论文标题
可行的牛顿对称张量Z-EIGENVALUE问题的方法
Feasible Newton's methods for symmetric tensor Z-eigenvalue problems
论文作者
论文摘要
找到对称张量的Z-eigenpair等同于找到球体的KKT点约束最小化问题。基于这种等效性，在本文中，我们首先提出了一类迭代方法，以获取对称张量的z-eigenpair。每种方法都可以生成一系列可行点，以便函数评估的序列正在减少。这些方法可以被视为无约束优化问题的下降方法的扩展。我们特别注意牛顿的方法。我们表明，在适当的条件下，牛顿的方法在全球和四边形上是收敛的。此外，经过许多有限的迭代，将始终接受单位级别的长度。我们还建议基于非线性方程的牛顿方法，并建立其全球和二次收敛。最后，我们进行了几项数值实验来测试所提出的牛顿的方法。结果表明，这两种牛顿的方法都非常有效。
Finding a Z-eigenpair of a symmetric tensor is equivalent to finding a KKT point of a sphere constrained minimization problem. Based on this equivalency, in this paper, we first propose a class of iterative methods to get a Z-eigenpair of a symmetric tensor. Each method can generate a sequence of feasible points such that the sequence of function evaluations is decreasing. These methods can be regarded as extensions of the descent methods for unconstrained optimization problems. We pay particular attention to the Newton's method. We show that under appropriate conditions, the Newton's method is globally and quadratically convergent. Moreover, after finitely many iterations, the unit steplength will always be accepted. We also propose a nonlinear equations based Newton's method and establish its global and quadratic convergence. In the end, we do several numerical experiments to test the proposed Newton's methods. The results show that both Newton's methods are very efficient.
