Second-order optimality conditions for problems with C1 data

In this paper we obtain second-order optimality conditions of Karush–Kuhn–Tucker type and Fritz John one for a problem with inequality constraints and a set constraint in nonsmooth settings using second-order directional derivatives. In the necessary conditions we suppose that the objective function and the active constraints are continuously differentiable, but their gradients are not necessarily locally Lipschitz. In the sufficient conditions for a global minimum we assume that the objective function is differentiable at and second-order pseudoconvex at , a notion introduced by the authors [I. Ginchev, V.I. Ivanov, Higher-order pseudoconvex functions, in: I.V. Konnov, D.T. Luc, A.M. Rubinov (Eds.), Generalized Convexity and Related Topics, in: Lecture Notes in Econom. and Math. Systems, vol. 583, Springer, 2007, pp. 247–264], the constraints are both differentiable and quasiconvex at . In the sufficient conditions for an isolated local minimum of order two we suppose that the problem belongs to the class C1,1. We show that they do not hold for C1 problems, which are not C1,1 ones. At last a new notion parabolic local minimum is defined and it is applied to extend the sufficient conditions for an isolated local minimum from problems with C1,1 data to problems with C1 one.