This paper considers an explicit continuation method and the trust-region updating strategy for the unconstrained optimization problem. Moreover, in order to improve its computational efficiency and robustness, the new method uses the switching preconditioning technique. In the well-conditioned phase, the new method uses the L-BFGS method as the preconditioning technique in order to improve its computational efficiency. Otherwise, the new method uses the inverse of the Hessian matrix as the pre-conditioner in order to improve its robustness. Numerical results also show that the new method is more robust and faster than the traditional optimization method such as the trust-region method and the line search method. The computational time of the new method is about one percent of that of the trust-region method (the subroutine fminunc.m of the MATLAB2019a environment is set by the trust-region method) or one fifth of that of the line search method (fminunc.m is set by the quasi-Newton method) for the large-scale problem. Finally, the global convergence analysis of the new method is also given.

本文考虑了无约束优化问题的显式连续方法和信任区域更新策略。此外，为了提高其计算效率和鲁棒性，新方法使用了开关预处理技术。在条件良好的阶段，该新方法使用L-BFGS方法作为预处理技术，以提高其计算效率。否则，新方法使用Hessian矩阵的逆作为预处理器，以提高其鲁棒性。数值结果还表明，与传统的优化方法如信任区域法和线搜索法相比，新方法具有更强的鲁棒性和更快的速度。新方法的计算时间约为信任区域方法（子例程fminunc）的计算时间的百分之一。对于大型问题，MATLAB2019a环境的m是通过信任区域方法设置的，或者是线搜索方法的五分之一（fminunc.m是通过拟牛顿方法设置的）。最后，给出了新方法的全局收敛性分析。