Об одноранговой аппроксимации положительных матриц с помощью методов тропической оптимизации

Translated title of the contribution: On rank-one approximation of positive matrices using methods of tropical optimization

Н.К. Кривулин, Е.Ю. Романова

Research output: Contribution to journalArticlepeer-review

Abstract

In the paper, an approach to the problem of rank-one approximation of positive matrices in the Chebyshev metric in logarithmic scale is developed based on the application of methods of tropical optimization. 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. For many practically important problems, methods of tropical optimization allow finding a complete solution explicitly in a closed form. In this work, the approximation problem under consideration is reduced to 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 more simple 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 positive matrix of the second order is given in analytical form.
Translated title of the contributionOn rank-one approximation of positive matrices using methods of tropical optimization
Original languageRussian
Pages (from-to)208-220
JournalВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. МАТЕМАТИКА. МЕХАНИКА. АСТРОНОМИЯ
Volume6 (64)
Issue number2
StatePublished - 2019

Scopus subject areas

  • Algebra and Number Theory
  • Control and Optimization

Keywords

  • tropical mathematics
  • tropical optimization
  • max-algebra
  • rank-one matrix approximation
  • log-Chebyshev distance

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