On the Rank-One Approximation of Positive Matrices Using Tropical Optimization Methods

Abstract

An approach to the problem of rank-one approximation of positive matrices in the Chebyshev metric in logarithmic scale is developed in this work, based on the application of tropical optimization methods. The theory and methods of tropical optimization constitute one of the areas of tropical mathematics that deals with semirings and semifields with idempotent addition and their applications. Tropical optimization methods allow finding a complete solution to many problems of practical importance explicitly in a closed form. In this paper, the approximation problem under consideration is reduced to a multidimensional tropical optimization problem, which has a known solution in the general case. A new solution to the problem in the case when the matrix has no zero columns or rows is proposed and represented in a simpler form. On the basis of this result, a new complete solution of the problem of rank-one approximation of positive matrices is developed. To illustrate the results obtained, an example of the solution of the approximation problem for an arbitrary two-dimensional positive matrix is given in an explicit form.