A note on scalar 'generalized' invexity

This paper aims to study how much “generalized” invex properties differ from invexity and to establish whether or not the use of more and more parameters and functionals in the definitions is really effective and helpful. In particular, both smooth and nonsmooth scalar functions are considered. As a conclusion, by means of some equivalence results not necessarily related to invexity, it is proved that several “generalized” invexity properties are actually equivalent to invexity, and that this happens in both the differentiable case and the nondifferentiable one. In other words, the introduction of parameters in defining scalar “generalized” invexity properties does not yield “a priori” any kind of generalization.