This work extends the ones of Hu et al. (2008) and Bai et al. (2013) of a logical Benders approach for globally solving Linear Programs with Complementarity Constraints. By interpreting the logical Benders method as a reversed branch-and-bound method, where the whole exploration procedure starts from the leaf nodes in an enumeration tree, we provide a new framework over which we can combine master problem and cut generation in a single process. We also present an optimization-based sparsification process which makes the cut generation more efficient. Numerical results are presented to show the effectiveness of this unified method. Results are also extended to larger complementarity dimensions, exceeding what the original method has been able to handle.