Nonsmooth Milyutin-Dubovitskii Theory and Clarke's Tangent Cone

Abstract

In optimization theory the idea of approximating nonconvex sets by convex cones which satisfy an abstract condition—the Intersection Principle—is due to Milyutin and Dubovitskii. This approach has been successfully applied to optimization problems with differentiable data. The validity of the Intersection Principle for the Clarke tangent cone, which is the apposite approximant for nonsmooth constraints, is established via an intersection theorem which relates the Clarke tangent cone of an intersection of sets to the intersection of the individual Clarke tangent cones, provided the latter are inseparable. This allows an extension of the Milyutin-Dubovitskii approach to nonsmooth optimization theory. A multiplier rule, of the type obtained by Clarke and Hiriart-Urruty, for a nonsmooth inequality constrained problem, is thereby encompassed within Milyutin-Dubovitskii theory.