  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 aslo 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, it is set by the trust-region method) or one fifth of
that 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.

Optimization and Control;Numerical Analysis;Dynamical Systems;Numerical Analysis