  In this paper, we study convex optimization methods for computing the trace
norm regularized least squares estimate in multivariate linear regression. The
so-called factor estimation and selection (FES) method, recently proposed by
Yuan et al. [22], conducts parameter estimation and factor selection
simultaneously and have been shown to enjoy nice properties in both large and
finite samples. To compute the estimates, however, can be very challenging in
practice because of the high dimensionality and the trace norm constraint. In
this paper, we explore a variant of Nesterov's smooth method [20] and interior
point methods for computing the penalized least squares estimate. The
performance of these methods is then compared using a set of randomly generated
instances. We show that the variant of Nesterov's smooth method [20] generally
outperforms the interior point method implemented in SDPT3 version 4.0 (beta)
[19] substantially . Moreover, the former method is much more memory efficient.
