A Generalized Convex Functions And Vector Optimization Problems In Banach Spaces

In this paper, we extend the scalar-valued B-semipreinvex functions and vector-valued preinvex functions to the cases of vector-valued B-semipreinvex functions in Banach spaces. We investigate the efficient solutions involving B-semipreinvex functions for vector optimization problems, and the results obtained are similiar to some properties of convex functions. At the same time, we also consider a class of Lipschitz vector-valued nonsmooth programming problems (CVOP) in which a constraint qualification is required. In terms of the Ralph vector sub-gradient, we obtain the generalized Kuhn-Tucker type sufficient optimality conditions and saddle point condition for (CVOP). Finally, we formulate a class of generalized Mond-Weir type dual and Wolf type dual and establish some duality theorems for the pair of primal and dual programs under regular B-semipreinvexity assumptions. The results presented in this paper generalize some main results of the existing work.