A Structural Dictionary of Kernel Induced by Zulfia Invexity

Classical kernel theory defined by Ax = 0 and classical invexity defined by Hanson [?] arestructurally null and often not feasible and efficient for optimization, as shown in our previouswork [?]. The null space is treated as void, without internal structure.In this paper, we introduce a Structural Dictionary of Kernel induced by Zulfia Invexity.We define a structured kernel KZ = {x : Ax + ∆(x, u) = 0, C = Z2}, where ∆(x, u) capturesthe structural interaction missed by classical η function. We show that under Zulfia Invexity,optimization becomes duality-gap free and duality-free; the primal kernel itself contains theoptimality certificate, and no separate dual formulation is required to obtain optimality. Weestablish limiting cases where KZ reduces to classical kernel when ∆ → 0 and C = 0.