Primal-Dual Symmetry and Scale Invariance of Interior-Point Algorithms for Convex Optimization

We present a definition of symmetric primal-dual algorithms for convex optimization problems expressed in the conic form. After describing a generalization of the υ-space approach for such optimization problems, we show that a symmetric υ-space approach can be developed for a convex optimization problem in the conic form if and only if the underlying cone is homogeneous and self-dual. We provide an alternative definition of self-scaled barriers and then conclude with a discussion of the scalings of the variables which keep the underlying convex cone invariant.