Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya Newton directions and their variants

Monteiro and Tsuchiya [22] have proposed two primal-dual Newton directions for semidefinite programming, referred to as the MT directions, and established polynomial convergence of path-following methods based on them. This paper reports some computational results on the performance of interior-point predictor-corrector methods based on the MT directions and a variant of these directions, called the S-Ch-MT direction. We discuss how to compute these directions efficiently and derive their corresponding computational complexities. A main feature of our analysis is that computational formulae for these directions are derived from a unified point of view which entirely avoids the use of Kronecker product. Using this unified approach, we also present schemes to compute the Alizadeh-Haeberly-Overton (AHO) direction, the Nesterov-Todd direction and the HRVW/KSH/M direction with computational complexities (for dense problems) better than previously reported in the literature. We present some computational results for small dense problems, which are quite promising. We have obtained better performance for the methods based on the AHO, NT and HRVW/KSH/M directions. We have also observed that the method based on the S-Ch-MT direction compares favorably with the new implementation of the methods based on the NT direction and the HRVW/KSH/M direction.