A 2-block semi-proximal ADMM for solving the H-weighted nearest correlation matrix problem. (2nd January 2017) - Chang, Xiaokai; Liu, Sanyang
Abstract : Higham considered two types of nearest correlation matrix (NCM) problems, namely the W -weighted case and the H -weighted case. Since there exists well-defined computable formula for the projection onto the symmetric positive semidefinite cone under the W -weighting, it has been well studied to make several Lagrangian dual-based efficient numerical methods available. But these methods are not applicable for the H -weighted case mainly due to the lack of a computable formula. The H -weighted case remains numerically challenging, especially for the highly ill-conditioned weight matrix H . In this paper, we aim to solve the dual form of the H -weighted NCM problem, which has three separable blocks in the objective function with the second part being linear. Based on the linear part, we reformulate it as a new problem with two separable blocks, and introduce the 2-block semi-proximal alternating direction method of multipliers to deal with it. The efficiency of the proposed algorithms is demonstrated on the random test problems, whose weight matrix H are highly ill-conditioned or rank deficient.