Near-subconvexlikeness in vector optimization with set-valued functions

A new class of generalized convex set-valued functions, termed nearly-subco nvexlike functions, is introduced. This class is a generalization of cone-s ubconvexlike maps, nearly-convexlike set-valued functions, and preinvex set -valued functions. Properties for the nearly-subconvexlike functions are de rived and a theorem of the alternative is proved. A Lagrangian multiplier t heorem is established and two scalarization theorems are obtained for vecto r optimization.