Abstract
Interval programming is a modern tool for dealing with uncertainty in practical optimization problems. In this paper, we consider a special class of interval linear programs with interval coefficients occurring only in the objective function and the right-hand-side vector, i.e. programs with a fixed (real) coefficient matrix. The main focus of the paper is on the complexity-theoretic properties of interval linear programs. We study the problems of testing weak and strong feasibility, unboundedness and optimality of an interval linear program with a fixed coefficient matrix. While some of these hard decision problems become solvable in polynomial time, many remain (co-)NP-hard even in this special case. Namely, we prove that testing strong feasibility, unboundedness and optimality remains co-NP-hard for programs described by equations with non-negative variables, while all of the weak properties are easy to decide. For inequality-constrained programs, the (co-)NP-hardness results hold for the problems of testing weak unboundedness and strong optimality. However, if we also require all variables of the inequality-constrained program to be non-negative, all of the discussed problems are easy to decide.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Ben-Tal, A., Ghaoui, L., Nemirovski, A.: Robust Optimization. In: Princeton Series in Applied Mathematics. Princeton University Press (2009)
Garajová, E.: The optimal solution set of interval linear programming problems. Master’s thesis, Charles University, Prague (2016). http://is.cuni.cz/webapps/zzp/detail/168259/?lang=en
Gerlach, W.: Zur Lösung linearer Ungleichungssysteme bei Störung der rechten Seite und der Koeffizientenmatrix. Math. Operationsforsch. Stat. Ser. Optim. 12(1), 41–43 (1981)
Hladík, M.: Complexity of necessary efficiency in interval linear programming and multiobjective linear programming. Optim. Lett. 6(5), 893–899 (2012)
Hladík, M.: Interval linear programming: a survey. In: Mann, Z.A. (ed.) Linear Programming—New Frontiers in Theory and Applications, Chap. 2, pp. 85–120. Nova Science Publishers, New York (2012)
Hladík, M.: Weak and strong solvability of interval linear systems of equations and inequalities. Linear Algebra Appl. 438(11), 4156–4165 (2013). https://doi.org/10.1016/j.laa.2013.02.012
Koníčková, J.: Strong unboundedness of interval linear programming problems. In: 12th GAMM—IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006), p. 26 (2006)
Li, W., Luo, J., Wang, Q., Li, Y.: Checking weak optimality of the solution to linear programming with interval right-hand side. Optim. Lett. 8(4), 1287–1299 (2014)
Rohn, J.: Interval linear programming. In: Linear Optimization Problems with Inexact Data, pp. 79–100. Springer, US (2006)
Rohn, J.: Solvability of systems of interval linear equations and inequalities. In: Linear Optimization Problems with Inexact Data, pp. 35–77. Springer, US (2006)
Acknowledgements
The first two authors were supported by the Czech Science Foundation under the project P402/13-10660S and by the Charles University, project GA UK No. 156317. The work of the first author was also supported by the grant SVV-2017-260452. The work of the third author was supported by the Czech Science Foundation Project no. 17-13086S.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Garajová, E., Hladík, M., Rada, M. (2017). On the Properties of Interval Linear Programs with a Fixed Coefficient Matrix. In: Sforza, A., Sterle, C. (eds) Optimization and Decision Science: Methodologies and Applications. ODS 2017. Springer Proceedings in Mathematics & Statistics, vol 217. Springer, Cham. https://doi.org/10.1007/978-3-319-67308-0_40
Download citation
DOI: https://doi.org/10.1007/978-3-319-67308-0_40
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-67307-3
Online ISBN: 978-3-319-67308-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)