  This paper considers sparsity in linear regression under the restriction that
the regression weights sum to one. We propose an approach that combines
$\ell_0$- and $\ell_1$-regularization. We compute its solution by adapting a
recent methodological innovation made by Bertsimas et al. (2016) for
$\ell_0$-regularization in standard linear regression. In a simulation
experiment we compare our approach to $\ell_0$-regularization and
$\ell_1$-regularization and find that it performs favorably in terms of
predictive performance and sparsity. In an application to index tracking we
show that our approach can obtain substantially sparser portfolios compared to
$\ell_1$-regularization while maintaining a similar tracking performance.
