Before revision:  We solve such a problem by exploiting its characterization as a nonlinear eigenvalue problem with eigenvector nonlinearity (NEPv) . We show that with the property of NEPv , any algorithm finding the positive stationary point of this optimization problem actually finds its global minimum, such as the regularized Newton (RN) method.
After revision:  We solve such a problem 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 finding the positive stationary point of this optimization problem actually finds its global minimum, such as the regularized Newton (RN) method.