A Trust-region Method with Two Subproblems and Backtracking Line Search

Unconstrained optimization problems occur frequently in many real world applications such as engineering and scientific computing. Under the trust-region framework, we combine an un- constrained subproblem with a trust-region subproblem, and propose a trust-region method with two subproblems for solving unconstrained optimization. A backtracking line search is carried out if the trust-region trail step fails since there is always a suffcient descent direction for the objective function. The global convergence and the local quadratic convergence rate are proved under standard assump- tions. Numerical results show that this algorithm is reliable and more effcient.