Infinitely Many Invex Functions Without Convexity

Invex functions form a broader class of differentiable functions than convex functions, ensuring global optimality at stationary points. However, the theoretical properties of invex functions, particularly those that drop convexity, remain underexplored. In this paper, we investigate these properties, providing examples that include both smooth and nonsmooth cases, and demonstrate the existence of infinitely many invex functions that are not necessarily convex. An illustrative example using a characteristic polynomial highlights the critical role of consistency in a 4th-order pairwise comparison matrix in determining invexity. Our findings expand the theoretical understanding of invexity and suggest potential enhancements in optimization techniques, enabling the analysis of a broader class of functions beyond convexity.