Solving the linear interval tolerance problem

Abstract

For the interval linear system Ax = b, the linear tolerance problem is considered that requires inner evaluation of the tolerable solution set Σ∀∃(A, b) = {x ∈ Rn ∣ (∀A ∈ A)(Ax ∈ b)} formed by all point vectors x such that the product Ax remains within b for all possible A ∈ A. Along with the simple incompatibility criterion, we develop comprehensive solvability theory for the linear tolerance problem that not only settles whether Σ∀∃ is empty or not, but also enables modification of the problem to ensure its desired properties. To conclude, we advance several numerical methods of various accuracy and complexity for construction of an interval solution to the linear tolerance problem around a given center.