Erratum to: A regularized Newton method without line search for unconstrained optimization

For unconstrained optimization, Newton-type methods have good convergence properties, and areused in practice. The Newton’s method combined with a trust-region method (the TR-Newtonmethod), the cubic regularization of Newton’s method and the regularized Newton method withline search methods are such Newton-type methods. The TR-Newton method and the cubic regu-larization of Newton’s method have to solve nonconvex subproblems at each iteration in order to geta search direction although these methods converge rapidly with fewer function evaluations. Thustheir total computational times may become large. On the other hand, the regularized Newtonmethod with line search methods gets its search direction by only solving linear equations. How-ever, it may evaluate the objective function value many times in a line search step. Therefore, itis significant to construct a solution method whose behavior is similar to the TR-Newton method,and whose subproblems can be solved easily.In this paper, we propose a regularized Newton method without line search. The proposedmethod controls a regularized parameter instead of a step size in order to guarantee the globalconvergence. We demonstrate that it is closely related to the TR-Newton method when the Hessianof the objective function is positive definite. Moreover, it does not solve nonconvex problems butlinear equations as subproblems at each iteration. Thus, the proposed algorithm is regarded as adesired solution method mentioned above. We show that the proposed algorithm has the followingconvergence properties. (a) The proposed algorithm has global convergence under appropriateconditions. (b) It has superlinear rate of convergence under the local error bound condition. (c)Its global complexity bound, which is the first iteration k such that‖∇f(xk)‖ ≤ 2, is O(2−2) whenf is nonconvex, O(2− 53) when f is convex, and O(2−1) when f is strongly convex. Moreover, wereport numerical results that show that the proposed algorithm is competitive with the existingNewton-type methods, and hence it is very promising.