Mathematical Programming

, Volume 99, Issue 2, pp 283–296 | Cite as

Safe bounds in linear and mixed-integer linear programming

  • Arnold Neumaier
  • Oleg Shcherbina


Current mixed-integer linear programming solvers are based on linear programming routines that use floating-point arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. An example is given where many state-of-the-art MILP solvers fail. It is then shown how, using directed rounding and interval arithmetic, cheap pre- and postprocessing of the linear programs arising in a branch-and-cut framework can guarantee that no solution is lost, at least for mixed-integer programs in which all variables can be bounded rigorously by bounds of reasonable size.


linear programming mixed-integer programming rounding errors directed rounding interval arithmetic branch-and-cut lower bounds mixed-integer rounding generalized Gomory cut safe cuts safe presolve certificate of infeasibility 


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  1. 1.
    Brearley, A.L., Mitra, G., Williams, H.P.: An analysis of mathematical programming problems prior to applying the simplex method. Math. Programming 8, 54–83 (1975)zbMATHGoogle Scholar
  2. 2.
    Ceria, S., Cornuejols, G., Dawande, M.: Combining and strengthening Gomory cuts. In: Integer Programming and Combinatorial Optimization, E. Balas, J. Clausen (eds.), Springer, Berlin, 1995, pp. 438–451Google Scholar
  3. 3.
    Czyzyk, J., Mesnier, M.,Moré, J.: The NEOS Server. IEEE J. Comput. Sci. Eng. 5, 68–75 (1998). Scholar
  4. 4.
    COCONUT, COntinuous CONstraints - Updating the Technology.∼neum/glopt/coconut/Google Scholar
  5. 5.
    Fourer, R., Gay, D.M.: Experience with a primal presolve algorithm. In: Large Scale Optimization: State of the Art, W.W. Hager, D.W. Hearn, P.M. Pardalos (eds.), Kluwer, Dordrecht, 1994, pp. 135–154Google Scholar
  6. 6.
    Gropp, W., Moré, J.: Optimization Environments and the NEOS Server. In: Approximation Theory and Optimization, M.D. Buhmann, A. Iserles (eds.), Cambridge University Press, Cambridge, 1997, pp. 167–182Google Scholar
  7. 7.
    ILOG CPLEX 7.1 User’s manual, ILOG, France 2001Google Scholar
  8. 8.
    Jansson, C.: Zur linearen Optimierung mit unscharfen Daten. Dissertation, Fachbereich Mathematik. Universität Kaiserslautern (1985)Google Scholar
  9. 9.
    Jansson, C.: Rigorous lower and upper bounds in linear programming. Manuscript, 2002Google Scholar
  10. 10.
    Jansson, C., Rump, S.M.: Rigorous solution of linear programming problems with uncertain data. ZOR – Methods and Models of Operations Research 35, 87–111 (1991)zbMATHGoogle Scholar
  11. 11.
    Krawczyk, R.: Fehlerabschätzung bei linearer Optimierung. In: Interval Mathematics, K. Nickel (ed.), lecture Notes in Computer Science 29, Springer, Berlin, 1975, pp. 215–222Google Scholar
  12. 12.
    Lagarias, J.C.: Bounds for local density of sphere packings and the Kepler conjecture. Discrete Comput. Geom. 27, 165–193 (2002)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Marchand, H., Wolsey, L.A.: Aggregation and mixed integer rounding to solve MIPs. Operations research 49, 363–371 (2001). Scholar
  14. 14.
    Moore, R.E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia 1981Google Scholar
  15. 15.
    Netlib Linear Programming Library. Scholar
  16. 16.
    Neumaier, A.: Interval Methods for Systems of Equations. Cambridge Univ. Press, Cambridge 1990Google Scholar
  17. 17.
    Neumaier, A.: Introduction to Numerical Analysis. Cambridge Univ. Press, Cambridge 2001Google Scholar
  18. 18.
    Neumaier, A.: A simple derivation of the Hansen-Bliek-Rohn-Ning-Kearfott enclosure for linear interval equations. Reliable Computing 5, 131–136 (1999), Erratum. Reliable Computing 6, 227 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Ordóñez, F., Freund, R.M.: Computational experience and the explanatory value of condition numbers for linear optimization. MIT Operations Research Center Working paper # OR361-02, submitted to SIAM J. Optimization. Scholar
  20. 20.
    Rump, S.M.: Verification methods for dense and sparse systems of equations. In: J. Herzberger (ed.), Topics in Validated Computations - Studies in Computational Mathematics, Elsevier, Amsterdam, 1994, pp. 63–136Google Scholar
  21. 21.
    Walster, W.: Interval Arithmetic Solves Nonlinear Problems While Providing Guaranteed Results. FORTE TOOLS Feature Stories, WWW-Manuscript, 2001 Scholar
  22. 22.
    Wolsey, L.A.: Integer Programming. Wiley, 1998Google Scholar

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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Institut für MathematikUniversität WienWienAustria

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