An Augmented Lagrangian Dual Approach for the H-Weighted Nearest Correlation Matrix Problem

In [15], Higham considered two types of nearest correlation matrix problem, namely the W -weighted case and the H-weighted case. While the W -weighted case has since then been well studied to make several Lagrangian dual based efficient numerical methods available, the H-weighted case remains numerically challenging. The difficulty of extending those methods from the W -weighted case to the H-weighted case lies in the fact that an analytic formula for the metric projection onto the positive semidefinite cone under the H-weight, unlike the case under the W -weight, is not available. In this paper, we introduce an augmented Lagrangian dual based approach, which avoids the explicit computation of the metric projection under the H-weight. This method solves a sequence of unconstrained strongly convex optimization problems, each of which can be efficiently solved by a semismooth Newton method combined with the conjugate gradient method. Numerical experiments demonstrate that the augmented Lagrangian dual approach is not only fast but also robust. AMS subject classifications. 49M45, 90C25, 90C33