  Recently, a new Lagrange multiplier approach was introduced by Cheng, Liu and
Shen in \cite{cheng2020new}, which has been broadly used to solve various
challenging phase field problems. To design original energy stable schemes,
they have to solve a nonlinear algebraic equation to determine the introduced
Lagrange multiplier, which can be computationally expensive, especially for
large-scale and long-time simulations involving complex nonlinear terms. This
paper presents an essential improved technique to modify this issue, which can
be seen as a semi-implicit combined Lagrange multiplier approach. In general,
the new constructed schemes keep all the advantages of the Lagrange multiplier
method and significantly reduce the computation costs. Besides, the new
proposed BDF2 scheme dissipates the original energy, as opposed to a modified
energy for the classical Lagrange multiplier approach in \cite{cheng2020new}.
We further construct high-order BDF$k$ schemes based on the new proposed
approach. In addition, we establish a general framework for extending our
constructed method to dissipative systems. Finally several examples have been
presented to demonstrate the effectiveness of the proposed approach.
