Cautious modified Newton method for unconstrained optimization problem

When Newton method is used to solve a nonconvex minimization problem,the Hessian matrix of the objective function at each iterate point must not be positive definite.For this,a cautious modified Newton method is proposed in this paper,where the first and the second information of the objective function at each iterate point are employed to determine a search direction.It is a hybrid method based on the steepest descent method,the Newton method and the existing modified Newton method.Under some mild assumptions,the global convergence theory is established for the devel-oped algorithm.Numerical experiments demonstrate the computational efficiency of the algorithm,particularly in comparison with the existing similar algorithms.