On D-preinvex-type Functions

Convexity and generalized convexity play a central role in mathematical economics, engineering, and optimization theory. Therefore, the research on convexity and generalized convexity is one of the most important aspects in mathematical programming (see [1–4, 6–11] and the references therein). Weir and Mond [7] and Weir and Jeyakumar [6] introduced the definition of preinvexity for the scalar function f : X ⊂ Rn → R. Recently, Yang and Li [9] gave some properties of preinvex function under Condition C. Yang and Li [9] introduced the definitions of strict preinvexity and semistrict preinvexity for the scalar function f : X ⊂ Rn → R and discussed the relationships among preinvexity, strictly preinvexity, and semistrictly preinvexity for the scalar functions. Yang [8] also obtained some properties of semistrictly convex function and discussed the interrelations among convex function, semistrictly convex function, and strictly convex function. Throughout this paper, we will use the following assumptions. Let X be a real topological vector space and Y a real locally convex vector space, let S⊂ X be a nonempty subset, letD ⊂ Y be a nonempty pointed closed convex cone, Y∗ is the dual space of Y , equipped with the weak∗ topology. The dual cone D∗ of cone D is defined by D∗ = { f ∈ Y∗ : f (y)= 〈 f , y〉 ≥ 0, ∀y ∈D. (1.1)