论文标题
SCF的本地单位NEPV和收敛分析
Locally Unitarily Invariantizable NEPv and Convergence Analysis of SCF
论文作者
论文摘要
我们考虑一类依赖于单一不变特性的特征向量依赖性非线性特征值问题（NEPV）。这些NEPV通常是在Stiefel歧管上特定类型的优化问题的一阶最佳条件，并且在文献中已经研究了特殊情况。在NEPV的特征矩阵上，有两个必要的条件，即确定性条件和保留等级的条件，这是相关优化问题的全球优化器，在此，在此之前已调查的特殊案例已知确定性条件。我们表明，在局部接近满足这两种必要条件的本地基质矩阵，NEPV可以通过基础对准操作（换句话说，NEPV都是本地不变的，可以将NEPV重新构成一个单位不变的NEPV，即所谓的Aligned NEPV。从数值上讲，NEPV通过SCF型迭代自然解决。通过利用对齐NEPV的系数矩阵的可不同性，我们为SCF型迭代建立了封闭形式的局部收敛速率，并分析了其水平变化的变体。数值实验证实了我们的理论结果。
We consider a class of eigenvector-dependent nonlinear eigenvalue problems (NEPv) without the unitary invariance property. Those NEPv commonly arise as the first-order optimality conditions of a particular type of optimization problems over the Stiefel manifold, and previously, special cases have been studied in the literature. Two necessary conditions, a definiteness condition and a rank-preserving condition, on an eigenbasis matrix of the NEPv that is a global optimizer of the associated optimization problem are revealed, where the definiteness condition has been known for the special cases previously investigated. We show that, locally close to the eigenbasis matrix satisfying both necessary conditions, the NEPv can be reformulated as a unitarily invariant NEPv, the so-called aligned NEPv, through a basis alignment operation -- in other words, the NEPv is locally unitarily invariantizable. Numerically, the NEPv is naturally solved by an SCF-type iteration. By exploiting the differentiability of the coefficient matrix of the aligned NEPv, we establish a closed-form local convergence rate for the SCF-type iteration and analyze its level-shifted variant. Numerical experiments confirm our theoretical results.
