  Sorted $L_1$ penalization estimator (SLOPE) is a regularization technique for
sorted absolute coefficients in high-dimensional regression. By arbitrarily
setting its regularization weights $\lambda$ under the monotonicity constraint,
SLOPE can have various feature selection and clustering properties. On weight
tuning, the selected features and their clusters are very sensitive to the
tuning parameters. Moreover, the exhaustive tracking of their changes is
difficult using grid search methods. This study presents a solution path
algorithm that provides the complete and exact path of solutions for SLOPE in
fine-tuning regularization weights. A simple optimality condition for SLOPE is
derived and used to specify the next splitting point of the solution path. This
study also proposes a new design of a regularization sequence $\lambda$ for
feature clustering, which is called the quasi-spherical and octagonal shrinkage
and clustering algorithm for regression (QS-OSCAR). QS-OSCAR is designed with a
contour surface of the regularization terms most similar to a sphere. Among
several regularization sequence designs, sparsity and clustering performance
are compared through simulation studies. The numerical observations show that
QS-OSCAR performs feature clustering more efficiently than other designs.
