Nonsymmetric potential-reduction methods for general cones

In this paper we propose two new nonsymmetric primal-dual potential-reduction methods for conic problems. The methods are based on the primal-dual lifting [5]. This procedure allows to construct a strictly feasible primal-dual pair related by an exact scaling relation even if the cones are not symmetric. It is important that all necessary elements of our methods can be obtained from the standard solvers for primal Newton system. The first of the proposed schemes is based on the usual affine-scaling direction. For the second one, we apply a new first-order affine-scaling direction, which incorporates in a symmetric way the gradients of primal and dual barriers. For both methods we prove the standard O( √ ν ln 12 ) complexity estimate, where ν is the parameter of the barrier and 2 is the required accuracy.