Dual Spectral Projected Gradient Method for Generalized Log-det Semidefinite Programming

TL;DR

This paper extends the dual spectral projected gradient method to solve a more general class of log-det semidefinite programming problems, demonstrating convergence and efficiency through numerical experiments.

Contribution

It introduces a unified extension of the DSPG method for generalized log-det SDP problems, covering more structures in Gaussian graphical models.

Findings

The extended DSPG method converges to the optimal value.

Numerical experiments show improved efficiency over existing methods.

The method effectively handles additional terms in the SDP problem.

Abstract

Log-det semidefinite programming (SDP) problems are optimization problems that often arise from Gaussian graphic models. A log-det SDP problem with an l1-norm term has been examined in many methods, and the dual spectral projected gradient (DSPG) method by Nakagaki et al.~in 2020 is designed to efficiently solve the dual problem of the log-det SDP by combining a non-monotone line-search projected gradient method with the step adjustment for positive definiteness. This paper extends the DSPG method for solving a generalized log-det SDP problem involving additional terms to cover more structures in Gaussian graphical models in a unified style. We establish the convergence of the proposed method to the optimal value. We conduct numerical experiments to illustrate the efficiency of the proposed method.