The Zulfia Kernel: Defect Signatures and Stratification of Vector-Valued Mappings

Let Z = η + ∆ be a mapping with values in a finite-dimensional inner productspace E = V1 ⊕ · · · ⊕ Vm (orthogonal sum). For each pair (x, u) we define the defectsignature σ(x, u) = {i : PiZ(x, u)̸ = 0}, where Pi is the orthogonal projector onto Vi.The Zulfia kernel is the set of pairs with empty signature, and the signatures partitionthe domain into strata. We prove two results:(i) for Z(x, u) = A(x − u) with A a block matrix, a signature I occurs if and onlyif the set of blocks [m] \ I is closed with respect to the rank function of A, andthe corresponding stratum is an open dense subset of a subspace of dimensionn − rank A[m]\I (Theorem 3.2);(ii) for an η-invex function f , the classical Hanson invexity inequality reduces to asum over the components in σ(x, u), which we call the Zulfia-invexity inequal-ity (Theorem 4.2), and which yields a partial stationarity criterion and a kernelinvariance property.