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RLT insights into lift-and-project closures

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Abstract

This paper examines relationships between the two methods for constructing polyhedral outer-approximations of mixed 0-1 linear programs known in the literature as the level-1 Reformulation–Linearization-Technique (RLT) and the elementary closure of lift-and-project (L&P) cuts based on the level-1 L&P. The latter can be obtained, via a transformation of variables, as a relaxed version of the former through the removal of constraints. Whereas both approaches derive formulations in lifted spaces by scaling problem constraints with suitable multipliers, a key difference is that the multipliers used by the RLT are defined in terms of the binary variables of the given problem. As a consequence, these multipliers, in contrast to those used in the L&P, portend important advantages, including the ability to handle nonlinear expressions, to recognize redundant inequalities, and to exploit special structures. This paper compares the two methods, and uses RLT constructs to provide insights into the L&P approach. Some particular insights are illustrated on the quadratic assignment and MAX-2SAT problems, with level-2 RLT and level-2 L&P comparisons also given for the latter problem.

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References

  1. Adams, W.P., Dearing, P.M.: On the equivalence between roof duality and Lagrangian duality for unconstrained 0-1 quadratic programming problems. Discret. Appl. Math. 48(1), 1–20 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams, W.P., Forrester, R.J., Glover, F.W.: Comparisons and enhancement strategies for linearizing mixed 0-1 quadratic programs. Discret. Optim. 1(2), 99–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Adams, W.P., Guignard, M., Hahn, P.M., Hightower, W.L.: A level-2 reformulation-linearization technique bound for the quadratic assignment problem. Eur. J. Oper. Res. 180(3), 983–996 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Adams, W.P., Hadavas, P.T.: A network approach for specially-structured linear programs arising in 0-1 quadratic optimization. Discret. Appl. Math. 156(11), 2142–2165 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Adams, W.P., Johnson, T.A.: Improved linear programming-based bounds for the quadratic assignment problem. In: Pardalos, P.M., Wolkowicz, H. (eds.) Quadratic Assignment and Related Problems. DIMACS Series, vol. 16, pp. 43–75. AMS (1994)

  6. Adams, W.P., Sherali, H.D.: A tight linearization and an algorithm for zero-one quadratic programming problems. Manag. Sci. 32(10), 1274–1290 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Adams, W.P., Sherali, H.D.: Linearization strategies for a class of zero-one mixed integer programming problems. Oper. Res. 38(2), 217–226 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Adams, W.P., Sherali, H.D.: Mixed-integer bilinear programming problems. Math. Progr. 59(3), 279–305 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Adams, W.P., Sherali, H.D.: A hierarchy of relaxations leading to the convex hull representation for general discrete optimization problems. Ann. Oper. Res. 140(1), 21–47 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Balas, E.: Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. SIAM J. Algebr. Discret. Methods 6(3), 466–486 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Math. Progr. 58(3), 295–324 (1993)

    Article  MATH  Google Scholar 

  12. Balas, E., Ceria, S., Cornuéjols, G.: Mixed 0-1 programming by lift-and-project in a branch-and-cut framework. Manag. Sci. 42(9), 1229–1246 (1996)

    Article  Google Scholar 

  13. Bonami, P., Minoux, M.: Exact MAX-2SAT solution via lift-and-project closure. Oper. Res. Lett. 34(4), 387–393 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bonami, P., Minoux, M.: Using rank-1 lift-and-project closures to generate cuts for 0-1 MIPs, a computational investigation. Discret. Optim. 2(4), 288–307 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Burkard, R.E., Stratmann, K.H.: Numerical investigations on quadratic assignment problems. Naval Res. Logist. Q. 25(1), 129–148 (1978)

    Article  MATH  Google Scholar 

  16. Cornuéjols, G., Li, Y.: Elementary closures for integer programs. Oper. Res. Lett. 28(1), 1–8 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Edwards, C.S.: A branch and bound algorithm for the Koopmans–Beckmann quadratic assignment problem. Math. Progr. Study 13, 35–53 (1980)

    Article  MATH  Google Scholar 

  18. Fortet, R.: L’algèbre de Boole et ses applications en recherche opérationnelle. Cahiers du Centre d’Études de Rec. Opérationnelle, vol. 1, pp. 5–36 (1959)

  19. Fortet, R.: Applications de l’algèbre de Boole en recherche opérationnelle. Rev. Française Informatique Recherche Opér, vol. 4, pp. 17–26 (1960)

  20. Frieze, A.M., Yadegar, J.: On the quadratic assignment problem. Discret. Appl. Math. 5(1), 89–98 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gilmore, P.C.: Optimal and suboptimal algorithms for the quadratic assignment problem. J. Soc. Ind. Appl. Math. 10(2), 305–313 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  22. Glover, F.: Improved linear integer programming formulations of nonlinear integer programs. Manag. Sci. 22(4), 455–460 (1975)

