Complete algebraic solution of multidimensional optimization problems in tropical semifield

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3 Scopus citations


We consider multidimensional optimization problems that are formulated in the framework of tropical mathematics to minimize functions defined on vectors over a tropical semifield (a semiring with idempotent addition and invertible multiplication). The functions, given by a matrix and calculated through multiplicative conjugate transposition, are nonlinear in the tropical mathematics sense. We start with known results on the solution of the problems with irreducible matrices. To solve the problems in the case of arbitrary (reducible) matrices, we first derive the minimum value of the objective function, and find a set of solutions. We show that all solutions of the problem satisfy a system of vector inequalities, and then use these inequalities to establish characteristic properties of the solution set. Furthermore, all solutions of the problem are represented 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 allows one to reduce the brute-force generation of sparsified matrices by skipping those, which cannot provide solutions, and thus offers an economical way to obtain all subsets in the family. Finally, the characteristic properties of the solution set are used to provide complete solutions in a closed form. We illustrate the results obtained with simple numerical examples.
Original languageEnglish
Pages (from-to)26-40
Number of pages15
JournalJournal of Logical and Algebraic Methods in Programming
Early online date18 May 2018
StatePublished - Oct 2018
EventThe 16th International Conference on Relational and Algebraic Methods in Computer Science - Université de Lyon, Lyon, France
Duration: 15 May 201718 May 2017
Conference number: 16

Scopus subject areas

  • Control and Optimization
  • Algebra and Number Theory
  • Management Science and Operations Research


  • tropical semifield
  • tropical optimization
  • matrix sparsification
  • complete solution
  • backtracking
  • Backtracking
  • Tropical semifield
  • Complete solution
  • Tropical optimization
  • Matrix sparsification

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