  We consider a convex relaxation of sparse principal component analysis
proposed by d'Aspremont et al. in (d'Aspremont et al. SIAM Rev 49:434-448,
2007). This convex relaxation is a nonsmooth semidefinite programming problem
in which the $\ell_1$ norm of the desired matrix is imposed in either the
objective function or the constraint to improve the sparsity of the resulting
matrix. The sparse principal component is obtained by a rank-one decomposition
of the resulting sparse matrix. We propose an alternating direction method
based on a variable-splitting technique and an augmented Lagrangian framework
for solving this nonsmooth semidefinite programming problem. In contrast to the
first-order method proposed in (d'Aspremont et al. SIAM Rev 49:434-448, 2007)
that solves approximately the dual problem of the original semidefinite
programming problem, our method deals with the primal problem directly and
solves it exactly, which guarantees that the resulting matrix is a sparse
matrix. Global convergence result is established for the proposed method.
Numerical results on both synthetic problems and the real applications from
classification of text data and senate voting data are reported to demonstrate
the efficacy of our method.
