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. 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.