Journal of Applied Analysis Vol. 4, No. 2 (1998), pp. 245–258 HIGHER–ORDER NECESSARY OPTIMALITY CONDITIONS FOR A NONSMOOTH EXTREMUM PROBLEM M. CASTELLANI Received January 15, 1997 and, in revised form, March 12, 1998 Abstract. By extending the definition of Dini–Hadamard directional derivatives, higher–order necessary optimality conditions for a non- smooth extremum problem are established, in the presence of both equality and inequality constraints. The presence of a regularity as- sumption for the inequality constraints strenghtens the optimality con- dition. 1. Introduction The calculus of first–order necessary optimality conditions for nonsmooth optimization is well–established. In the last years, taking into account the pioneering works of [2] and [9], the theory of higher–order necessary opti- mality conditions for nonsmooth optimization has been the object of much development with the introduction of various kinds of higher–order direc- tional derivatives which enable to avoid the assumptions of higher–order differentiability (see [1, 4, 8, 14, 16–18] and refences therein). Our purpose is to state new higher–order necessary optimality conditions for a nonsmooth extremum problem with a finite number of constraints with or without a 1991 Mathematics Subject Classification. 49K27, 90C48, 90C30. Key words and phrases. Nonsmooth extremum problem, higher–order directional derivatives, higher–order necessary optimality conditions. ISSN 1425-6908 c Heldermann Verlag.