  We introduce a financial portfolio optimization framework that allows us to
automatically select the relevant assets and estimate their weights by relying
on a sorted $\ell_1$-Norm penalization, henceforth SLOPE. Our approach is able
to group constituents with similar correlation properties, and with the same
underlying risk factor exposures. We show that by varying the intensity of the
penalty, SLOPE can span the entire set of optimal portfolios on the
risk-diversification frontier, from minimum variance to the equally weighted.
To solve the optimization problem, we develop a new efficient algorithm, based
on the Alternating Direction Method of Multipliers. Our empirical analysis
shows that SLOPE yields optimal portfolios with good out-of-sample risk and
return performance properties, by reducing the overall turnover through more
stable asset weight estimates. Moreover, using the automatic grouping property
of SLOPE, new portfolio strategies, such as SLOPE-MV, can be developed to
exploit the data-driven detected similarities across assets.
