Newton´s method and the goldstein step length rule for constrained minimization

A relaxed form of Newton´s method is analyzed for the problem, min??F, with ?? a convex subset of a real Banach space X, and F:X ?? R1 twice differentiable in Fr??chet´s sense. Feasible directions are obtained by minimizing local quadratic approximations Q to F, and step lengths are determined by Goldstein´s rule. The results established here yield two significant extensions of an earlier theorem of Goldstein for the special case ?? = X = a Hilbert space. Connections are made with a recently formulated classification scheme for singular and nonsingular extremals.