Sufficient Conditions for the Joined Set of Solutions of the Overdetermined Interval System of Linear Algebraic Equations Membership to Only One Orthant

TL;DR

This paper establishes polynomial-time verifiable conditions ensuring the solution set of overdetermined interval linear systems lies entirely within one orthant, enabling efficient solution and stable coefficient signs in linear models.

Contribution

It introduces new polynomial-time conditions for the solution set of ISLAE to be confined to a single orthant, facilitating efficient solution and sign stability of model coefficients.

Findings

Solutions can be found in polynomial time using linear programming.

The solution set forms a convex bounded polyhedron within one orthant.

Coefficients of the resulting linear model maintain consistent signs within the admissible domain.

Abstract

Interval systems of linear algebraic equations (ISLAE) are considered in the context of constructing of linear models according to data with interval uncertainty. Sufficient conditions for boundedness and convexity of an admissible domain (AD) of ISLAE and its belonging to only one orthant of an $n$-dimensional space are proposed, which can be verified in polynomial time by the methods of computational linear algebra. In this case, AD ISLAE turns out to be a convex bounded polyhedron, entirely lying in the corresponding ortant. These properties of AD ISLAE allow, firstly, to find solutions to the corresponding ISLAE in polynomial time by linear programming methods (while finding a solution to ISLAE of a general form is an NP-hard problem). Secondly, the coefficients of the linear model obtained by solving the corresponding ISLAE have an analogue of the significance property of the coefficient of the linear model, since the coefficients of the linear model do not change their sign within the limits of the AD. The formulation and proof of the corresponding theorem are presented. The error estimation and convergence of an arbitrary solution of ISLAE to the normal solution of a hypothetical exact system of linear algebraic equations are also investigated. An illustrative numerical example is given.