Optimality Conditions for Lower Semi-continuous Functions*

Let f be a lower semi-continuous and bounded below function from a Banach space X into y`,q`x where X is assumed to admit a Lipschitz smooth ‘‘bump-function.’’ Generalizing results of Chaney, we study optimality conditions for xgX to be a local minimum point of f. These conditions are described in terms of generalized Chaney’s subdifferentials and second-order derivatives