  The exclusive lasso (also known as elitist lasso) regularization has become
popular recently due to its superior performance on group sparsity. Compared to
the group lasso regularization which enforces the competition on variables
among different groups, the exclusive lasso regularization also enforces the
competition within each group. In this paper, we propose a highly efficient
dual Newton based preconditioned proximal point algorithm (PPDNA) to solve
machine learning models involving the exclusive lasso regularizer. As an
important ingredient, we provide a rigorous proof for deriving the closed-form
solution to the proximal mapping of the weighted exclusive lasso regularizer.
In addition, we derive the corresponding HS-Jacobian to the proximal mapping
and analyze its structure --- which plays an essential role in the efficient
computation of the PPA subproblem via applying a semismooth Newton method on
its dual. Various numerical experiments in this paper demonstrate the superior
performance of the proposed PPDNA against other state-of-the-art numerical
algorithms.
