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We consider a special nonconvex quartic minimization problem over a singlespherical constraint, which includes the discretized energy functionalminimization problem of non-rotating Bose-Einstein condensates (BECs) as one ofthe important applications. Such a problem is studied by exploiting itscharacterization as a nonlinear eigenvalue problem with eigenvectornonlinearity (NEPv), which admits a unique nonnegative eigenvector, and thiseigenvector is exactly the global minimizer to the quartic minimization. Withthese properties, any algorithm converging to the nonnegative stationary pointof this optimization problem finds its global minimum, such as the regularizedNewton (RN) method. In particular, we obtain the global convergence to globaloptimum of the inexact alternating direction method of multipliers (ADMM) forthis problem. Numerical experiments for applications in non-rotating BECvalidate our theories.

Finding the Global Optimum of a Class of Quartic Minimization Problem