Solvability of systems of interval linear equations and inequalities

This chapter deals with solvability and feasibility (i.e., nonnegative solvability) of systems of interval linear equations and inequalities. After a few preliminary sections, we delineate in Section 2.6 eight decision problems (weak solvability of equations through strong feasibility of inequalities) that are then solved in eight successive sections 2.7 to 2.14. It turns out that four problems are solvable in polynomial time and four are NP-hard. Some of the results are easy (Theorem 2.13)) some difficult to prove (Theorem 2.14), and some are surprising (Theorem 2.24). Although solutions of several of them are already known, the complete classification of the eight problems given here is new. Some special cases (tolerance, control and algebraic solutions, systems with square matrices) are treated in Sections 2.16 to 2.19. The last, Section 2.21 contains additional notes and references to the material of this chapter. Some of the results find later applications in interval linear programming (Chapter 3) . We use the following notations. The ith row of a matrix A is denoted by A,. and the j t h column by A.j. For two matrices A, B of the same size, inequalities like A I B or A < B are understood componentwise. A is called nonnegative if 0 5 A; AT is the transpose of A. The absolute value of a matrix A = (aij) is defined by lA1 = (laijl). We use the following easy-to-prove properties valid whenever the respective operations and inequalities are defined.