On solution of a multidimensional tropical optimization problem using matrix sparsification

Abstract

In this work a complete solution for a problem of minimizing a function defined on vectors with elements from a tropical (idempotent) semifield is proposed. The tropical optimization problem under consideration arises, for instance, when one needs to find the best, in the sense of Chebyshev metric, approximate solution for tropical vector equations, and occurs in various applications, including scheduling, location and decision-making problems. To solve the problem, first, the minimum value of the objective function is obtained, a characterization of the solution set in the form of a system of inequalities is proposed, and one of the solutions is presented. Next, by using a sparsification of the matrix in the problem, an extended set of solutions, and then a complete solution in the form of a family of subsets are derived. Procedures are described that allow reducing the number of subsets, which one needs to examine when constructing the complete solution. It is shown how the complete solution can be represented in parametric way in a compact vector form. The solution obtained extends known results, which are usually confined to the derivation of one solution and do not allow to find the entire solution set. To illustrate the main results of the work numerical examples are given for the solution of a problem in the set of three-dimensional vectors.