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

A relaxed form of Newton's method is analyzed for the problem, minΩF, with Ω a convex subset of a real Banach space X, and F:X → R1 twice differentiable in Fréchet'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 Ω = X = a Hilbert space. Connections are made with a recently formulated classification scheme for singular and nonsingular extremals.