Generalized convex functions and vector variational inequalities

In this paper, (α, φ, Q)-invexity is introduced, where α:X ×X → int Rm+, φ:X ×X →X, X is a Banach space, Q is a convex cone of Rm. This unifies the properties of many classes of functions, such as Q-convexity, pseudo-linearity, representation condition, null space condition, and V-invexity. A generalized vector variational inequality is considered, and its equivalence with a multi-objective programming problem is discussed using (α, φ, Q)-invexity. An existence theorem for the solution of a generalized vector variational inequality is proved. Some applications of (α, φ, Q)-invexity to multi-objective programming problems and to a special kind of generalized vector variational inequality are given.