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Safe bounds in linear and mixed-integer linear programming

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Abstract.

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.

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References

  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)

    MATH  Google Scholar 

  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. 438–451

  3. Czyzyk, J., Mesnier, M.,Moré, J.: The NEOS Server. IEEE J. Comput. Sci. Eng. 5, 68–75 (1998). http://www-neos.mcs.anl.gov/neos/

    Article  Google Scholar 

  4. COCONUT, COntinuous CONstraints - Updating the Technology. http://www.mat.univie.ac.at/∼neum/glopt/coconut/

  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. 135–154

  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. 167–182

  7. ILOG CPLEX 7.1 User’s manual, ILOG, France 2001

  8. Jansson, C.: Zur linearen Optimierung mit unscharfen Daten. Dissertation, Fachbereich Mathematik. Universität Kaiserslautern (1985)

  9. Jansson, C.: Rigorous lower and upper bounds in linear programming. Manuscript, 2002

  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)

    MATH  Google Scholar 

  11. Krawczyk, R.: Fehlerabschätzung bei linearer Optimierung. In: Interval Mathematics, K. Nickel (ed.), lecture Notes in Computer Science 29, Springer, Berlin, 1975, pp. 215–222

  12. Lagarias, J.C.: Bounds for local density of sphere packings and the Kepler conjecture. Discrete Comput. Geom. 27, 165–193 (2002)

    MathSciNet  MATH  Google Scholar 

  13. Marchand, H., Wolsey, L.A.: Aggregation and mixed integer rounding to solve MIPs. Operations research 49, 363–371 (2001). http://www.core.ucl.ac.be/wolsey/mir.ps

    Article  MathSciNet  Google Scholar 

  14. Moore, R.E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia 1981

  15. Netlib Linear Programming Library. http://www.netlib.org/lp

  16. Neumaier, A.: Interval Methods for Systems of Equations. Cambridge Univ. Press, Cambridge 1990

  17. Neumaier, A.: Introduction to Numerical Analysis. Cambridge Univ. Press, Cambridge 2001

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. http://web.mit.edu/rfreund/www/CVfreund.pdf

  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. 63–136

  21. Walster, W.: Interval Arithmetic Solves Nonlinear Problems While Providing Guaranteed Results. FORTE TOOLS Feature Stories, WWW-Manuscript, 2001 http://www.sun.com/forte/info/features/intervals.html

  22. Wolsey, L.A.: Integer Programming. Wiley, 1998

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  1. Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090, Wien, Austria

    Arnold Neumaier & Oleg Shcherbina

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  1. Arnold Neumaier
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  2. Oleg Shcherbina
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Correspondence to Arnold Neumaier.

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Mathematics Subject Classification (2000): primary 90C11, secondary 65G20

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Neumaier, A., Shcherbina, O. Safe bounds in linear and mixed-integer linear programming. Math. Program., Ser. A 99, 283–296 (2004). https://doi.org/10.1007/s10107-003-0433-3

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