2023 , Volume 28, № 5, p.87-100

The article considers the solution of interval systems of linear algebraic equations (ISLAE) and presents two simple, fast and reliable ways to find the unconstrained maximum of the recognizing functional of the tolerance set of solutions.

The recognizing functional is a special function that, using the sign and magnitude of its values, makes it possible to judge whether a point belongs to the tolerable solution set and gives a measure of “tolerable compatibility” of the interval system of equations at the point. The technique based on the use of the recognizing functional is one of the powerful tools for studying interval systems of equations and their tolerable solution sets. The paper presents two methods for computing the maximum of the recognizing functional, united by the general idea of reducing the original maximization problem to solving one or more special linear programming problems built on a given interval system. The theoretical basis of the proposed methods is presented. The issues of software implementation are discussed, as well as computational experiments with model problems.

The reduction is based on the Rohn theorem on the characterization of points of the tolerable solution sets for ISLAE, which allows us to representing it as solutions to a system of linear inequalities. This opens up the possibility of using linear programming methods for unconstrained maximization of the recognition functional and for studying whether the tolerable solution set is empty or notnempy.

The methods proposed in this article naturally complement the currently existing algorithms for computing the maximum of the recognizing functional, which use numerical methods of non- smooth optimization. Our new methods are both technologically advanced and easy to implement, since they allow us to utilize developed theory and ready-made software products for solving linear programming problems.