The New Retrospective Trust-region Algorithm For Solving Unconstrained Optimization Problem

Trust-region method is a very reliable and efficient method for solving the general unconstrained optimization problems and it's special solution,thus this method has gained widely attention and has been used in many problems.The traditional algorithm for unconstrained problems will finally show that the trust-region radius be larger than a positive constant which is different from the new method.In this paper,I propose a new trust-region algorithm in which the trust-region will play the role,that means the radius will converge to zero.In the meanwhile,you can find the update of the radius in this algorithm is performed depending on the performance of how close of the value of the objective function and the value which is retrospectively predicted by the current model.We show that the new algorithm can be global convergent to the first-order critical points under classical assumptions and super-linear convergence is also proved under certain conditions.Numerical experiments can show that this algorithm is efficient for small size problems and preliminary examples on CUTEst problems indicate this algorithm is competitive with the new trust-region method.