  We consider a special nonconvex quartic minimization problem over a single
spherical constraint, which includes the discretized energy functional
minimization problem of non-rotating Bose-Einstein condensates (BECs) as one of
the important applications. Such a problem is studied by exploiting its
characterization as a nonlinear eigenvalue problem with eigenvector
nonlinearity (NEPv), which admits a unique nonnegative eigenvector, and this
eigenvector is exactly the global minimizer to the quartic minimization. With
these properties, any algorithm converging to the nonnegative stationary point
of this optimization problem finds its global minimum, such as the regularized
Newton (RN) method. In particular, we obtain the global convergence to global
optimum of the inexact alternating direction method of multipliers (ADMM) for
this problem. Numerical experiments for applications in non-rotating BEC
validate our theories.
