On relaxation methods for systems of linear inequalities

In their classical papers Agmon and Motzkin and Schoenberg introduced a relaxation method to find a feasible solution for a system of linear inequalities. So far the method was believed to require infinitely many iterations on some problem instances since it could (depending on the dimension of the set of feasible soltions) converge asymptotically to a feasible solution, if one exists. Hence it could not be used to determine infeasibility.Using two lemma's basic to Khachian's polynomially bounded algorithm we can show that the relaxation method is finite in all cases and thus can handle infeasible systems as well. In spite of more refined stopping criteria the worst case behaviour of the relaxation method is not polynomially bounded as examplified by a class of problems constructed here.     

On nonnegative solutions of random systems of linear inequalities

The set of nonnegative solutions of a system of linear equations or inequalities is a convex polyhedron. If the coefficients of the system are chosen at random, the number of vertices of this polyhedron is a random variable. Its expected value, dependent on the probability distribution of the coefficients, which are assumed to be nonnegative throughout, is investigated, and a distribution-independent upper bound for this expected value is established.     

Topological stability of linear semi-infinite inequality systems

In this note, we analyze the relationship between the lower semicontinuity of the feasible set mapping for linear semi-infinite inequality systems and the so-called topological stability, which is held when the solution sets of all the systems obtained by sufficiently small perturbations of the data are homeomorphic to each other. This topological stability and its relation with the Mangasarian-Fromovitz constraints qualification have been studied deeply by Jongen et al. in Ref. 1. The main difference of our approach is that we are not assuming any kind of structure for the index set and, consequently, any particular property for the functional dependence between the inequalities and the associated indices. In addition, we deal with systems whose solution sets are not necessarily bounded.This work has been supported partially by the DGICYT of Spain, Grant PB93-0943, by Generalitat Valenciana, Grant GV-2219/94, and by IVEI, Grant 003/026.The authors would like to thank J. E. Martínez Legaz for his valuable comments.     

Irreducibly inconsistent systems of linear inequalities

Irreducibly inconsistent systems of linear inequalities are considered from the point of view of applying simplex-like methods for investigation.We give necessary and sufficient conditions, for a system to be irreducibly inconsistent, which can be found by applying a simplex algorithm. Next we consider general inconsistent systems. The simplex method can be used to identify irreducibly inconsistent subsystem. Such subsystems can give structural insight in the inconsistency and are used in trying to make the overall system consistent. This can be done in an easy way e.g. by right-hand side manipulations. Finally the problem of finding a ‘cheapest’ way of doing this can be formulated and solved as an extended linear program.     

On inequalities of Lyapunov for linear Hamiltonian systems on time scales

In this paper, we establish several new Lyapunov type inequalities for linear Hamiltonian systems on an arbitrary time scale T when the end-points are not necessarily usual zeroes, but rather, generalized zeroes, which generalize and improve all related existing ones including the continuous and discrete cases.     

Some preconditioners for systems of linear inequalities

We show that a combination of two simple preprocessing steps would generally improve the conditioning of a homogeneous system of linear inequalities. Our approach is based on a comparison among three different but related notions of conditioning for linear inequalities.     

On the non-polynomiality of the relaxation method for systems of linear inequalities

A class of linear programs is given in which the relaxation method for inequalities, under the same operating rules as Khacian's method, is not polynomial in the length of the input. This result holds for any value of the relaxation parameter.This research was supported in part by the D.G.E.S. (Quebec), the N.S.E.R.C. of Canada under grant A 4152, and the S.S.H.R.C. of Canada.     

Flexible solutions to systems of linear inequalities

In this paper we present two methods for calculating solutions to a system of linear inequalities which have certain prescribed distance properties to the constraints (referred to as flexibility properties). The first method (based on the calculation of the centre of an inscribed sphere, which is a well-known problem in linear programming) allows changes in the solution in any direction over a certain distance without becoming infeasible. The second method consists in the calculation of a solution with flexibility properties with respect to a prescribed coordinate system. It comprises an iterative procedure inside the area of feasible solutions using simple arithmetic operations. The major part of the paper will be devoted to the second method, since it is new and has attractive properties. Both methods are particularly applicable to the analysis of linear programming models with uncertain and inaccurate constraint parameters and in cases where operational adjustments are desired.     

On stable uniqueness in linear semi-infinite optimization

This paper is intended to provide conditions for the stability of the strong uniqueness of the optimal solution of a given linear semi-infinite optimization (LSIO) problem, in the sense of maintaining the strong uniqueness property under sufficiently small perturbations of all the data. We consider LSIO problems such that the family of gradients of all the constraints is unbounded, extending earlier results of Nürnberger for continuous LSIO problems, and of Helbig and Todorov for LSIO problems with bounded set of gradients. To do this we characterize the absolutely (affinely) stable problems, i.e., those LSIO problems whose feasible set (its affine hull, respectively) remains constant under sufficiently small perturbations.     

Fuzzification of linear systems of equalities and inequalities

Translated from Aktual'nye Voprosy Prikladnoi Matematiki, pp. 73–80, 1989.     

Iterative solution of inconsistent systems of linear inequalities

In 1980 S.P. Han proposed a finitely terminating (in exact arithmetic) algorithm for solving an inconsistent system of linear inequalities in a least squares sense. This algorithm uses a singular value decomposition of a submatrix of the problem matrix on each iteration, making it impractical for all but only smaller problems. In this paper we show that a modification of Han's algorithm allows us to introduce an iterative approximation to the singular value decomposition solution. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A discretization method for systems of linear inequalities

We investigate a method for approximating a convex domainCR ⁿ described by a (possibly infinite) set of linear inequalities. In contrast to other methods, the approximating domains (polyhedrons) lie insideC. We discuss applications to semi-infinite programming and present numerical examples.The paper was written at the Institut für Angewandte Mathematik, Universität Hamburg, Hamburg, West Germany. The author thanks Prof. U. Eckhardt, Dr. K. Roleff, and Prof. B. Werner for helpful discussions.     

Projecting systems of linear inequalities with binary variables

We investigate methods for projecting out 0–1 variables from a system of linear inequalities. After reviewing some special cases like monotone polyhedra and the satisfiability problem, we examine why Fourier elimination cannot be applied to the general case. Finally, we give a procedure based on disjunctive programming for solving the general case. We also discuss a simpler procedure applicable only under certain conditions.     

Lyapunov inequalities and stability for linear Hamiltonian systems

In this paper, we will establish several Lyapunov inequalities for linear Hamiltonian systems, which unite and generalize the most known ones. For planar linear Hamiltonian systems, the connection between Lyapunov inequalities and estimates of eigenvalues of stationary Dirac operators will be given, and some optimal stability criterion will be proved.