  In sparse linear regression, the SLOPE estimator generalizes LASSO by
penalizing different coordinates of the estimate according to their magnitudes.
In this paper, we present a precise performance characterization of SLOPE in
the asymptotic regime where the number of unknown parameters grows in
proportion to the number of observations. Our asymptotic characterization
enables us to derive the fundamental limits of SLOPE in both estimation and
variable selection settings. We also provide a computational feasible way to
optimally design the regularizing sequences such that the fundamental limits
are reached. In both settings, we show that the optimal design problem can be
formulated as certain infinite-dimensional convex optimization problems, which
have efficient and accurate finite-dimensional approximations. Numerical
simulations verify all our asymptotic predictions. They demonstrate the
superiority of our optimal regularizing sequences over other designs used in
the existing literature.
