Journal of Global Optimization

, Volume 68, Issue 3, pp 677–683 | Cite as

A review of computation of mathematically rigorous bounds on optima of linear programs

  • Jared T. Guilbeau
  • Md. Istiaq Hossain
  • Sam D. Karhbet
  • Ralph Baker Kearfott
  • Temitope S. Sanusi
  • Lihong Zhao
Short Communication

Abstract

Linear program solvers sometimes fail to find a good approximation to the optimum value, without indicating possible failure. However, it may be important to know how close the value such solvers return is to an actual optimum, or even to obtain mathematically rigorous bounds on the optimum. In a seminal 2004 paper, Neumaier and Shcherbina, propose a method by which such rigorous lower bounds can be computed; we now have significant experience with this method. Here, we review the technique. We point out typographical errors in two formulas in the original publication, and illustrate their impact. Separately, implementers and practitioners can also easily make errors. To help implementers avoid such problems, we cite a technical report where we explain things to mind and where we present rigorous bounds corresponding to alternative formulations of the linear program.

Keywords

Linear program Interval computations Mathematically rigorous lower bounds Branch and bound Relaxation Global optimization 

References

  1. 1.
    Althaus, E., Dumitriu, D.: Certifying feasibility and objective value of linear programs. Op. Res. Lett. 40(4), 292–297 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Applegate, D.L., Cook, W., Dash, S., Espinoza, D.G.: Exact solutions to linear programming problems. Op. Res. Lett. 35(6), 693–699 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Audet, C., Hansen, P., Messine, F., Ninin, J.: The small octagons of maximal width. Discrete Comput. Geom. 49(3), 589–600 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Baharev, A., Kolev, L., Rév, E.: Computing multiple steady states in homogeneous azeotropic and ideal two-product distillation. AIChE J. 57(6), 1485–1495 (2011)CrossRefGoogle Scholar
  5. 5.
    Baharev, A., Rév, E.: Reliable computation of equilibrium cascades with affine arithmetic. AIChE J. 54(7), 1782–1797 (2008)CrossRefGoogle Scholar
  6. 6.
    Benhamou, F., Granvilliers, L.: Chapter 16—continuous and interval constraints. Foundations of artificial intelligence. In: van Beek, P., Rossi, F., Walsh, T. (eds.) Handbook of Constraint Programming, vol. 2, pp. 571–603. Elsevier, New York (2006)CrossRefGoogle Scholar
  7. 7.
    Boccia, M., Sforza, A., Sterle, C., Vasilyev, I.: A cut and branch approach for the capacitated \(p\)-median problem based on Fenchel cutting planes. J. Math. Modell. Algorithms 7(1), 43–58 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chindelevitch, L., Trigg, J., Regev, A., Berger, B.: An exact arithmetic toolbox for a consistent and reproducible structural analysis of metabolic network models. Nat. Commun. 5(4893), 1–10 (2014)Google Scholar
  9. 9.
    Cook, W., Dash, S., Fukasawa, R., Goycoolea, M.: Numerically safe gomory mixed-integer cuts. INFORMS J. Comput. 21(4), 641–649 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cook, W., Koch, T., Steffy, D.E., Wolter, K.: A hybrid branch-and-bound approach for exact rational mixed-integer programming. Math. Program. Comput. 5(3), 305–344 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cornuéjols, G., Margot, F., Nannicini, G.: On the safety of Gomory cut generators. Math. Program. Comput. 5(4), 345–395 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Domes, F., Neumaier, A.: Rigorous filtering using linear relaxations. J. Glob. Optim. 53(3), 441–473 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fukasawa, R.: Gomory cuts. In: Cochran, J.J., Cox, L.A., Keskinocak, P., Kharoufeh, J.P., Cole Smith, J. (eds.) Wiley Encyclopedia of Operations Research and Management Science. Wiley, New York (2011)Google Scholar
  14. 14.
    Gouttefarde, M., Daney, D., Merlet, J.-P.: Interval-analysis-based determination of the wrench-feasible workspace of parallel cable-driven robots. IEEE Trans. Robot. 27(1), 1–13 (2011)CrossRefMATHGoogle Scholar
