(untitled) (Contemporary Mathematics, 2025)
In this article, we generalize the notion of fuzzy (F) invex sets and preinvex functions, and introduce the ideas of intuitionistic fuzzy (IF) invex sets and preinvex functions. These functions are also generalized forms of F-convex functions when IF-invex functions are differentiable. With the support of some results, we explain the definitions like invex, incave, and quasi-IF-sets differently and IF-invex sets. Similarly, different preinvex functions are defined with the help of IF-mappings (IFMs) on the invex set and characterize some properties. Finally, new proofs of some known important results are offered. These important outcomes are based on optimization in which the minimum of the IF-preinvex function is distinguished by IF-variational-like (VL) inequalities and VL-inequalities. These results define the relationship of IF-preinvex functions with IF-mixed-VL-inequalities. These concepts are more important than F-nonconvexity because the F-theory depend upon the membership function and IF-theory contains the membership and non-membership functions. Some of the non-trivial examples are also provided to validate the given results.