Second order monotonicities and second order variational-like inequality problems

The purpose of this paper is to introduce the concept of generalized second order invex monotonicities and investigate the relations between generalized second order invex functions with the generalized second order invex monotonicities of their gradient functions. We also introduce the notion of second order variational-like inequality problem and explore the conditions for existence and uniqueness of their solutions. Furthermore, we relate the solution of a mathematical programming problem involving second order invex function with the solution of second order variational-like inequality problem. Also, we provide examples to verify our results.