  We show that the $A$-optimal design optimization problem over $m$ design
points in $\mathbb{R}^n$ is equivalent to minimizing a quadratic function plus
a group lasso sparsity inducing term over $n\times m$ real matrices. This
observation allows to describe several new algorithms for $A$-optimal design
based on splitting and block coordinate decomposition. These techniques are
well known and proved powerful to treat large scale problems in machine
learning and signal processing communities. The proposed algorithms come with
rigorous convergence guaranties and convergence rate estimate stemming from the
optimization literature. Performances are illustrated on synthetic benchmarks
and compared to existing methods for solving the optimal design problem.