    Article  MathSciNet  Google Scholar 

  23. Glover, F., Woolsey, E.: Converting the 0-1 polynomial programming problem to a 0-1 linear program. Oper. Res. 22(1), 180–182 (1974)

    Article  MATH  Google Scholar 

  24. Hahn, P., Grant, T.: Lower bounds for the quadratic assignment problem based upon a dual formulation. Oper. Res. 46(6), 912–922 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hahn, P.M., Grant, T.L., Hall, N.: A branch-and-bound algorithm for the quadratic assignment problem based on the Hungarian method. Eur. J. Oper. Res. 108(3), 629–640 (1998)

    Article  MATH  Google Scholar 

  26. Hahn, P., Hightower, W., Johnson, T.A., Guignard-Spielberg, M.: Tree elaboration strategies in branch and bound algorithms for solving the quadratic assignment problem. Yugosl. J. Oper. Res. 11(1), 41–60 (2001)

    MathSciNet  MATH  Google Scholar 

  27. Hahn, P., Krarup, J.: A hospital facility problem finally solved. J. Intell. Manufact. 12(5/6), 487–496 (2001)

    Article  Google Scholar 

  28. Hahn, P.M., Zhu, Y.-R., Guignard, M., Hightower, W.L., Saltzman, M.J.: A level-3 reformulation-linearization technique bound for the quadratic assignment problem. INFORMS J. Comput. 24(2), 202–209 (2012)

    Article  MathSciNet  Google Scholar 

  29. Koopmans, T.C., Beckmann, M.: Assignment problems and the location of economic activities. Econometrica 25(1), 53–76 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lawler, E.: The quadratic assignment problem. Manag. Sci. 19(4), 586–599 (1963)

    Article  MathSciNet  Google Scholar 

  31. Lee, Y., Pardalos, P.M., Ramakrishnan, K.G., Resende, M.G.C.: Lower bounds for the quadratic assignment problem. Ann. Oper. Res. 50(1), 387–410 (1994)

    Article  MathSciNet  Google Scholar 

  32. Lougee-Heimer, R., Adams, W.P.: A conditional logic approach for strengthening mixed 0-1 linear programs. Ann. Oper. Res. 139(1), 289–320 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Minoux, M., Ouzia, H.: DRL*: a hierarchy of strong block-decomposable linear relaxations for 0-1 MIPs. Discret. Appl. Math. 158(18), 2031–2048 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nugent, C.E., Vollman, T.E., Ruml, J.: An experimental comparison of techniques for the assignment of facilities to locations. Oper. Res. 6(1), 150–173 (1968)

    Article  Google Scholar 

  35. Roucairol, C.: A reduction method for quadratic assignment problems. Oper. Res. Verfahr. Methods Oper. Res. 32, 185–187 (1979)

    Google Scholar 

  36. Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discret. Math. 3(3), 411–430 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sherali, H.D., Adams, W.P.: A Reformulation–Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer Academic Publishers, Dordrecht (1999)

    Book  MATH  Google Scholar 

  38. Sherali, H.D., Adams, W.P.: A hierarchy of relaxations and convex hull characterizations for mixed-integer zero-one programming problems. Discret. Appl. Math. 52(1), 83–106 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  39. Sun, X: Combinatorial algorithms for Boolean and pseudo-Boolean functions. PhD Dissertation, Rutgers University (1992)

  40. Sherali, H.D., Adams, W.P., Driscoll, P.J.: Exploiting special structures in constructing a hierarchy of relaxations for 0-1 mixed integer problems. Oper. Res. 46(3), 396–405 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sherali, H.D., Lee, Y.: Tighter representations for set partitioning problems. Discret. Appl. Math. 68(1–2), 153–167 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  42. Trotter, L.E.: A class of facet producing graphs for vertex packing polyhedra. Discret. Math. 12(4), 373–388 (1975)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors gratefully acknowledge support from the National Science Foundation under grant numbers CMMI-0968909 and CMMI-0969169. We also thank two anonymous referees for providing valuable suggestions that improved the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, O-327 Martin Hall, Clemson University, Clemson, SC, 29634-0975, USA

    Warren P. Adams

  2. Grado Department of Industrial and Systems Engineering (0118), Virginia Tech, 250 Durham Hall, Blacksburg, VA, 24061, USA

    Hanif D. Sherali

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  1. Warren P. Adams
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Correspondence to Warren P. Adams.

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Adams, W.P., Sherali, H.D. RLT insights into lift-and-project closures. Optim Lett 9, 19–39 (2015). https://doi.org/10.1007/s11590-014-0763-5

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