基于Newton-PCG算法的实现探讨

Research on implementation of Newton-PCG method

理论上,Newton-PCG算法适于求解大规模无约束优化问题,并且Newton-PCG算法优于牛顿法。为了使Newton-PCG算法能更好地用于科学计算,对该算法的实现进行了探讨,给出了一个使用Newton-PCG算法求解无约束优化问题的软件包。软件给出了牛顿法和Newton-PCG算法2种求解问题的方法,实验表明:对于绝大多数无约束优化问题,Newton-PCG算法比牛顿法求解时间短,尤其当问题的维数增大时,比率逐渐减小,说明Newton-PCG算法的优势更加明显。因此,它是数值软件库的一个有益的补充。

Theoretically, Newton-PCG method is fit for large scale unconstrained optimization problems, and it is more efficient than Newton method. A software package is presented for finding the unconstrained minimizer of a nonlinear function of n variables with Newton-PCG method. The software allows the user to choose between Newton-PCG method and Newton method. In the experiment, the CPU times spent by Newton-PCG method is less than that by Newton method at the same dimension for most of unconstrained optimization problems; and the ratio of the run-time become quite small with the increasing of the problem＇ s dimension. The result shows that Newton-PCG method is very efficient. Therefore, it could be a useful supplement to numerical libraries.