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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 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 s ..."
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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 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
A Note on a Boundedness Property of Normal Barriers of Convex Cones
"... Let K be a closed convex pointed cone in a finite dimensional Hilbert space E. Assume K has a nonempty interior K # . Let r = inf{#x, y# : x, y # K, #x# = #y# = 1}, where #·, ·# denotes the inner product and # · # the corresponding induced norm. The quantity r gives a measure of obtusenes ..."
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Let K be a closed convex pointed cone in a finite dimensional Hilbert space E. Assume K has a nonempty interior K # . Let r = inf{#x, y# : x, y # K, #x# = #y# = 1}, where #·, ·# denotes the inner product and # · # the corresponding induced norm. The quantity r gives a measure of obtuseness of K and it necessarily lies in (1, 1]. Let F be any #normal barrier for K # . We prove that for each d # K # , the operator D # # 2 F (d) 1/2 satisfies the inequality #D# # ##d#, where # # 1 + q 2 1+r . In particular, if K is acute, i.e., #x, y# # 0, #x, y # K, then # # 1 + # 2. We prove the above using a property from the theory of selfconcordance of Nesterov and Nemirovskii [3], and a result in Kalantari [2]. The existence of this bound implies that the polynomialtime potentialreduction and pathfollowing algorithms which were described in [2], for selfconcordant homogeneous programming and for a generalizations of the diagonal matrix scaling problem, are a...
A Simple PathFollowing Algorithm for the Feasibility Problem in Semidefinite Programming and for Matrix Scaling over the Semidefinite Cone
"... Let E be the Hilbert space of symmetric matrices of the form diag(A, M ), where A is n n, and M is an l l diagonal matrix, and the inner product #x, y# # T race(xy). Given x # E, we write x # 0 (x > 0) if it is positive semidefinite (positive definite). Let Q : E # E be a symmetri ..."
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Let E be the Hilbert space of symmetric matrices of the form diag(A, M ), where A is n n, and M is an l l diagonal matrix, and the inner product #x, y# # T race(xy). Given x # E, we write x # 0 (x > 0) if it is positive semidefinite (positive definite). Let Q : E # E be a symmetric positive semidefinite linear operator, and = min{#(x) = 0.5T race(xQx) : #x# = 1, x # 0}. The feasibility problem in SDP can be formulated as the problem of testing if = 0 for some Q. Let # # (0, 1) be a given accuracy, u = Qe  e, e the identity matrix in E, and N = n + l. We describe a simple pathfollowing algorithm that in case = 0, in O( # N ln[N#u#/#]) Newton iterations produces x # 0, #x# = 1 such that T race(xQx) # #. If > 0, in O( # N ln[N#u#/#]) Newton iterations the algorithm produces d > 0 such that #DQDe  e# # #, where D is the operator that maps w # E to d 1/2 wd 1/2 . Moreover, we use the algorithm to prove: > 0, if and only if there exists d > 0...
On The ArithmeticGeometric Mean Inequality And Its Relationship To Linear Programming, Matrix Scaling, And Gordan's Theorem
, 1998
"... It is a classical inequality that the minimum of the ratio of the (weighted) arithmetic mean to the geometric mean of a set of positive variables is equal to one, and is attained at the center of the positivity cone. While there are numerous proofs of this fundamental homogeneous inequality, in the ..."
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It is a classical inequality that the minimum of the ratio of the (weighted) arithmetic mean to the geometric mean of a set of positive variables is equal to one, and is attained at the center of the positivity cone. While there are numerous proofs of this fundamental homogeneous inequality, in the presence of an arbitrary subspace, and/or the replacement of the arithmetic mean with an arbitrary linear form, the new minimization is a nontrivial problem. We prove a generalization of this inequality, also relating it to linear programming, to the diagonal matrix scaling problem, as well as to Gordan's theorem. Linear programming is equivalent to the search for a nontrivial zero of a linear or positive semidefinite quadratic form over the nonnegative points of a given subspace. The goal of this paper is to present these intricate, surprising, and significant relationships, called scaling dualities, and via an elementary proof. Also, to introduce two conceptually simple polynomialtime alg...