Characterizations of the Lagrange-Karush-Kuhn-Tucker Property

In this note, we revisit the classical first order necessary condition in mathematical programming in infinite dimension. The constraint set being defined by C = g−1(K) where g is a smooth map between Banach spaces, and K a closed convex cone, we show that existence of Lagrange-Karush-Kuhn-Tucker multipliers is equivalent to metric subregularity of the multifunction defining the constraint, and is also equivalent to a generalized Abadie’s qualification condition. These results extend widely previous ones like [10, 11, 12, 16] by removing convexity type assumptions on the data. Key-words: Lagrange and Karush-Kuhn-Tucker multipliers, metric subregularrity, error bounds, Abadie’s qualification condition.