Abstract
The Reformulation Linearization Technique (RLT) of Sherali and Adams (Manag Sci 32(10):1274–1290, 1986; SIAM J Discrete Math 3(3):411–430, 1990), when applied to a pure 0–1 quadratic optimization problem with linear constraints (P), constructs a hierarchy of LP (i.e., continuous and linear) models of increasing sizes. These provide monotonically improving continuous bounds on the optimal value of (P) as the level, i.e., the stage in the process, increases. When the level reaches the dimension of the original solution space, the last model provides an LP bound equal to the IP optimum. In practice, unfortunately, the problem size increases so rapidly that for large instances, even computing bounds for RLT models of level k (called RLTk) for small k may be challenging. Their size and their complexity increase drastically with k. To our knowledge, only results for bounds of levels 1, 2, and 3 have been reported in the literature. We are proposing, for certain quadratic problem types, a way of producing stronger bounds than continuous RLT1 bounds in a fraction of the time it would take to compute continuous RLT2 bounds. The approach consists in applying a specific decomposable Lagrangean relaxation to a specially constructed RLT1-type linear 0–1 model. If the overall Lagrangean problem does not have the integrality property, and if it can be solved as a 0–1 rather than a continuous problem, one may be able to obtain 0–1 RLT1 bounds of roughly the same quality as standard continuous RLT2 bounds, but in a fraction of the time and with much smaller storage requirements. If one actually decomposes the Lagrangean relaxation model, this two-step procedure, reformulation plus decomposed Lagrangean relaxation, will produce linear 0–1 Lagrangean subproblems with a dimension no larger than that of the original model. We first present numerical results for the Crossdock Door Assignment Problem, a special case of the Generalized Quadratic Assignment Problem. These show that just solving one Lagrangean relaxation problem in 0–1 variables produces a bound close to or better than the standard continuous RLT2 bound (when available) but much faster, especially for larger instances, even if one does not actually decompose the Lagrangean problem. We then present numerical results for the 0–1 quadratic knapsack problem, for which no RLT2 bounds are available to our knowledge, but we show that solving an initial Lagrangean relaxation of a specific 0–1 RLT1 decomposable model drastically improves the quality of the bounds. In both cases, solving the fully decomposed rather than the decomposable Lagrangean problem to optimality will make it feasible to compute such bounds for instances much too large for computing the standard continuous RLT2 bounds.
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Notes
In the RLT theory, problems RLTk are normally meant to be LP problems. In this paper, however, they may be either continuous or 0–1 problems and to avoid confusion, we will refer to them as continuous RLTk or 0–1 RLTk. If we talk about just RLTk, we mean 0–1 RLTk, because this is what we are advocating.
In the paper, the term “Lagrangean relaxation problem” always means that it is solved as an integer programming problem, not as an LP. If it is not the case, we will explicitly talk of a continuous Lagrangean relaxation, or a continuous Lagrangean bound.
We want to thank Jongwoo Park for suggesting this option.
We want to thank Peter M. Hahn for graciously providing us with unpublished continuous RLT2 bounds and running times for the CDAP when available.
We want to thank Jongwoo Park for making available for this paper a small sample of his computational experiment on the CDAP using RLT1 + LR + ILP. See Park and Guignard (2018).
References
Adams, W. P., Guignard, M., Hahn, P. M., & Hightower, W. L. (2007). A Level-2 Reformulation-Linearization technique bound for the quadratic assignment problem. European Journal of Operational Research,180(3), 983–996.
Adams, W. P., & Johnson, T. A. (1994). Improved linear programming-based lower bounds for quadratic assignment problems. Discrete Mathematics & Theoretical Computer Science,16, 43–77.
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.
Beltran, C., & Heredia, F. J. (2005). An effective line search for the subgradient method. Journal of Optimization Theory and Applications,125, 1–18.
Billionnet, A., & Calmels, F. (1996). Linear programming for the 0–1 quadratic knapsack problem. European Journal of Operational Research,92, 310–325.
Bragin, M. A., Luh, P. B., Yan, J. H., Yu, N., & Stern, G. A. (2015). Convergence of the surrogate Lagrangian relaxation method. Journal of Optimization Theory and Applications,164, 173–201.
Caprara, A., Pisinger, D., & Toth, P. (1999). Exact solution of the quadratic knapsack problem. INFORMS Journal on Computing,11(2), 125–137.
