  In Part I of this paper, we introduced a 2D eigenvalue problem (2DEVP) and
presented theoretical results of the 2DEVP and its intrinsic connetion with the
eigenvalue optimizations. In this part, we devise a Rayleigh quotient iteration
(RQI)-like algorithm, 2DRQI in short, for computing a 2D-eigentriplet of the
2DEVP. The 2DRQI performs $2\times$ to $8\times$ faster than the existing
algorithms for large scale eigenvalue optimizations arising from the minmax of
Rayleigh quotients and the distance to instability of a stable matrix.
