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 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 ¿ = X = a Hilbert space. Connections are made with a recently formulated classification scheme for singular and nonsingular extremals.