A Class of Generalized Zulfia Invex Functions with In-Built Structural Emergence

Convexity was extended to invexity by replacing the fixed direction with a mapping,and Zulfia invexity further structured admissible mappings. However, a single admissiblemapping is insufficient to capture regime-dependent behavior.We introduce generalized Zulfia invexity by considering a class of admissible mappingsZi(x, u) = η(x, u) + ∆(x, u) + I(x, u) + Si(x, u), i ∈ I,together with the conditionf (x) − f (u) ≥ ∇f (u)T Zi(x, u), ∀x, u ∈ X.This formulation incorporates in-built structural emergence and allows admissible di-rections to vary across regimes. Numerical justification and a KKT-type sufficiency resultshow that the proposed framework extends admissible direction-based optimization beyondsingle-mapping structures.