The author considers a finite-dimensional nonlinear programming problem of standard form, i.e. having a finite number of equality and inequality constraints. All the functions appearing in the problem are assumed to be semismooth in the sense of \textit{R. Mifflin} [ibid. 15, 959-972 (1977; Zbl 0376.90081)]. The main theorem provides first- and second-order necessary optimality conditions which are stated in terms of the generalized gradients of F. H. Clarke and the upper second-order directional derivatives introduced by the author in his earlier paper [Second-order directional derivatives for nonsmooth functions. J. Math. Anal. Appl. (to appear)]. These conditions do not require any constraint qualification. If all the functions are of class \(C^ 2\), then the theorem reduces to Theorem 3.2 of \textit{A. Ben-Tal} [J. Optimization Theory Appl. 31, 143-165 (1980; Zbl 0416.90062)]. The author also gives an example showing that the upper second-order directional derivative appearing in the main theorem cannot be replaced by the lower one.