Primal-dual symmetry and scale invariance of interior-point algorithms forconvex optimization

We present a definition of symmetric primal-dual algorithms for convex opti mization problems expressed in the conic form. After describing a generaliz ation of the v-space approach for such optimization problems, we show that a symmetric v-space approach can be developed for a convex optimization pro blem in the conic form if and only if the underlying cone is homogeneous an d self-dual. We provide an alternative definition of self-scaled barriers a nd then conclude with a discussion of the scalings of the variables which k eep the underlying convex cone invariant.