## Abstract

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.

Original language | English |
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Title of host publication | MAT TRIAD 2019 |

Subtitle of host publication | International Conference on Matrix Analysis and its Applications: Book of Abstracts |

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

Place of Publication | Prague |

Publisher | Charles University in Prague |

Pages | 38-38 |

Publication status | Published - 2019 |

Event | International Conference on Matrix Analysis and its Applications - Liblice Duration: 8 Sep 2019 → 13 Sep 2019 Conference number: 8 https://mattriad.math.cas.cz/ |

### Conference

Conference | International Conference on Matrix Analysis and its Applications |
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Abbreviated title | MAT TRIAD 2019 |

Country | Czech Republic |

City | Liblice |

Period | 8/09/19 → 13/09/19 |

Internet address |

## Scopus subject areas

- Algebra and Number Theory