In this paper, we consider estimating sparse inverse covariance of a Gaussian graphical model whose conditional independence is assumed to be partially known. Similarly as in , we formulate it as an $l_1$-norm penalized maximum likelihood estimation problem. Further, we propose an algorithm framework, and develop two first-order methods, that is, adaptive spectral projected gradient (ASPG) method and adaptive Nesterov's smooth (ANS) method, for solving this estimation problem. Finally, we compare the performance of these two methods on a set of randomly generated instances. Our computational results demonstrate that both methods are able to solve problems of size at least a thousand and number of constraints of nearly a half million within a reasonable amount of time. We also observe that the ASPG method generally outperforms the ANS method.
Manuscript, Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada, November 2008