  15. 15.
    Guilbeau, J.T., Hossain, M.I., Karhbet, S.D., Kearfott, R.B., Sanusi, T.S., Zhao, L.: Advice for mathematically rigorous bounds on optima of linear programs. Technical report, University of Louisiana at Lafayette. http://interval.louisiana.edu/preprints/reformulations-for-rigorous (2016)
  16. 16.
    Jansson, C.: Rigorous lower and upper bounds in linear programming. SIAM J. Optim. 14(3), 914–935 (2004)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Jansson, C., Chaykin, D., Keil, C.: Rigorous error bounds for the optimal value in semidefinite programming. SIAM J. Numer. Anal. 46(1), 180–200 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jansson, C.: On verified numerical computations in convex programming. Jpn. J. Ind. Appl. Math. 26(2–3), 337–363 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Jordan, N., Messine, F., Hansen, P.: 4OR. Reliab. Affine Relax. Method Glob. Optim. 13(3), 247–277 (2015)Google Scholar
  20. 20.
    Kearfott, R.B.: Discussion and empirical comparisons of linear relaxations and alternate techniques in validated deterministic global optimization. Optim. Methods Softw. 21, 715–731 (2006)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kearfott, R.B., Castille, J., Tyagi, G.: GlobSol user guide. Optim. Methods Softw. 24(4–5), 687–708 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lebbah, Y., Michel, C., Rueher, M., Daney, D., Merlet, J.: Efficient and safe global constraints for handling numerical constraint systems. SIAM J. Numer. Anal. 42(5), 2076–2097 (2005)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lebbah, Y., Michel, C., Rueher, M.: A rigorous global filtering algorithm for quadratic constraints. Constraints 10(1), 47–65 (2005)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Margot, F.: Testing cut generators for mixed-integer linear programming. Math. Program. Comput. 1(1), 69–95 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Neumaier, A., Shcherbina, O., Huyer, W., Vinkó, T.: A comparison of complete global optimization solvers. Math. Program. 103(2), 335–356 (2005)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Neumaier, A., Shcherbina, O.: Safe bounds in linear and mixed-integer programming. Math. Program. 99(2), 283–296 (2004)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Porta, J.M., Ros, L., Thomas, F.: A linear relaxation technique for the position analysis of multiloop linkages. IEEE Trans. Robot. 25(2), 225–239 (2009)CrossRefGoogle Scholar
  28. 28.
    Prodan, I., Zio, E.: A model predictive control framework for reliable microgrid energy management. Int. J. Electr. Power Energy Syst. 61, 399–409 (2014)CrossRefGoogle Scholar
  29. 29.
    Ralph Baker, K., Castille, J., Tyagi, G.: A general framework for convexity analysis in deterministic global optimization. J. Glob. Optim. 56(3), 765–785 (2013)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Ratschan, S., She, Z.: Providing a basin of attraction to a target region of polynomial systems by computation of Lyapunov-like functions. SIAM J. Control Optim. 48(7), 4377–4394 (2010)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Smith, A.P.: Fast construction of constant bound functions for sparse polynomials. J. Glob. Optim. 43(2–3), 445–458 (2009)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    van Nooijen, R.R., Kolechkina, A.: Speed of discrete optimization solvers for real time sewer control. Urban Water J. 10(5), 354–363 (2013)CrossRefGoogle Scholar
  33. 33.
    Xiang, Y., Lan, T.: Smart Pricing Cloud Res. Wiley, New York (2014)Google Scholar
  34. 34.
    Xuan-Ha, V., Sam-Haroud, D., Faltings, B.: Enhancing numerical constraint propagation using multiple inclusion representations. Ann. Math. Artif. Intell. 55(3–4), 295–354 (2009)MathSciNetMATHGoogle Scholar
  35. 35.
    Yi, X., Shunze, W., Zang, H., Hou, G.: An interval joint-probabilistic programming method for solid waste management: a case study for the city of Tianjin, China. Front. Environ. Sci. Eng. 8(2), 239–255 (2014)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.University of LouisianaLafayetteUSA

Personalised recommendations