Necessary Optimality Conditions in Terms of Convexificators in Lipschitz Optimization

This study is devoted to constraint qualifications and Kuhn-Tucker type necessary optimality conditions for nonsmooth optimization problems involving locally Lipschitz functions. The main tool of the study is the concept of convexificators. First, the case of a minimization problem in the presence of an arbitrary set constraint is considered by using the contingent cone and the adjacent cone to the constraint set. Then, in the case of a minimization problem with inequality constraints, Abadie type constraint qualifications and several other qualifications are proposed; Kuhn-Tucker type necessary optimality conditions are derived under the qualifications.Communicated by S. SchaibleThe authors thank the referees for bringing to their attention some papers closely related to this study and for helpful comments and constructive suggestions that have greatly improved the original version of the paper. Further, they are indebted to Professors H. W. Sun and F. Y. Lu, who suggested an example for this paper. The first author thanks S. Schaible for encouragement during this research.