Regularized Newton method for unconstrained convex optimization

We introduce the regularized Newton method (rnm) for unconstrained convex optimization. For any convex function, with a bounded optimal set, the rnm generates a sequence that converges to the optimal set from any starting point. Moreover the rnm requires neither strong convexity nor smoothness properties in the entire space. If the function is strongly convex and smooth enough in the neighborhood of the solution then the rnm sequence converges to the unique solution with asymptotic quadratic rate. We characterized the neighborhood of the solution where the quadratic rate occurs.