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Theoretical and computational study of several linearisation techniques for binary quadratic problems

  • Fabio FuriniEmail author
  • Emiliano Traversi
Original Research

Abstract

We perform a theoretical and computational study of the classical linearisation techniques (LT) and we propose a new LT for binary quadratic problems (BQPs). We discuss the relations between the linear programming (LP) relaxations of the considered LT for generic BQPs. We prove that for a specific class of BQP all the LTs have the same LP relaxation value. We also compare the LT computational performance for four different BQPs from the literature. We consider the Unconstrained BQP and the maximum cut of edge-weighted graphs and, in order to measure the effects of constraints on the computational performance, we also consider the quadratic extension of two classical combinatorial optimization problems, i.e., the knapsack and stable set problems.

Keywords

Linearisation techniques Binary quadratic problems Max cut problem Quadratic knapsack problem Quadratic stable set problem 

Notes

Acknowledgements

Thanks are due to two anonymous referees for careful reading and useful comments.

References

  1. Adams, W. P., Forrester, R. J., & Glover, F. W. (2004). Comparisons and enhancement strategies for linearizing mixed 0–1 quadratic programs. Discrete Optimization, 1(2), 99–120.CrossRefGoogle Scholar
  2. Adams, W. P., & Sherali, H. D. (1986). A tight linearization and an algorithm for zero-one quadratic programming problems. Management Science, 32(10), 1274–1290.CrossRefGoogle Scholar
  3. Billionnet, A., & Soutif, E. (2004). An exact method based on lagrangian decomposition for the 01 quadratic knapsack problem. European Journal of Operational Research, 157(3), 565–575.CrossRefGoogle Scholar
  4. Caprara, A. (2008). Constrained 0–1 quadratic programming: Basic approaches and extensions. European Journal of Operational Research, 187, 1494–1503.CrossRefGoogle Scholar
  5. Caprara, A., Pisinger, D., & Toth, P. (1999). Exact solution of the quadratic knapsack problem. INFORMS Journal on Computing, 11(2), 125–137.CrossRefGoogle Scholar
  6. Chaovalitwongse, W., Pardalos, P. M., & Prokopyev, O. A. (2004). A new linearization technique for multi-quadratic 01 programming problems. Operations Research Letters, 32(6), 517–522.CrossRefGoogle Scholar
  7. Conforti, M., Cornuéjols, G., & Zambelli, G. (2010). Extended formulations in combinatorial optimization. 4OR, 8(1), 1–48.CrossRefGoogle Scholar
  8. Forrester, R., & Greenberg, H. (2008). Quadratic binary programming models in computational biology. Algorithmic Operations Research, 3(2), 110–129.Google Scholar
  9. Fortet, R. (1960). L’algebre de boole et ses applications en recherche operationnelle. Trabajos de Estadistica, 11(2), 111–118.CrossRefGoogle Scholar
  10. Furini, F., & Traversi, E. (2013). Hybrid SDP bounding procedure. Lecture Notes in Computer Science, 7933, 248–259.CrossRefGoogle Scholar
  11. Glover, F. (1975). Improved linear integer programming formulations of nonlinear integer programs. Management Science, 22(4), 455–460.CrossRefGoogle Scholar
  12. Glover, F., & Woolsey, E. (1973). Further reduction of zero-one polynomial programming problems to zero-one linear programming problems. Operations Research, 21(1), 156–161.CrossRefGoogle Scholar
  13. Glover, F., & Woolsey, E. (1974). Converting the 0–1 polynomial programming problem to a 0–1 linear program. Operations Research, 22(1), 180–182.CrossRefGoogle Scholar
  14. Gueye, S., & Michelon, P. (2009). A linearization framework for unconstrained quadratic (0–1) problems. Discrete Applied Mathematics, 157(6), 1255–1266.CrossRefGoogle Scholar
  15. Hansen, P., & Meyer, C. (2009). Improved compact linearizations for the unconstrained quadratic 01 minimization problem. Discrete Applied Mathematics, 157(6), 1267–1290.CrossRefGoogle Scholar
  16. ILOG IBM. (2017). Cplex optimizer.Google Scholar
  17. Jaumard, B., Marcotte, O., & Meyer, C. (1998). Estimation of the Quality of Cellular Networks Using Column Generation Techniques. Cahiers du GÉRAD. Groupe d’études et de recherche en analyse des décisions.Google Scholar
  18. Krislock, N., Malick, J., & Roupin, F. (2014). Improved semidefinite bounding procedure for solving Max-Cut problems to optimality. Mathematical Programming, 143(1), 61–86.CrossRefGoogle Scholar
  19. Lodi, A. (2010). Mixed integer programming computation. In M. Jünger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, & L. A. Wolsey (Eds.), 50 Years of integer programming 1958–2008 (pp. 619–645). Berlin: Springer.CrossRefGoogle Scholar
  20. Padberg, M. (1989). The boolean quadric polytope: Some characteristics, facets and relatives. Mathematical Programming, 45(1), 139–172.CrossRefGoogle Scholar
  21. Pisinger, D. (2007). The quadratic knapsack problem: A survey. Discrete Applied Mathematics, 155(5), 623–648.CrossRefGoogle Scholar
  22. Rendl, F., Rinaldi, G., & Wiegele, A. (2010). Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Mathematical Programming, 121(2), 307–335.CrossRefGoogle Scholar
  23. Sherali, H. D., & Adams, W. P. (1998). A reformulation-linearization technique for solving discrete and continuous nonconvex problems. Berlin: Springer.Google Scholar
  24. Sherali, H. D., & Smith, J. C. (2007). An improved linearization strategy for zero-one quadratic programming problems. Optimization Letters, 1(1), 33–47.CrossRefGoogle Scholar
  25. Wang, H., Kochenberger, G., & Glover, F. (2012). A computational study on the quadratic knapsack problem with multiple constraints. Computers & Operations Research, 39(1), 3–11.CrossRefGoogle Scholar
  26. Wiegele, A. (2007). Biq mac library—A collection of max-cut and quadratic 01 programming instances of medium size. Technical report, Alpen-Adria-Universität Klagenfurt, Austria.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.PSL, CNRS, LAMSADE UMR 7243Université Paris DauphineParis Cedex 16France
  2. 2.Laboratoire d’Informatique de Paris NordUniversité de Paris 13VilletaneuseFrance

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