Second-order necessary conditions in locally Lipschitz optimization with inequality constraints

In this paper we obtain primal and dual second-order necessary conditions for a scalar optimization problem with inequality constraints in a nonsmooth setting using second-order directional derivatives. We generalize results of Ben-Tal (J Optim Theory Appl 31(2):143–165, 1980 ) given for scalar problems with twice differentiable data, of Ginchev and Ivanov (J Math Anal Appl 34:646–657, 2008 ) and of Ivanov (J Math Anal Appl 356:30–41, 2009 ) given for scalar problems with continuously differentiable data, and of Ivanov (Optim Lett 4:597–608, 2010 ) given for scalar problems with locally Lipschitz and second-order Hadamard differentiable data. We suppose that the objective function and the active constraints are only locally Lipschitz in the primal necessary conditions and only locally Lipschitz, regular in the sense of Clarke and Gâteaux differentiable at the extremum point in the dual necessary conditions. To illustrate the efficiency of our results, we provide some examples.