  This paper considers an explicit continuation method with the trusty
time-stepping scheme and the limited-memory BFGS (L-BFGS) updating formula
(Eptctr) for the linearly constrained optimization problem. At every iteration,
Eptctr only involves three pairs of the inner product of vector and one
matrix-vector product, other than the traditional and representative
optimization method such as the sequential quadratic programming (SQP) or the
latest continuation method such as Ptctr \cite{LLS2020}, which needs to solve a
quadratic programming subproblem (SQP) or a linear system of equations (Ptctr).
Thus, Eptctr can save much more computational time than SQP or Ptctr. Numerical
results also show that the consumed time of EPtctr is about one tenth of that
of Ptctr or one fifteenth to 0.4 percent of that of SQP. Furthermore, Eptctr
can save the storage space of an $(n+m) \times (n+m)$ large-scale matrix, in
comparison to SQP. The required memory of Eptctr is about one fifth of that of
SQP. Finally, we also give the global convergence analysis of the new method
under the standard assumptions.
