  The optimization of the variance supplemented by a budget constraint and an
asymmetric $\ell_1$ regularizer is carried out analytically by the replica
method borrowed from the theory of disordered systems. The asymmetric
regularizer allows us to penalize short and long positions differently, so the
present treatment includes the no-short-constrained portfolio optimization
problem as a special case. Results are presented for the out-of-sample and the
in-sample estimator of the regularized variance, the relative estimation error,
the density of the assets eliminated from the portfolio by the regularizer, and
the distribution of the optimal portfolio weights. We have studied the
dependence of these quantities on the ratio $r$ of the portfolio's dimension
$N$ to the sample size $T$, and on the strength of the regularizer. We have
checked the analytic results by numerical simulations, and found general
agreement. Regularization extends the interval where the optimization can be
carried out, and suppresses the large sample fluctuations, but the performance
of $\ell_1$ regularization is rather disappointing: if the sample size is large
relative to the dimension, i.e. $r$ is small, the regularizer does not play any
role, while for $r$'s where the regularizer starts to be felt the estimation
error is already so large as to make the whole optimization exercise pointless.
We find that the $\ell_1$ regularization can eliminate at most half the assets
from the portfolio, corresponding to this there is a critical ratio $r=2$
beyond which the $\ell_1$ regularized variance cannot be optimized: the
regularized variance becomes constant over the simplex. These facts do not seem
to have been noticed in the literature.
