Characterization of generalized FJ and KKT conditions in nonsmooth nonconvex optimization

In this paper, we investigate new generalizations of Fritz John (FJ) and Karush–Kuhn–Tucker (KKT) optimality conditions for nonconvex nonsmooth mathematical programming problems with inequality constraints and a geometric constraint set. After defining generalized FJ and KKT conditions, we provide some alternative-type characterizations for them. We present characterizations of KKT optimality conditions without assuming traditional Constraint Qualification (CQ), invoking strong duality for a sublinear approximation of the problem in question. Such characterizations will be helpful when traditional CQs fail. We present the results with more details for a problem with a single-inequality constraint, and address an application of the derived results in mathematical programming problems with equilibrium constraints. The objective function and constraint functions of the dealt with problem are nonsmooth and we establish our results in terms of the Clarke generalized directional derivatives and generalized gradient. The results of the current paper cover classic optimality conditions existing in the literature and extend the outcomes of Flores-Bazan and Mastroeni (SIAM J Optim 25:647–676, 2015).