Revisiting Invex Functions: Explicit Kernel Constructions and Applications

TL;DR

This work addresses the challenge of proving invexity by constructing explicit kernel functions and develops systematic methods to generate kernels for a broad class of functions. It clarifies the relationships between invex, pseudoconvex, and quasiconvex notions in both smooth and nonsmooth settings, and provides concrete templates—convex transformations, fractional programming, concave-convex composites, separable sums, and perturbations—for building invex functions with explicit kernels. The results yield simpler, constructive proofs of invexity for nonsmooth, non-pseudoconvex regularizers encountered in signal processing and learning, with implications for constrained optimization and algorithm design. By enabling explicit kernel-based proofs and illustrating practical examples, the paper enhances the applicability of invex theory to real-world problems and lays a foundation for kernel-driven optimization methods.

Abstract

An invex function generalizes a convex function in the sense that every stationary point is a global minimizer. Recently, invex functions and related concepts have attracted attention in signal processing and machine learning. However, proving that a function is invex is not straightforward, because the definition involves an unknown function called a kernel function. This paper develops several methods for constructing explicit kernel functions, which have been missing from the literature. These methods support proving invexity of new functions, and they would also be useful in the development of optimization algorithms for invex problems. We also clarify connections to pseudoconvex functions and present examples of nonsmooth, non-pseudoconvex invex functions that arise in signal processing.

Figures (6)

• Figure 1: Graphs of an invex function $f(x)=x^2/(x^2+1)$ (solid line) and its tangent curve at $x=2$: $f(2)+f'(2)\eta(2,x)=4/5+4(x-2)/(x^2+1)^2$ (dashed line). The invexity and a kernel function of $f$ is given by Corollary \ref{['crl_frac']}.
• Figure 2: Contour lines of $f(x,y)=x-y^2$, which has no stationary points, and hence is invex. A convex box constraint (dashed line) can destroy the invexity, i.e., it can generate a non-global local minimum.
• Figure 3: The Venn diagram of convex, pseudoconvex, quasiconvex, and invex functions under the assumption of locally Lipschitz continuity. Ex. A-C are shown in Figure \ref{['fig_ex']}. Note that, under the assumption of local Lipschitz continuity, the class of pseudoconvex functions coincides with the intersection of the classes of invex and quasiconvex functions (Theorem \ref{['thm_quasi_invex']}).
• Figure 4: Examples of functions shown in Figure \ref{['fig_venn']}. (a) is pseudoconvex. (b) is invex but not pseudoconvex (has nonconvex sublevel sets). (c) is quasiconvex but not invex (has stationary points that are not global minimizers).
• Figure 8: Examples of nonconvex pseudoconvex functions generated by fraction, composition, and their sum.

Theorems & Definitions (45)

• Definition 1: smooth invex functions (e.g., mishra08)
• Proposition 2.1: mishra08
• proof
• Definition 2: Clarke subdifferential and stationarity
• Definition 3: nonsmooth invex functions (e.g., mishra08)
• Proposition 2.2: mishra08
• proof
• Remark 1
• Proposition 2.3: Extension of mishra08 to the nonsmooth case
• proof