Computing optimal scalings by parametric network algorithms

Abstract

Asymmetric scaling of a square matrixA ≠ 0 is a matrix of the formXAX ⁻¹ whereX is a nonnegative, nonsingular, diagonal matrix having the same dimension ofA. Anasymmetric scaling of a rectangular matrixB ≠ 0 is a matrix of the formXBY ⁻¹ whereX andY are nonnegative, nonsingular, diagonal matrices having appropriate dimensions. We consider two objectives in selecting a symmetric scaling of a given matrix. The first is to select a scalingA′ of a given matrixA such that the maximal absolute value of the elements ofA′ is lesser or equal that of any other corresponding scaling ofA. The second is to select a scalingB′ of a given matrixB such that the maximal absolute value of ratios of nonzero elements ofB′ is lesser or equal that of any other corresponding scaling ofB. We also consider the problem of finding an optimal asymmetric scaling under the maximal ratio criterion (the maximal element criterion is, of course, trivial in this case). We show that these problems can be converted to parametric network problems which can be solved by corresponding algorithms.