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On the Properties of Interval Linear Programs with a Fixed Coefficient Matrix

  • Elif Garajová
  • Milan Hladík
  • Miroslav Rada
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 217)

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.

Keywords

Interval linear programming Computational complexity 

Notes

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.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics and Physics, Department of Applied MathematicsCharles UniversityPragueCzech Republic
  2. 2.Department of Financial Accounting and AuditingUniversity of Economics, PraguePragueCzech Republic

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