  We consider the computations of the action ground state for a rotating
nonlinear Schr\"odinger equation. It reads as a minimization of the action
functional under the Nehari constraint. In the focusing case, we identify an
equivalent formulation of the problem which simplifies the constraint. Based on
it, we propose a normalized gradient flow method with asymptotic Lagrange
multiplier and establish the energy-decaying property. Popular optimization
methods are also applied to gain more efficiency. In the defocusing case, we
prove that the ground state can be obtained by the unconstrained minimization.
Then the direct gradient flow method and unconstrained optimization methods are
applied. Numerical experiments show the convergence and accuracy of the
proposed methods in both cases, and comparisons on the efficiency are
discussed. Finally, the relation between the action and the energy ground
states are numerically investigated.
