w -Invexity and optimality problem

As the development of a family of preinvex functions, we define the w-invex set and a class of real functions. These are functions with the names w-preinvex, w-strictly preinvex, w-prequasi-invex, w-strictly prequasi-invex, w-semi-strictly prequasi-invex, and w-pre-pseudo-invex. The fundamental characteristics of these functions are thoroughly examined. Several examples are provided here that serve to clarify the concept in this context. Several of these preinvex functions are addressed to investigate the optimization problems further.