On the solution set of a linear equation with the right-hand side and operator given by intervals

Quantifier-free descriptions for quantifier solutions to interval linear systems of relations

“…Usually, the solution sets to interval systems of equations and inequalities are described using real arithmetic in R [3, chapter 2; 2, pp. 93-95; 5,10,11,15,16,22], since it is simple, familiar, has good properties, and we can apply developed numerical methods in R. For various subclasses of interval quantifier systems of linear equations, a number of quantifier-free descriptions have been obtained in interval arithmetics [1,14,20,21]. Despite poor algebraic properties of the interval arithmetics (the absence of the distributivity, etc.…”

“…Lakeyev and S.I. Noskov in [10], and they also presented, as an evident one, a description, in IR, for the set of controllable solutions to the equation Ax = b (see also [17]). The remaining descriptions for the basic types of quantifier solutions to the equation Ax = b in the interval arithmetics IR and KR are also known, for example, as obvious corollaries of the statement (11), proved by S.P Shary in [18,19].…”

AE solutions and AE solvability to general interval linear systems

“…• weak solutions to Ax = b; see [3,11,12] • weak solutions to Ax ≤ b; see [3] • controllable solutions to Ax = b; see [3,12]…”

Checking solvability of systems of interval linear equations and inequalities via mixed integer programming

“…In this paper, we consider the problems of checking strong solvability and feasibility of linear interval equations; checking weak solvability of linear interval equations and inequalities; and finding control solutions of linear interval equations. It is known that these problems are NP-hard [7,8,13,14,16]. The reader is referred to the classical book [3] for background on computational complexity theory.…”