The primal-dual second-order cone approximations algorithm for symmetric cone programming - Chua, Chek Beng. (2009)
Given any open convex cone K, a logarithmically homogeneous, self-concordant barrier for K and any positive real number r < 1, we associate, with each direction x 2 K, a second-order cone ˆKr(x) containing K. We show that K is the interior of the intersection of the second-order cones ˆKr(x), as x ranges over all directions in K. Using these second-order cones as approximations to cones of symmetric, positive definite matrices, we develop a new polynomial-time primal-dual interior-point algorithm for semidefinite programming. The algorithm is extended to symmetric cone programming via the relation between symmetric cones and Euclidean Jordan algebras.