This paper studies convergence properties of regularized
Newton methods for minimizing a convex function whose Hessian matrix may be singular everywhere. We show that
if the objective function is LC^2, then the methods possess local quadratic convergence under a local error bound condition without the requirement of isolated nonsingular solutions. By using a backtracking line search, we globalize an inexact regularized Newton method. We show that the unit stepsize is accepted eventually.