TY - JOUR

T1 - Solution of a multidimensional tropical optimization problem using matrix sparsification

AU - Krivulin, N. K.

AU - Sorokin, V. N.

N1 - Krivulin, N.K., Sorokin, V.N. Solution of a Multidimensional Tropical Optimization Problem Using Matrix Sparsification. Vestnik St.Petersb. Univ.Math. 51, 66–76 (2018). https://doi.org/10.3103/S1063454118010065

PY - 2018

Y1 - 2018

N2 - A complete solution is proposed for the problem of minimizing a function defined on vectors with elements in a tropical (idempotent) semifield. The tropical optimization problem under consideration arises, for example, when we need to find the best (in the sense of the Chebyshev metric) approximate solution to tropical vector equations and occurs in various applications, including scheduling, location, and decision-making problems. To solve the problem, the minimum value of the objective function is determined, the set of solutions is described by a system of inequalities, and one of the solutions is obtained. Thereafter, an extended set of solutions is constructed using the sparsification of the matrix of the problem, and then a complete solution in the form of a family of subsets is derived. Procedures that make it possible to reduce the number of subsets to be examined when constructing the complete solution are described. It is shown how the complete solution can be represented parametrically in a compact vector form. The solution obtained in this study generalizes known results, which are commonly reduced to deriving one solution and do not allow us to find the entire solution set. To illustrate the main results of the work, an example of numerically solving the problem in the set of three-dimensional vectors is given.

AB - A complete solution is proposed for the problem of minimizing a function defined on vectors with elements in a tropical (idempotent) semifield. The tropical optimization problem under consideration arises, for example, when we need to find the best (in the sense of the Chebyshev metric) approximate solution to tropical vector equations and occurs in various applications, including scheduling, location, and decision-making problems. To solve the problem, the minimum value of the objective function is determined, the set of solutions is described by a system of inequalities, and one of the solutions is obtained. Thereafter, an extended set of solutions is constructed using the sparsification of the matrix of the problem, and then a complete solution in the form of a family of subsets is derived. Procedures that make it possible to reduce the number of subsets to be examined when constructing the complete solution are described. It is shown how the complete solution can be represented parametrically in a compact vector form. The solution obtained in this study generalizes known results, which are commonly reduced to deriving one solution and do not allow us to find the entire solution set. To illustrate the main results of the work, an example of numerically solving the problem in the set of three-dimensional vectors is given.

KW - idempotent semifield

KW - tropical optimization

KW - Chebyshev approximation

KW - complete solution

KW - matrix sparsification

KW - ALGEBRA

UR - http://www.scopus.com/inward/record.url?scp=85045076747&partnerID=8YFLogxK

U2 - 10.3103/S1063454118010065

DO - 10.3103/S1063454118010065

M3 - Article

VL - 51

SP - 66

EP - 76

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 1

ER -