Gap Function And Error Bounds For Set-valued Quasi-variational Inequalities

Based on variational inequalities and smooth convex optimization problems, therelationships between quasi-variational inequalities and optimization problems are presented.Firstly, the set-valued quasi-variational inequalities are established in the different space, and thecharacteristics of their solutions are discussed. Secondly, different gap functions are introducedto show a necessary and sufficient condition for existence of solutions. In case of solutions areexist unique of their solutions is given. Thirdly, error bounds and applications of set-valuedquasi-variational inequalities are considered. At last, convex sets are extended to E-convex sets,and the properties of geodesic E-quasiconvex functions and geodesic E-epigraph are studied．Themain contents are as follows.1. In the strongly convex smooth linear norm space, the quasi-variational inequalities forset-valued maps were established. Gap function for set-valued quasi-variational inequalitiesmaps was introduced by an axiomatic approach and the properties of the gap function wereprovided.2. Error bounds for QVI (T, S)and conditions of unique solution are prensented.Applications to generalized Nash equilibrium problems are considered.3. Taking advantage of the new definition of projective operator S（x） in the Banach. weabtain the relationships of the solutions between QVI (T, S)and （S x） and the confident ofthe unique solution for QVI (T, S).4. The definition of geodesic E-quasiconvex functions is established in a geodesic metricspace. Meanwhile, a relation of geodesic E-quasiconvex functions, geodesic E-convex functionsand geodesic E-almostconvex functions is given. Furthermore，the notion of E-Epigraphs isgeneralized to geodesic E-Epigraphs and a characterization of geodesic E-quasiconvex functionsin terms of its geodesic E-Epigraphs is showed.