On Pair Invexity in Optimization — A Coupled Mapping Generalization of Invexity

This paper introduces a Pair Invexity framework based on coupled mappings for constrainedoptimization problems. The framework is built upon the coupled mapping;P (x, u) = Ψ(η1(x, u), η2(x, u)),and Pair Invexity is defined throughf (x) − f (u) ≥ ∇f (u)T P (x, u), ∀x, u ∈ X.The proposed framework is developed for both objective and constraint functions in constrainedoptimization. Under Pair Invexity assumptions, Karush-Kuhn-Tucker (KKT) conditions, suffi-ciency and induced sufficiency theorems are established. Limiting cases and relationship struc-tures are also investigated, showing that classical invexity appears as a special realization ofthe proposed coupled mapping framework.