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International Conference on Relational and Algebraic Methods in Computer Science

RAMICS 2017: Relational and Algebraic Methods in Computer Science pp 226–241Cite as

Complete Solution of an Optimization Problem in Tropical Semifield

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10226))

Abstract

We consider a multidimensional optimization problem that is formulated in the framework of tropical mathematics to minimize a function defined on vectors over a tropical semifield (a semiring with idempotent addition and invertible multiplication). The function, given by a matrix and calculated through a multiplicative conjugate transposition, is nonlinear in the tropical mathematics sense. We show that all solutions of the problem satisfy a vector inequality, and then use this inequality to establish characteristic properties of the solution set. We examine the problem when the matrix is irreducible. We derive the minimum value in the problem, and find a set of solutions. The results are then extended to the case of arbitrary matrices. Furthermore, we represent all solutions of the problem as a family of subsets, each defined by a matrix that is obtained by using a matrix sparsification technique. We describe a backtracking procedure that offers an economical way to obtain all subsets in the family. Finally, the characteristic properties of the solution set are used to provide a complete solution in a closed form.

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Acknowledgments

This work was supported in part by the Russian Foundation for Humanities (grant number 16-02-00059). The author is very grateful to three referees for their extremely valuable comments and suggestions, which have been incorporated into the revised version of the manuscript.

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Authors and Affiliations

  1. Saint Petersburg State University, Saint Petersburg, 199034, Russia

    Nikolai Krivulin

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  1. Nikolai Krivulin
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Correspondence to Nikolai Krivulin .

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Editors and Affiliations

  1. Data61, CSIRO, Sydney, New South Wales, Australia

    Peter Höfner

  2. CNRS, Lyon, France

    Damien Pous

  3. University of Sheffield, Sheffield, United Kingdom

    Georg Struth

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Krivulin, N. (2017). Complete Solution of an Optimization Problem in Tropical Semifield. In: Höfner, P., Pous, D., Struth, G. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2017. Lecture Notes in Computer Science(), vol 10226. Springer, Cham. https://doi.org/10.1007/978-3-319-57418-9_14

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  • DOI: https://doi.org/10.1007/978-3-319-57418-9_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-57417-2

  • Online ISBN: 978-3-319-57418-9

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