Asymptotical good behavior on inequalities with completely approximate K-T concept

We characterize a wide class of regular convex functionals that are asymptotically well behaved on a convex set given by (infinite) inequalities, namely, those restricted functions whose stationary sequences (bounded or not) are minimizing ones. After showing the equivalence with the Kuhn-Tucker type stationarity, we prove that the class of such functions remains unchanged when the Kuhn-Tucker system is completely relaxed. This allows us to proceed for enlarging the scope of convergence of certain penalty (exterior as well as interior) methods including a new exterior penalization for infinite inequalities.