Minimax Theorem For Vector-valued Function And Error Bound For Vector Variational Inequality

Minimax theorem for vector-valued function and error bound for weak vectorvariational inequality (WVVI) are studied in this thesis.In Chapter2, similar to the definition of efficient point in vector optimization, theminimax set and maximin set for vector valued function, which contain more than onepoint, are defined. The reputable Sion minimax theorem asserts that the minimax valueand maximin value for real-valued functions are equal under certain conditions, whilethe situation is totally different for vector valued function. Hence, some generalconvexity of vector-valued function and their relationship with minimax set isconsidered. Finally, by using a linear scalarization function and minimax equalities inscalar case, two types of minimax inequalities for vector-valued functions wereestablished under natural quasi cone convex and properly quasi cone convexassumptions. An example was given to illustrate that the result is a generalization of thecorresponding one in reference.In Chapter3, by using a scalarization approach, the equivalence among weakvector variational inequality (WVVI) and scalar variational inequality with set valuedmapping is established under suitable conditions and the relations among their gapfunctions are analyzed. Since the Auslender type gap function is not guaranteed to befinite-valued without the compactness of feasible set, we consider the regularization ofgap function. On the other hand, we obtain another type of gap function in terms ofHiriart-Urruty function, and the dual representation of this gap function is given. Anerror bound for weak vector variational inequality is presented in terms of these gapfunctions.