  It is well known that the out-of-sample performance of Markowitz's
mean-variance portfolio criterion can be negatively affected by estimation
errors in the mean and covariance. In this paper we address the problem by
regularizing the mean-variance objective function with a weighted elastic net
penalty. We show that the use of this penalty can be motivated by a robust
reformulation of the mean-variance criterion that directly accounts for
parameter uncertainty. With this interpretation of the weighted elastic net
penalty we derive data driven techniques for calibrating the weighting
parameters based on the level of uncertainty in the parameter estimates. We
test our proposed technique on US stock return data and our results show that
the calibrated weighted elastic net penalized portfolio outperforms both the
unpenalized portfolio and uniformly weighted elastic net penalized portfolio.
  This paper also introduces a novel Adaptive Support Split-Bregman approach
which leverages the sparse nature of $\ell_{1}$ penalized portfolios to
efficiently compute a solution of our proposed portfolio criterion. Numerical
results show that this modification to the Split-Bregman algorithm results in
significant improvements in computational speed compared with other techniques.
