Convergence properties of the regularized Newton method for the unconstrained nonconvex optimization

The regularized Newton method (RNM) is one of the efficient solution methods for the unconstrained convex optimization. It is well-known that the RNM has good convergence properties as compared to the steepest descent method and the pure Newton’s method. For example, Li, Fukushima, Qi and Yamashita showed that the RNM has a quadratic rate of convergence under the local error bound condition. Recently, Polyak showed that the global complexity bound of the RNM, which is the first iteration k such that krf(xk)k · , is O(−4), where f is the objective function and is a given positive constant. In this paper, we consider the RNM for the unconstrained “nonconvex ” optimization. We show that the RNM has the following properties. (a) The RNM has a global convergence property under appropriate conditions. (b) The global complexity bound of the RNM is O(−2) if r2f is Lipschitz continuous on a certain compact set. (c) The RNM has a superlinear rate of convergence under the local error bound condition