Higher-order Optimality Conditions of Strict Local Minima in Optimization Problems

In this paper, a nonsmooth optimization problem with generalized inequality constraints and a set constraint is studied. Some necessary optimality conditions for a point to be a strict local minimizer are given by using higher-order Studniarski derivatives and the contingent cone to the constraint set. In the same line, some sufficient optimality conditions are developed in terms of the lower Studniarski derivative of the Lagrangian function when the initial space is finite dimensional. These optimality conditions are important in the study of the convergence of numerical procedures and in stability analysis for optimization problems.