A quadratically convergent Newton method for computing the nearest correlation matrix
A quadratically convergent Newton method for computing the nearest correlation matrix
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DOI：
10.1137/050624509
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发表时间：
2006-01-01
影响因子：
1.5
通讯作者：
Sun, Defeng
中科院分区：
文献类型：
--
作者：
Qi, Houduo;Sun, Defeng
最近相关矩阵问题是找到一个在弗罗贝尼乌斯范数下与给定对称矩阵最接近的相关矩阵。经过充分研究的对偶方法是将此问题重新表述为一个无约束的连续可微凸优化问题。梯度方法和拟牛顿方法（如BFGS）已被直接用于获得全局收敛的方法。由于对偶方法中的目标函数不是二次连续可微的，这些方法最多线性收敛。在本文中，我们研究一种用于最近相关矩阵问题的牛顿型方法。基于强半光滑矩阵值函数的最新进展，我们证明了所提出的牛顿方法的二次收敛性。数值实验证实了该方法的快速收敛性和高效性。
The nearest correlation matrix problem is to find a correlation matrix which is closest to a given symmetric matrix in the Frobenius norm. The well-studied dual approach is to reformulate this problem as an unconstrained continuously differentiable convex optimization problem. Gradient methods and quasi-Newton methods such as BFGS have been used directly to obtain globally convergent methods. Since the objective function in the dual approach is not twice continuously differentiable, these methods converge at best linearly. In this paper, we investigate a Newton-type method for the nearest correlation matrix problem. Based on recent developments on strongly semismooth matrix valued functions, we prove the quadratic convergence of the proposed Newton method. Numerical experiments confirm the fast convergence and the high efficiency of the method.