Conditioning of convex programs from a primal–dual perspective

Given a convex program and its dual, we analyze the conditioning of the primal–dual system of constraints, obtained by putting the primal and dual constraints together. We show that the conditioning of the primal–dual system can be estimated in terms of the conditioning of the primal and dual systems. In particular, provided both the primal and dual systems are well-conditioned. The primal-dual system is well-conditioned, the primal–dual system is well-conditioned. We also investigate how the conditioning of the primal–dual system relates to properties of both the primal and dual problems, providing in this way a primal–dual viewpoint for the theory of condition numbers for convex programming.