On Maximal Inner Estimation of the Solution Sets of Linear Systems with Interval Parameters

The purpose of this paper is to inquire the connection between maximal inner interval estimates of the solution sets to interval linear system and solutions of the dualization equation in Kaucher interval arithmetic. The results of our work are as follows: 1) a criterion of inner interval estimate of the solution set, 2) a criterion for a solution of dualization equation to be a maximal inner interval estimate of the solution set, 3) a criterion for multiplication by an interval matrix to be upper strictly isotone. 1. Notation We use Latin letters for real objects: small for numbers and vectors (a, b, c,...) and capital for matrices (A,B,C,...). By interval, we call an object of the form [x, x] with x, x ∈ R (not necessarily x ≤ x). If x ≤ x, then [x, x] is said to be a proper interval. At the same time, we think of a proper interval [x, x] as the set of all real numbers between the points x and x, i.e., {x ∈ R | x ≤ x ≤ x}. A vector (matrix) with interval components is called an interval vector (matrix). An interval vector (matrix) is said to be proper if all its components are proper intervals. Similar to the one-dimensional case, we think of a proper interval vector [x, x] := ([x1, x1],..., [xn, xn]) as the set of all real vectors bounded by the vectors x and x, i.e., {x ∈ R n | xj ≤ xj ≤ xj, j = 1,..., n}. We use boldface Latin letters for interval objects: small for intervals and interval vectors (a, b, c,...) and capital for interval matrices (A,B,C,...). The bold symbol 0 designates the interval [0, 0].