论文标题
找到一类四级最小化问题的全局最佳
Finding the Global Optimum of a Class of Quartic Minimization Problem
论文作者
论文摘要
我们考虑在单个球形约束上考虑一个特殊的非凸Quartic最小化问题，其中包括非旋转Bose-Einstein冷凝物（BEC）的离散能量功能最小化问题是重要的应用之一。通过利用其特征为特征向量非线性（NEPV），该问题是通过将其表征作为非线性特征值问题（NEPV）来研究的，该问题承认了独特的非负特征向量，而该特征向量正是Quartic最小化的全球最小化器。借助这些属性，任何算法都会收敛到此优化问题的非负固定点，都可以找到其全局最小值，例如正则化牛顿（RN）方法。特别是，我们为此问题获得了乘数（ADMM）的全局融合到全局最佳的交替方向方法。用于非旋转BEC应用的数值实验验证了我们的理论。
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.
