  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.
