Revisiting Invex Functions: Explicit Kernel Constructions and Characterizations

An invex function generalizes a convex function in the sense that every stationary point is a global minimizer. Recently, invex functions and their subclasses have attracted attention in signal processing and machine learning. However, verifying invexity is often difficult because its definition involves an unknown function called a kernel function. This paper studies kernel functions associated with invex functions, which have received relatively limited attention in the literature. In particular, we develop several methods for constructing explicit kernel functions and establish a characterization of pseudoconvexity in terms of kernel functions. These results provide constructive tools for proving invexity of new functions and for clarifying their structural properties. We also present examples of nonsmooth, non-pseudoconvex invex functions arising in signal processing.