@MISC{Kalantari05matrixscaling, author = {Bahman Kalantari}, title = {MATRIX SCALING DUALITIES IN CONVEX PROGRAMMING }, year = {2005} }

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Abstract

We consider convex programming problems in a canonical homogeneous format, a very general form of Karmarkarâ€™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