Emergence of Sufficiency in Optimization: A Zulfia Mapping Approach

Any feasible direction mapping admits the Zulfia decompositionZ(x, u) = η(x, u) + ∆(x, u), (1)where η encodes structural invariance and ∆ is the remainder. The purpose is toshow that sufficiency of optimality emerges from structure, without convexity orclassical invexity. For the constrained problem min{f (x) : gi(x) ≤ 0}, define ∆-energy ∆E(x, x∗) := f (x) − fZ (x) with fZ generated by η. If fZ is Zulfia-invexw.r.t. η at x∗ with ∇fZ (x∗) = 0, thenx∗ is globally optimal ⇐⇒ ∆E(x, x∗) ≥ 0, ∀x feasible. (2)When ∆ ≡ 0, the framework recovers Hanson’s theorem and KKT sufficiency.Optimality is interpreted as η active, ∆ silent, giving a new view for nonconvexproblems.