For a tangent cone A, an extended-real-value function f is said to admit an ‘A upper DSL approximate’ at x if its ‘A directional derivative’ at x is majorized by a difference of lower semicontinuous sublinear functions. By means of such approximates, the authors establish necessary optimality conditions of Fritz John and Kuhn-Tucker type for a nonsmooth, inequality-constrained mathematical program. Optimality conditions involving the quasidifferentials of Demyanov, the upper convex approximates of Pshenichnyi, and the upper DSL approximates of A. Shapiro are among the special cases of these general optimality conditions.
