  Estimation of Gaussian graphical models is important in natural science when
modeling the statistical relationships between variables in the form of a
graph. The sparsity and clustering structure of the concentration matrix is
enforced to reduce model complexity and describe inherent regularities. We
propose a model to estimate the sparse Gaussian graphical models with hidden
clustering structure, which also allows additional linear constraints to be
imposed on the concentration matrix. We design an efficient two-phase algorithm
for solving the proposed model. We develop a symmetric Gauss-Seidel based
alternating direction method of the multipliers (sGS-ADMM) to generate an
initial point to warm-start the second phase algorithm, which is a proximal
augmented Lagrangian method (pALM), to get a solution with high accuracy.
Numerical experiments on both synthetic data and real data demonstrate the good
performance of our model, as well as the efficiency and robustness of our
proposed algorithm.
