Abstract
An algorithm is presented that produces an integer vector nearly parallel to a given vector. The algorithm can be used to discover exact rational solutions of homogeneous or inhomogeneous linear systems of equations, given a sufficiently accurate approximate solution.
As an application, we show how to verify rigorously the feasibility of degenerate vertices of a linear program with integer coefficients, and how to recognize rigorously certain redundant linear constraints in a given system of linear equations and inequalities. This is a first step towards the handling of degeneracies and redundandies within rigorous global optimization codes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
COCONUT, COntinuous CONstraints—Updating the Technology, an IST Project funded by the European Union, http://www.mat.univie.ac.at/~neum/glopt/coconut/
Golub, G. H. and van Loan, C. F.: Matrix Computations, 2nd ed., Johns Hopkins Univ. Press, Baltimore, 1989.
Hansen, R. E.: Global Optimization Using Interval Analysis, Dekker, New York, 1992.
Kearfott, R. B.: Rigorous Global Search: Continuous Problems, Kluwer Academic Publishers, Dordrecht, 1996.
Neumaier, A.: Interval Methods for Systems of Equations, Cambridge Univ. Press, Cambridge, 1990.
Neumaier, A.: Introduction to Numerical Analysis, Cambridge Univ. Press, Cambridge, 2001.
Ratschek, H. and Rokne, J.: New Computer Methods for Global Optimization, Wiley, New York, 1988.
Van Hentenryck, P., Michel, L. and Deville, Y.: Numerica. A Modeling Language for Global Optimization, MIT Press, Cambridge, 1997.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Huyer, W., Neumaier, A. Integral Approximation of Rays and Verification of Feasibility. Reliable Computing 10, 195–207 (2004). https://doi.org/10.1023/B:REOM.0000032108.23609.bc
Issue Date:
DOI: https://doi.org/10.1023/B:REOM.0000032108.23609.bc