一类四次极小化问题的全局最优值的求法
Finding the Global Optimum of a Class of Quartic Minimization Problem
摘要：
考虑了一类特殊的单球约束非凸四次极小化问题，其中非旋转玻色-爱因斯坦凝聚体（BECs）的离散能量泛函极小化问题是其重要应用之一.将该问题描述为一个具有非线性特征向量（NEPv）的非线性本征值问题，其中存在唯一的非负特征向量，且该特征向量恰好是四次极小化的全局极小元.有了这些性质，任何收敛到这个最优化问题的非负稳定点的算法都会找到它的全局极小值，例如正则化牛顿（RN）方法。特别地，我们得到了该问题的不精确交替方向法（ADMM）到全局最优值的整体收敛。非旋转BEC中的应用的数值实验验证了我们的理论。
英文摘要：
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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