  We examine the problem of approximating a positive, semidefinite matrix
$\Sigma$ by a dyad $xx^T$, with a penalty on the cardinality of the vector $x$.
This problem arises in sparse principal component analysis, where a
decomposition of $\Sigma$ involving sparse factors is sought. We express this
hard, combinatorial problem as a maximum eigenvalue problem, in which we seek
to maximize, over a box, the largest eigenvalue of a symmetric matrix that is
linear in the variables. This representation allows to use the techniques of
robust optimization, to derive a bound based on semidefinite programming. The
quality of the bound is investigated using a technique inspired by Nemirovski
and Ben-Tal (2002).
