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

This paper introduces a Pair Invexity framework based on coupled mappings for constrained
optimization problems. The framework is built upon the coupled mapping;
P (x, u) = Ψ(η1(x, u), η2(x, u)),
and Pair Invexity is defined through
f (x) − f (u) ≥ ∇f (u)T P (x, u), ∀x, u ∈ X.
The proposed framework is developed for both objective and constraint functions in constrained
optimization. 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 of
the proposed coupled mapping framework.