  We study preconditioned proximal point methods for a class of saddle point
problems, where the preconditioner decouples the overall proximal point method
into an alternating primal--dual method. This is akin to the Chambolle--Pock
method or the ADMM. In our work, we replace the squared distance in the dual
step by a barrier function on a symmetric cone, while using a standard
(Euclidean) proximal step for the primal variable. We show that under
non-degeneracy and simple linear constraints, such a hybrid primal--dual
algorithm can achieve linear convergence on originally strongly convex problems
involving the second-order cone in their saddle point form. On general
symmetric cones, we are only able to show an $O(1/N)$ rate. These results are
based on estimates of strong convexity of the barrier function, extended with a
penalty to the boundary of the symmetric cone.
