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Abstract
An approach to the problem of rankone 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 rankone approximation of positive matrices is developed. To illustrate the results obtained, an example of the solution of the approximation problem for an arbitrary twodimensional positive matrix is given in an explicit form.
Original language  English 

Pages (fromto)  145153 
Journal  Vestnik St. Petersburg University: Mathematics 
Volume  52 
Issue number  2 
Early online date  11 Jun 2019 
DOIs  
State  Published  2019 
Scopus subject areas
 Algebra and Number Theory
 Control and Optimization
Keywords
 tropical mathematics
 tropical optimization
 maxalgebra
 rankone matrix approximation
 logChebyshev distance function
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Activities

Using tropical optimization in rankone approximation of positive matrices
Николай Кимович Кривулин (Speaker)
28 Jun 2018Activity: Talk types › Oral presentation

The 6th IMA Conference on Numerical Linear Algebra and Optimization
Николай Кимович Кривулин (Participant)
27 Jun 2018 → 29 Jun 2018Activity: Attendance types › Participating in a conference, workshop, ...