Solution of mathematical programming problems using methods of tropical optimization

Abstract

A class of mathematical programming problems, which includes linear and non-linear programming problems of a particular form is considered. First, a linear programming problem is examined and the possibility to derive a direct complete solution of the problem, which does not use known iterative computational procedures and algorithms of linear programming, such as the simplex algorithm is investigated. A direct solution of the problem with a reduced set of constraints and of the minimal dimension is given, and it is shown that, as the dimension increases, the derivation of such solutions becomes an extremely hard task, and thus is hardly feasible. Examples of other problems of linear and non-linear programming, which can be obtained by isomorphic transformations of the problem considered above are presented. Furthermore, an overview of main definitions and preliminary results of tropical mathematics, which are necessary in the subsequent description and application of methods of tropical mathematics is given. A tropical optimization problem is formulated, and direct complete solutions for the problem and its special cases are described. The above problems of linear and non-linear programming reduce to the tropical optimization problem, which provides a direct complete solution to the former problems in terms of tropical mathematics. For the linear programming problem with a reduced set of constraints a representation of the solution in the usual mathematical terms is demonstrated. Refs 17.