Matrix scaling, entropy minimization, and conjugate duality. I. existence conditions

AbstractWe have four primary objectives in this paper. First, we introduce a problem called truncated matrix scaling that generalizes two well-studied matrix scaling problems—diagonal similarity scaling and fixed row-column equivalence scaling. Second, we derive necessary and sufficient conditions for the attainment of the infimum in the Fenchel dual of a class of convex optimization problems. Third, we show that existence of a solution for truncated matrix scaling is equivalent to the attainment of the infimum in a corresponding dual optimization problem. We thereby derive necessary and sufficient conditions for the existence of a solution for truncated matrix scaling. Fourth, we derive known existence conditions for similarity and equivalence scaling from the conditions for truncated matrix scaling