Abstract.
Given a feasible solution to a Mixed Integer Programming (MIP) model, a natural question is whether that solution can be improved using local search techniques. Local search has been applied very successfully in a variety of other combinatorial optimization domains. Unfortunately, local search relies extensively on the notion of a solution neighborhood, and this neighborhood is almost always tailored to the structure of the particular problem being solved. A MIP model typically conveys little information about the underlying problem structure. This paper considers two new approaches to exploring interesting, domain-independent neighborhoods in MIP. The more effective of the two, which we call Relaxation Induced Neighborhood Search (RINS), constructs a promising neighborhood using information contained in the continuous relaxation of the MIP model. Neighborhood exploration is then formulated as a MIP model itself and solved recursively. The second, which we call guided dives, is a simple modification of the MIP tree traversal order. Loosely speaking, it guides the search towards nodes that are close neighbors of the best known feasible solution. Extensive computational experiments on very difficult MIP models show that both approaches outperform default CPLEX MIP and a previously described approach for exploring MIP neighborhoods (local branching) with respect to several different metrics. The metrics we consider are quality of the best integer solution produced within a time limit, ability to improve a given integer solution (of both good and poor quality), and time required to diversify the search in order to find a new solution.
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References
Aboudi, R., Jörnsten, K.: Tabu Search for General Zero-One Integer Programs Using the Pivot and Complement Heuristic. ORSA J. Comput. 6 (1), 82–93 (1994)
Adams, J., Balas, E., Zawack, D.: The Shifting Bottleneck Procedure for Job-Shop Scheduling. Manage. Sci. 34 (3), 391–401 (1988)
Anbil, R., Gelman, E., Patty, B., Tanga, R.: Recent advances in crew-pairing optimization at American Airlines. Interfaces 21, 62–74 (1991)
Applegate, D., Cook, W.: A Computational Study of the Job-Shop Scheduling Problem. ORSA J. Comput. 3 (2), 149–156 (1991)
Balas, E., Ceria, S., Dawande, M., Margot, F., Pataki, G.: OCTANE: A New Heuristic for Pure 0-1 Programs. Oper. Res. 49 (2), 207–225 (2001)
Balas, E., Martin, C.: Pivot and Complement – A Heuristic for 0-1 Programming. Manage. Sci. 26 (1), 86–96 (1980)
Baptiste, P., Le Pape, C., Nuijten, W.: Incorporating Efficient Operations Research Algorithms in Constraint-Based Scheduling. In: Proceedings of the First International Joint Workshop on Artificial Intelligence and Operations Research, 1995
Bixby, R.E., Ceria, S., McZeal, C.M., Savelsbergh, M.W.P.: An updated mixed integer programming library: MIPLIB 3.0 Optima 58, 12–15 (1998)
Bixby, R.E., Fenelon, M., Gu, Z., Rothberg, E., Wunderling, R.: MIP: Theory and practice – closing the gap. Kluwer Academic Publishers, 2000, pp. 19–49
Caseau, Y., Laburthe, F.: Disjunctive Scheduling with Task Intervals. Technical report, École Normale Supérieure, 1995
Caseau, Y., Laburthe, F.: SaLSA Specification language for search algorithms. Technical report, École Normale Supérieure, LIENS-97-11, 1997
Chabrier, A., Danna, E., Le Pape, C., Perron, L.: Solving a Network Design Problem. To appear in Annals of Operations Research, Special Issue following CP-AI-OR’2002, 2004
Danna, E.: Intégration des techniques de recherche locale à la programmation linéaire en nombres entiers (in French). PhD thesis, Université d’Avignon, 2004
Danna, E., Le Pape, C.: Accelerating branch-and-price with local search: A case study on the vehicle routing problem with time windows. Technical Report, ILOG, 03-006, 2003
