  The problems of Lasso regression and optimal design of experiments share a
critical property: their optimal solutions are typically \emph{sparse}, i.e.,
only a small fraction of the optimal variables are non-zero. Therefore, the
identification of the support of an optimal solution reduces the dimensionality
of the problem and can yield a substantial simplification of the calculations.
It has recently been shown that linear regression with a \emph{squared}
$\ell_1$-norm sparsity-inducing penalty is equivalent to an optimal
experimental design problem. In this work, we use this equivalence to derive
safe screening rules that can be used to discard inessential samples. Compared
to previously existing rules, the new tests are much faster to compute,
especially for problems involving a parameter space of high dimension, and can
be used dynamically within any iterative solver, with negligible computational
overhead. Moreover, we show how an existing homotopy algorithm to compute the
regularization path of the lasso method can be reparametrized with respect to
the squared $\ell_1$-penalty. This allows the computation of a Bayes
$c$-optimal design in a finite number of steps and can be several orders of
magnitude faster than standard first-order algorithms. The efficiency of the
new screening rules and of the homotopy algorithm are demonstrated on different
examples based on real data.
