无约束最优化的信赖域BB法

梯度法是求解无约束最优化的一类重要方法.步长选取的好坏与梯度法的数值表现息息相关.注意到BB步长隐含了目标函数的二阶信息,本文将BB法与信赖域方法相结合,利用BB步长的倒数去近似目标函数的Hesse矩阵,同时利用信赖域子问题更加灵活地选取梯度法的步长,给出求解无约束最优化问题的单调和非单调信赖域BB法.在适当的假设条件下,证明了算法的全局收敛性.数值试验表明,与已有的求解无约束优化问题的BB类型的方法相比,非单调信赖域BB法中e_k=‖x_k-x~*‖的下降呈现更明显的阶梯状和单调性,因此收敛速度更快.

TRUST REGION BB METHODS FOR UNCONSTRAINED MINIMIZATION

It is well known that the numerical performances of the gradient methods are closely dependent on how to select the step-lengths in iterations. Notice that the BB step-length implies some second-order information of the objective function, by using the BB step in the trust region method, we propose a monotone trust region BB method and its nonmonotone version for unconstrained optimization problems. Our methods use the inverse of the BB step-lengths to approximate the Hesse matrix of objective function and select step-lengths of the gradient method by the trust region subproblems flexibly. Under suitable conditions, the new methods are proved to be globally convergent. Numerical tests show that the measure of error e_k=‖x_k-x*‖ obtained by the nonmonotone trust region BB method can decrease faster in comparison with some existing BB methods.