  A large portfolio of independent returns is optimized under the variance risk
measure with a ban on short positions. The no-short selling constraint acts as
an asymmetric $\ell_1$ regularizer, setting some of the portfolio weights to
zero and keeping the out of sample estimator for the variance bounded, avoiding
the divergence present in the non-regularized case. However, the
susceptibility, i.e. the sensitivity of the optimal portfolio weights to
changes in the returns, diverges at a critical value $r=2$. This means that a
ban on short positions does not prevent the phase transition in the
optimization problem, it merely shifts the critical point from its
non-regularized value of $r=1$ to $2$. At $r=2$ the out of sample estimator for
the portfolio variance stays finite and the estimated in-sample variance
vanishes. We have performed numerical simulations to support the analytic
results and found perfect agreement for $N/T<2$. Numerical experiments on
finite size samples of symmetrically distributed returns show that above this
critical point the probability of finding solutions with zero in-sample
variance increases rapidly with increasing $N$, becoming one in the large $N$
limit. However, these are not legitimate solutions of the optimization problem,
as they are infinitely sensitive to any change in the input parameters, in
particular they will wildly fluctuate from sample to sample. We also calculate
the distribution of the optimal weights over the random samples and show that
the regularizer preferentially removes the assets with large variances, in
accord with one's natural expectation.
