially an open question by Ferro, an Italy mathematician. We obtain an unified necessary and sufficient optimality condition. For set valued optimization problems, by using original contingint derivative of set valued maps, we establish the vector versions and set valued versions of Ekeland’s variational principle, Caristi Kirt fixed point theorem, Flower Petal theorem and Drop theorem and prove the equivalence among these theorems. We establish firstly the alternative theorem of set valued optimization problems. Due to these research works we win "ISI Citation Classic Award" and First Class of National Science Award of CAS.