A Quadratically Convergent Newton Method for Computing the Nearest Correlation Matrix

Abstract

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 like 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.

Anderson acceleration of the alternating projections method for computing the nearest correlation matrix

“…The Newton algorithm of [39] which solves the original nearest correlation matrix problem does not generalize to the fixed elements variant. According to Qi and Sun [41, p. 509], the Newton method that solves the so-called H -weighted nearest correlation matrix problem…”

“…The Newton algorithm [39] for the original nearest correlation matrix problem can be used to compute the solution to the problem with the constraint on λ min . We discuss this modification of the alternating projections method because it further demonstrates the flexibility of the method, which can easily incorporate both the fixed elements constraint and the eigenvalue constraint, unlike the existing Newton methods.…”

“…Although a faster Newton algorithm was subsequently developed by Qi and Sun [39], and practical improvements to it were made by Borsdorf and Higham [8], the alternating projections method remains widely used in applications. Major reasons for its popularity are its ease of coding and the availability of implementations in MATLAB, Python, R, and SAS [25].…”

An augmented Lagrangian dual approach for the H-weighted nearest correlation matrix problem

“…The W -weighted problem (1.1) has been well studied since Higham (2002) and now there are several good methods for it, including the alternating projection method (Higham, 2002), the gradient and quasi-Newton methods (Malick, 2004;Boyd & Xiao, 2005), the inexact semismooth Newton method combined with the conjugate gradient (CG) solver (Qi & Sun, 2006) and its modified version with several (preconditioned) iterative solvers (Borsdorf, 2007;Borsdorf & Higham, 2009) and the inexact interior-point methods (IPMs) with iterative solvers (Toh et al, 2007;Toh, 2008). All of these methods, except the inexact IPMs, crucially rely on the fact that the projection of a given matrix X ∈ S n onto S n + under the W -weighting, denoted by Π W S n + (X ), which is the optimal solution of the following problem:…”

“…Solving the W -weighted problem (1.1) is equivalent to solving a problem of the following type (cf. Qi & Sun, 2006, Section 4.1): min 1 2 X − G 2 such that diag(W −1/2 X W −1/2 ) = e, X ∈ S n + ,…”