### Аннотация

We consider linear vector inequalities defined in the framework of a linearly ordered tropical semifield (a semiring with idempotent addition and invertible multiplication). The problem is to solve two-sided inequalities, which have an unknown vector included in both sides, each taking the form of a given matrix multiplied by this unknown vector. Observing that the set of solutions is closed under vector addition and scalar multiplication, we reduce the problem to finding a matrix whose columns generate the entire solution set.

We represent the solution as a family of subsets, each defined by a matrix that is obtained from the given matrices by using a matrix sparsification technique. The technique exploits sparsified matrices to derive a series of new inequalities, which admit a direct solution in the form of matrices that generate their solutions. We describe a

backtracking procedure that reduces the brute-force search of sparsified matrices by skipping those, which cannot provide solutions, and thus offers an economical way to obtain all subsets in the family. The columns in the generating matrices for subsets are combined together to form a matrix, which is further reduced to have only columns that constitute a minimal generating system of the solution. We use the reduced matrix to represent a complete exact solution of the two-sided inequality under consideration in a compact vector form.

We illustrate the results with numerical examples. Extension of the approach to solve two-sided equations is also discussed.

We represent the solution as a family of subsets, each defined by a matrix that is obtained from the given matrices by using a matrix sparsification technique. The technique exploits sparsified matrices to derive a series of new inequalities, which admit a direct solution in the form of matrices that generate their solutions. We describe a

backtracking procedure that reduces the brute-force search of sparsified matrices by skipping those, which cannot provide solutions, and thus offers an economical way to obtain all subsets in the family. The columns in the generating matrices for subsets are combined together to form a matrix, which is further reduced to have only columns that constitute a minimal generating system of the solution. We use the reduced matrix to represent a complete exact solution of the two-sided inequality under consideration in a compact vector form.

We illustrate the results with numerical examples. Extension of the approach to solve two-sided equations is also discussed.

Язык оригинала | английский |
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Название основной публикации | MAT TRIAD 2019 |

Подзаголовок основной публикации | International Conference on Matrix Analysis and its Applications: Book of Abstracts |

Редакторы | Jan Bok, David Hartman, Milan Hladík, Miroslav Rozložník |

Место публикации | Prague |

Издатель | Charles University in Prague |

Страницы | 38-38 |

Состояние | Опубликовано - 2019 |

Событие | International Conference on Matrix Analysis and its Applications - Liblice, Чехия Продолжительность: 8 сен 2019 → 13 сен 2019 Номер конференции: 8 https://mattriad.math.cas.cz/ |

### Конференция

Конференция | International Conference on Matrix Analysis and its Applications |
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Сокращенный заголовок | MAT TRIAD 2019 |

Страна | Чехия |

Город | Liblice |

Период | 8/09/19 → 13/09/19 |

Адрес в сети Интернет |

### Fingerprint

### Предметные области Scopus

- Алгебра и теория чисел

### Цитировать

Krivulin, N. (2019). Complete solution of tropical vector inequalities using matrix sparsification. В J. Bok, D. Hartman, M. Hladík, & M. Rozložník (Ред.),

*MAT TRIAD 2019: International Conference on Matrix Analysis and its Applications: Book of Abstracts*(стр. 38-38). Prague: Charles University in Prague.