Zulfia Invexity: Structural Origin and Emergence of Invexity

Invexity generalizes convexity by replacing linear displacement with an admissible direction map-
ping, ensuring global optimality of stationary points beyond convex settings. However, the admissible
direction is externally specified, and its structural origin remains unexplained.
In this paper, we introduce Zulfia Invexity as a source-generative framework in which admissible
directions are derived from an underlying structural object associated with the function. A residual
formulation
R(x, u) = f (x) − f (u) − ∇f (u) · η(x, u) ≥ 0
is developed, providing an intrinsic certificate of optimality and yielding sufficiency results without
restrictive geometric assumptions.
We establish that invexity arises as a directional consequence of this structure and demonstrate that
Zulfia Invexity defines a broader class of functions for which global optimality holds. Structural com-
parisons, dominance results, and applications to failure and survival models are presented, providing
a unified perspective in which optimality is generated rather than imposed.