On Pair Invexity and Compromise Duality in Mathematical Programming

Classical invexity ensures sufficiency of the Karush–Kuhn–Tucker (KKT) conditionsfor nonlinear programming, but it relies on a single mapping η(x, u) and fails when feasibledirections depend on interacting components. In this paper we introduce Pair Invexity, ageneralization defined via a coupled mapping P : X × X → Rn. A differentiable functionf : X → R is Pair Invex at u ∈ X with respect to P iff (x) − f (u) ≥ ∇f (u)T P (x, u), ∀x ∈ X, (1)where P (x, u) = Ψ(η1(x, u), η2(x, u)) combines two directional mappings η1, η2 throughΨ. Under Pair Invexity, we prove that the KKT conditions are sufficient for global opti-mality and establish a single duality theorem that provides both a valid lower bound andzero duality gap at optimality. The framework unifies and extends invexity, Φ-invexity, andadditive invexity, and is illustrated with nonconvex examples where classical invexity fails.