Jiří Rohn

Checking robust nonsingularity is NP-hard

We consider the following problem: givenk+1 square matrices with rational entries,A0,A1,...,Ak, decide ifA0+r1A1+···+rkAk is nonsingular for all possible choices of real numbersr1, ...,rk in the interval [0, 1]. We show that this question, which is closely related to the robust stability problem, is NP-hard. The proof relies on the new concept ofradius of nonsingularity of a square matrix and on the relationship between computing this radius and a graph-theoretic problem.

Enclosing solutions of overdetermined systems of linear interval equations

A method for enclosing solutions of overdetermined systems of linear interval equations is described. Several aspects of the problem (algorithm, enclosure improvement, optimal enclosure) are studied.AbstractПредставлен метод нахождення длд решений переопределенных линейных интервальных систем уравненнй. Онисано несколько аснектов заачи — сам алгорнтм, сужение оболючек, нахождение онтимальных оболочек.

Computing the norm ∥A∥∞,1 is NP-hard ∗

It is proved that computing the subordinate matrix norm ∥A∥∞1 is NP-hard, Even more, existence of a polynomial-time algorithm for computing this norm with relative accuracy less than 1/(4n2 ), where n is matrix size, implies P = NP.

DUALITY IN INTERVAL LINEAR PROGRAMMING

Publisher Summary This chapter presents a duality theorem for interval linear programming and also presents various forms of optimality criteria. It also presents a proof of the duality theorem for the problems (P) and (D). It is found that if both (P) and (D) are feasible, then they both have optimal solutions and have a common optimal value. If one of the problems (P) or (D) is infeasible, then the second one is either infeasible or unbounded. It is found that if (P) is infeasible, then so is (Po), and the duality theorem states that (Do) is infeasible or unbounded. A Parkas-type theorem can be derived directly from the duality theorem.

Complexity of Some Linear Problems with Interval Data

During the recent years, a number of linear problems with interval data have been proved to be NP-hard. These results may seem rather obscure as regards the ways in which they were obtained. This survey paper is aimed at demonstrating that in fact it is not so, since many of these results follow easily from the recently established fact that for the subordinate matrix norm ‖ · ‖∞,1 it is NP-hard to decide whether ‖A‖∞,1 ≥ 1 holds, even in the class of symmetric positive definite rational matrices. After a brief introduction into the basic topics of the complexity theory in Section 1 and formulation of the underlying norm complexity result in Section 2, we present NP-hardness results for checking properties of interval matrices (Section 3), computing enclosures (Section 4), solvability of rectangular linear interval systems (Section 5), and linear and quadratic programming (Section 6). Due to space limitations, proofs are mostly only ketched to reveal the unifying role of the norm complexity result; technical details are omitted.

On the invariance of the sign pattern of matrix eigenvectors under perturbation

Abstract We study how much perturbation δ A in a real matrix A is allowed for the i th real eigenvector not to change sign.

Overestimations in bounding solutions of perturbed linear equations

Abstract It is proved that some classical bounds on solutions of perturbed systems of linear equations may yield arbitrarily large overestimations for arbitrarily narrow perturbations. The proofs are constructive.

Interval linear systems with prescribed column sums

Abstract Nonnegative solutions of an interval linear system A I x = b I ( A I being an interval matrix and b I an interval vector) with additional column sum restrictions of the type σ i α ij a ij ∈[ c -j , c j ] ∀j are described by a system of linear inequalities with auxiliary variables.

Checking bounds on solutions of linear interval equations is NP-hard

Abstract We prove that it is NP-hard to decide whether the solution set of a system of linear interval equations is contained in a given interval vector, even in the case that the system matrix is strongly regular.

Linear Interval Equations: Computing Enclosures with Bounded Relative Overestimation is NP-Hard

It is proved that if there exists a polynomial-time algorithm for enclosing solutions of linear interval equations with relative overestimation better than \(\frac{4}{{{n^2}}}\) (where n is the number of equations), then P=NP. The result holds for the symmetric case as well.