Finding the global optimum of a class of quartic minimization problem

Abstract

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). Firstly, we show that the NEPv has a unique nonnegative eigenvector, corresponding to the smallest nonlinear eigenvalue of NEPv, which is exactly the global minimizer to the optimization problem. Secondly, with these properties, we obtain that any algorithm converging to the nonnegative stationary point of this optimization problem finds its global optimum, such as the regularized Newton method. In particular, we obtain the convergence to the global optimum of the inexact alternating direction method of multipliers for this problem. Numerical experiments for applications in non-rotating BECs validate our theories.