  We propose and analyze a new numerical method for computing the ground state
of the modified Gross-Pitaevskii equation for modeling the Bose-Einstein
condensate with a higher order interaction by adapting the density function
formulation and the accelerated projected gradient method. By reformulating the
energy functional $E(\phi)$ with $\phi$, the wave function, in terms of the
density $\rho=|\phi|^2$, the original non-convex minimization problem for
defining the ground state is then reformulated to a convex minimization
problem. In order to overcome the semi-smoothness of the function $\sqrt{\rho}$
in the kinetic energy part, a regularization is introduced with a small
parameter $0<\varepsilon\ll1$. Convergence of the regularization is established
when $\varepsilon\to0$. The regularized convex optimization problem is
discretized by the second order finite difference method. The convergence rates
in terms of the density and energy of the discretization are established. The
accelerated projected gradient method is adapted for solving the discretized
optimization problem. Numerical results are reported to demonstrate the
efficiency and accuracy of the proposed numerical method. Our results show that
the proposed method is much more efficient than the existing methods in the
literature, especially in the strong interaction regime.