Drezner, Z., Hahn, P., & Taillard, E. (2005). Recent advances for the quadratic assignment problem with special emphasis on instances that are difficult for meta-heuristic methods. Annals of Operations Research,139, 65–94.
Fortet, R. (1959). L’Algèbre de Boole et ses applications en recherche opérationnelle. Cahiers Centre Etudes Rech Opér,4, 5–36.
Geoffrion, A. M. (1974). Lagrangean relaxation for integer programming. Mathematical Programming Study,2, 82–114.
Geoffrion, A. M., & McBride, R. (1978). Lagrangean relaxation applied to capacitated facility location problems. AIIE Transactions,10(1), 40–47.
Guignard, M. (2003). “Lagrangean relaxation”, invited survey. TOP,11(2), 151–228.
Guignard, M. (2006). RLT1 and LR for the GQAP. OPIM Department Research Report 06-06-01, the Wharton School, University of Pennsylvania.
Guignard, M., Hahn, P., Pessoa, A. A., & DaSilva, D. C. (2012). Algorithms for the cross-dock door assignment problem. In Proceedings of 4th international workshop model-based metaheuristics, Angra dos Reis, Brazil (pp. 1–12).
Guignard, M., & Kim, S. (1987). Lagrangean decomposition: A model yielding stronger bounds. Mathematical Programming,39(2), 215–228.
Hahn, P. M., Kim, B.-J., Guignard, M., Smith, J., & Zhu, Y.-R. (2008). An algorithm for the generalized quadratic assignment problem. Computational Optimization and Applications,40, 351–372.
Hahn, P. M., Zhu, Y.-R., Guignard, M., & Hightower, W. L. (2012). A level-3 reformulation-linearization technique bound for the quadratic assignment problem. INFORMS Journal on Computing,24(2), 202–209.
Kim, B.-J. (2006). Investigation of methods for solving new classes of quadratic assignment problems (QAPs). Doctoral Dissertation, University of Pennsylvania, ESE Department.
Lee, C.-G., & Ma, Z. (2004). The generalized quadratic assignment problem. Research Report, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, M5S 3G8, Canada.
Létocart, L., Nagih, A., & Plateau, G. (2012). Reoptimization in Lagrangian methods for the 0–1 quadratic knapsack problem. Computers & OR,39(1), 12–18.
Loiola, E. M., de Abreu, N. M. M., Boaventura-Netto, P. O., Hahn, P. M., & Querido, T. (2007). A survey for the quadratic assignment problem. European Journal of Operational Research,176, 657–690.
McCormick, G. P. (1976). Computability of global solutions to factorable nonconvex programs: Part 1 Convex underestimating problems. Mathematical Programming,10, 147–175.
Padberg, M., & Rijal, M. P. (1996). Location, scheduling, design and integer programming. Norwell, MA: Kluwer Academic.
Park, J. (2014). Generalized quadratic assignment problem: Combining level 1 Lagrangean decomposition and reformulation-linearization technique. Independent Study Report, University of Pennsylvania, ESE Department.
Park, J., & Guignard, M. (2018). Computing bounds for CDAP problems using RLT1+LR+ILP decomposition. Research Report, OID Department, University of Pennsylvania, 2018.
Pessoa, A. A., Hahn, P. M., Guignard, M., & Zhu, Y.-R. (2010). Algorithms for the generalized quadratic assignment problem combining Lagrangean decomposition and the Reformulation-Linearization Technique. European Journal of Operational Research,206(1), 54–63.
Sherali, H. D., & Adams, W. P. (1990). A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics,3(3), 411–430.
Tsui, L. Y., & Chang, C.-H. (1992). An optimal solution to a dock door assignment problem. Computer and Industrial Engineering,23(1), 283–286.
Zhao, X., Luh, P. B., & Wang, J. (1999). Surrogate gradient algorithm for Lagrangian Relaxation. Journal of Optimization Theory and Applications,100(3), 699–712.
Zhu, Y.-R. (2007). Recent advances and challenges in quadratic assignment and related problems. Doctoral Dissertation, University of Pennsylvania, ESE Department.
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The author would like to thank the three anonymous referees whose many insightful comments and suggestions helped to significantly improve the organization and contents of the paper.
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Guignard, M. Strong RLT1 bounds from decomposable Lagrangean relaxation for some quadratic 0–1 optimization problems with linear constraints. Ann Oper Res 286, 173–200 (2020). https://doi.org/10.1007/s10479-018-3092-8
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DOI: https://doi.org/10.1007/s10479-018-3092-8