Kuhn-Tucker multipliers and nonsmooth programs

In this paper we consider a nonconvex and nondifferentiable (but locally Lipschitz) program with inequality constraints. We study the question of existence of bounded Kuhn-Tucker multipliers for this program in not assuming that a constraint qualification such as Hiriart-Urruty’s condition (U) is satisfied. The present work may be viewed as a complement of the investigations concerned with optimality conditions in nonsmooth optimization by Clarke and Hiriart-Urruty. Our results are shown, when specialized to the case of convexity, to yield naturally a complete characterization of optimality of Ben-Israel, Ben-Tal and Zlobec.

Our approach here is new and enables us to treat the general class of locally Lipschitz problems. The newness of the approach is demonstrated to be effective for the convex case.