We study a special nonconvex optimization problem with a single spherical constraint to find a global minimizer of it. One important application of this problem is the discretized energy functional minimization problem of non-rotating Bose-Einstein condensate (BEC) . 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. In particular, we can obtain the global convergence to global optimum of the  alternating direction method of multipliers (ADMM) for this problem. Numerical experiments for applications in non-rotating BEC validate our theories.
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.
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study
consider
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optimization problem with
quartic minimization problem over
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condensate (BEC) . We solve such a problem
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. We show that with the property of NEPv
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finding the positive
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