In this paper, we give some new generalized convexities with the tool-right upper-Dini-derivative which is an extension of directional derivative. Next, we establish not only Karush--Kuhn--Tucker necessary but also sufficient optimality conditions for mathematical programming involving new generalized convex functions. In the end, weak, strong and converse duality results are proved to relate weak Pareto (efficient) solutions of the multi-objective programming problems (VP), (MVD) and (MWD).