Generalized Invex Functions (Φ-Univexity) in Mathematical Programming

This paper generalizes the theory of invex functions by introducing an operator-based framework termed Φ-Univexity (Φ-Univexity). Φ-Univex functions are de-fined through an increasing transformation operator and a generalized directionalmapping, providing a unified inequality structure that encompasses convex, invex,pseudo-invex, quasi-invex, hybrid-invex, and ratio-invex functions as special cases.The framework explicitly accommodates non-smooth optimization via subgradientsand yields sufficient global optimality conditions under standard constraint quali-fications. The proposed generalization offers a coherent and flexible foundation formodern optimization problems arising in engineering, physics, and applied sciences.