Zulfia Mapping as a Common Framework for Convexity and Invexity

We propose the Zulfia mapping Z(x, u) = η(x, u)+∆(x, u), with a direction componentη and a correction component ∆, as a common language for convexity, strong convexityand invexity. Rather than assuming a gradient, we ask which vectors s satisfyf (x) − f (u) ≥ ⟨s, Z(x, u)⟩ for all x.Three theorems describe the framework:(1) the classical tangent is forced by the linear part of Z;(2) the size of the direction η controls how fast gradient descent converges;(3) the spread of admissible vectors is bounded by the kink of f .Known results are recovered rather than claimed as new, applications are labeled bystatus, and the limits of the framework are stated before the conclusion