Matrix Scaling Dualities in Convex Programming∗

We consider convex programming problems in a canonical homogeneous format, a very general form of Karmarkar’s canonical linear programming problem. More specifically, by homogeneous programming we shall refer to the problem of testing if a homogeneous convex function has a nontrivial zero over a subspace and its intersection with a pointed convex cone. To this canonical problem, endowed with a normal barrier for the underlying cone, we associate dual problems and prove several matrix scaling dualities. We make use of these scaling dualities to derive new and conceptually simple potential-reduction and path-following algorithms, applicable to self-concordant homogeneous programming, as well as three dual problems defined as: the scaling problem, the homogeneous scaling problem, and the algebraic scaling problem. The simplest of the scaling dualities is the following equivalent of the classic separation theorem of Gordan: a positive semidefinite symmetric matrix Q either has a nontrivial nonnegative zero, or there exists a positive definite diagonal matrix D such that DQDe > 0, where e is the vector of ones. This duality is a key ingredient in the very simple path-following algorithm of Khachiyan and Kalantari for linear programming, as well as for quasi doubly stochastic scaling of Q, i.e. computing D such that DQDe = e. Our general results here give nontrivial extensions of our previous work on the role of matrix scaling in linear or semidefinite programming, when formulated as a homogeneous program. To establish the general results we associate a cone of linear operators induced by the normal barrier, called operator-cone and use it to reveal the intrinsic nature of scaling dualities corresponding to homogeneous programming formulation of convex programs. Our general algorithms although make use of some basic properties from the self-concordance theory of Nesterov and Nemirovskii, offer new algorithms for linear programming, quadratic programming, semidefinite programming, self-concordant programming itself, as well as for the corresponding scaling problems. Scaling dualities also result in generalizations of the classic arithmetic-geometric mean and the trace-determinant inequalities. Our collective results reveal the ubiquitous nature of matrix scaling in convex programming.