Solving the Global Optimum of a Class of Minimization Problem
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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. In particular, we can obtain the global convergence to global optimum of alternating direction method of multipliers (ADMM) for this problem. Numerical experiments for applications in BEC validate our theories and demonstrate the effectiveness of ADMM for solving this problem.
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