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MATRIX SCALING DUALITIES IN CONVEX PROGRAMMING
, 2005
"... 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 s ..."
Abstract

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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 potentialreduction and pathfollowing algorithms, applicable to selfconcordant 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 pathfollowing 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