A variant of trust-region methods for unconstrained optimization

In traditional trust-region methods, one, in practice, always employs quadratic model or conic model as the local approximation of the objective function, and there are lots of theoretical results and ripe algorithms. In this paper, we develop a practical trust-region algorithm with a linear model for unconstrained optimization problems. In particular, we combine a special weighted norm with the linear model so that the subproblem contains the information of Hessian matrix of the objective function, which successfully overcome the drawbacks of linear model, and we further complete the trust-region methods with three main types of models, namely, linear model, quadratic model and conic model. We show that the new method preserves the strong global convergence. Moreover, under the linear model, it unveils independently that the line-search algorithms can be viewed as a special case of trust-region methods. Numerical results indicate that the new method is effective and practical.