Relationships between pre-invex concepts

It is well known that some properties of convex optimization problems, especially sufficient conditions for optimality, and duality conditions, are preserved when convexity is weakened to pre-invexity. However, there are several distinct pre-invex concepts-B-vex, B-pre-invex, pre-univex, (p,r)-pre-invex-and these are not equivalent. The pre-invexity concepts mentioned above have also some properties which are characteristic and invariant for functions belonging to these classes of functions.