Danna, E., Le Pape, C.: Two generic schemes for efficient and robust cooperative algorithms. Constraint and integer programming, Michela Milano (ed.), Kluwer Academic Publishers, 2003, pp. 33–57
Faaland, B.H., Hillier, F.S.: Interior path methods for heuristic integer programming procedures. Oper. Res. 27 (6), 1069–1087 (1979)
Fischetti, M., Lodi, A.: Local Branching. Math. Program. Ser. B 98, 23–47 (2003)
Glover, F., Laguna, M.: Tabu Search. Kluwer Academic Publishers, 1997
Glover, F., Laguna, M., Martí, F.: Fundamentals of Scatter Search and Path Relinking. Control and Cybernetics 29 (3), 653–684 (2000)
Glover, F., Løkketangen, A., Woodruff, D.L.: Scatter Search to Generate Diverse MIP Solutions. In: M. Laguna, J.L. González-Velarde, (eds.), OR Computing Tools for Modeling, Optimization and Simulation: Interfaces in Computer Science and Operations Research, Kluwer Academic Publishers, 2000, pp. 299–317
Hillier, F.S.: Efficient heuristic procedures for integer linear programming with an interior. Oper. Res. 17 (4), 600–637 (1969)
Ibaraki, T., Ohashi, T., Mine, H.: A heuristic algorithm for mixed-integer programming problems. Math. Program. Study 2, 115–136 (1974)
Junker, U., Nuijten, W.: Preference-based Search for Minimizing Changes in Rescheduling Problems. In: IJCAI-99 Workshop on scheduling and planning meet real-time: Monitoring in a dynamic and uncertain world, 1999, pp. 39–45
Jussien, N., Lhomme, O.: Local search with constraint propagation and conflict-based heuristics. Artificial Intelligence 139, 21–45 (2002)
Le Pape, C., Baptiste, P.: Heuristic Control of a Constraint-Based Algorithm for the Preemptive Job-Shop Scheduling Problem. J. Heuristics 5 (3), 305–325 (1999)
Løkketangen, A., Woodruff, D.L.: Integrating Pivot Based Search with Branch and Bound for Binary MIP’s. Control and Cybernetics, Special issue on Tabu Search 29 (3), 741–760 (2001)
Minton, S., Johnston, M.D., Philips, A.B., Laird, P.: Minimizing conflicts: a heuristic repair method for constraint satisfaction and scheduling. Artificial Intelligence 58, 161–205 (1992)
Mitchell, M.: An Introduction to Genetic Algorithms. MIT Press, 1996
Nediak, M., Eckstein, J.: Pivot, cut and dive: A Heuristic for 0-1 Mixed Integer Programming. Technical Report, Rutgers Center for Operations Research, RRR 53-2001, 2001
Palpant, M., Artigues, C., Michelon, P.: A heuristic for solving the frequency assignment problem. In: XI Latin-Iberian American Congress of Operations Research (CLAIO), 2002
Shaw, P.: Using Constraint Programming and Local Search Methods to Solve Vehicle Routing Problems. In: Proceedings of the Fourth International Conference on Principles and Practice of Constraint Programming (CP’98), 1998, pp. 417–431
Van Hentenryck, P., Le Provost, T.: Incremental Search in Constraint Logic Programming. New Generation Comput. 9 (3), 257–275 (1991)
Van Laarhoven, P.J.M., Aarts, E.H.L.: Simulated Annealing: Theory and Practice. Kluwer Academic Publishers, 1987
Van Vyve, M.: A solution approach of production planning problems based on compact formulations for single-item lot-sizing models. PhD thesis, Université catholique de Louvain-la-Neuve, 2003
Vásquez, M., Whitley, L.D.: A comparison of Genetic Algorithms for the Dynamic Job Shop Scheduling Problem. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2000), 2000, pp. 1011–1018
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Mathematics Subject Classification (2000):20E28, 20G40, 20C20
Acknowledgement We wish to thank the two anonymous referees for their helpful comments.
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Danna, E., Rothberg, E. & Pape, C. Exploring relaxation induced neighborhoods to improve MIP solutions. Math. Program. 102, 71–90 (2005). https://doi.org/10.1007/s10107-004-0518-7
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DOI: https://doi.org/10.1007/s10107-004-0518-7