A dual spectral projected gradient method for log-determinant semidefinite problems

TL;DR

This paper introduces a spectral projected gradient method tailored for log-determinant semidefinite problems, demonstrating improved efficiency and convergence properties over existing methods through theoretical analysis and numerical experiments.

Contribution

It extends Birgin et al.'s spectral projected gradient method to log-determinant SDPs with linear constraints, proposing a dual approach with proven convergence and superior performance.

Findings

Method attains optimal value or terminates finitely.

Outperforms other methods in speed and accuracy.

Effective on gene expression and synthetic data.

Abstract

We extend the result on the spectral projected gradient method by Birgin et al. in 2000 to a log-determinant semidefinite problem (SDP) with linear constraints and propose a spectral projected gradient method for the dual problem. Our method is based on alternate projections on the intersection of two convex sets, which first projects onto the box constraints and then onto a set defined by a linear matrix inequality. By exploiting structures of the two projections, we show the same convergence properties can be obtained for the proposed method as Birgin's method where the exact orthogonal projection onto the intersection of two convex sets is performed. Using the convergence properties, we prove that the proposed algorithm attains the optimal value or terminates in a finite number of iterations. The efficiency of the proposed method is illustrated with the numerical results on randomly generated synthetic/deterministic data and gene expression data, in comparison with other methods including the inexact primal-dual path-following interior-point method, the adaptive spectral projected gradient method, and the adaptive Nesterov's smooth method. For the gene expression data, our results are compared with the quadratic approximation for sparse inverse covariance estimation method. We show that our method outperforms the other methods in obtaining a better optimal value fast.