A modified trust region algorithm

In this article, we propose an algorithm which solves unconstrained optimization problems by combining the trust region method with the quasi-Newton line search method. During consecutive trust region steps, when the quasi-Newton matrix Bk approximates the Hessian matrix of the objective function at xk well, the algorithm tries a full quasi-Newton step and, if this step is successful, continues trying quasi-Newton steps. On the other hand, when employing the line search method, if the quasi-Newton direction is nearly orthogonal to the gradient of the objective function or the step size becomes too small, the algorithm switches back to trust region steps. Convergence properties of the algorithm are proved, and numerical results are presented.