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Benders decomposition, Lagrangean relaxation and metaheuristic design

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Abstract

Large part of combinatorial optimization research has been devoted to the study of exact methods leading to a number of very diversified solution approaches. Some of those older frameworks can now be revisited in a metaheuristic perspective, as they are quite general frameworks for dealing with optimization problems. In this work, we propose to investigate the possibility of reinterpreting decompositions, with special emphasis on the related Benders and Lagrangean relaxation techniques. We show how these techniques, whose heuristic effectiveness is already testified by a wide literature, can be framed as a “master process that guides and modifies the operations of subordinate heuristics”, i.e., as metaheuristics. Obvious advantages arise from these approaches, first of all the runtime evolution of both upper and lower bounds to the optimal solution cost, thus yielding both a high-quality heuristic solution and a runtime quality certificate of that same solution.

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References

  • Agar, M., Salhi, S.: Lagrangean heuristics applied to a variety of large capacitated plant location problems. J. Oper. Res. Soc. 49, 1072–1084 (1998)

    Article  MATH  Google Scholar 

  • Ahuja, R.K., Orlin, J.B., Pallottino, S., Scaparra, M.P., Scutellà, M.G.: A multi-exchange heuristic for the single source capacitated facility location problem. Manag. Sci. 50 (2003)

  • Barahona, F., Anbil, R.: The volume algorithm: producing primal solutions with a subgradient method. Math. Program. 87, 385–399 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Barcelo, J., Casanova, J.: A heuristic Lagrangean algorithm for the capacitated plant location problem. Eur. J. Oper. Res. 15, 212–226 (1984)

    Article  MATH  Google Scholar 

  • Beasley, J.E.: Lagrangean relaxation. In: Reeves, C.R. (ed.) Modern Heuristic Techniques for Combinatorial Problems, pp. 243–303. Blackwell Scientific, Oxford (1993)

    Google Scholar 

  • Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. Numer. Math. 4, 280–322 (1962)

    Article  MathSciNet  Google Scholar 

  • Boschetti, M.A., Jelasity, M., Maniezzo, V.: A local approach to membership overlay design. Working paper, Department of Computer Science, University of Bologna (2006)

  • Bouleimen, K., Lecocq, H.: A new efficient simulated annealing algorithm for the resource-constrained project scheduling problem. In: Barbarosoglu, G., Karabati, S., Ozdamar, L., Ulusoy, G. (eds.) Proceedings of the Sixth International Workshop on Project Management and Scheduling, pp. 1922. Bogazici University (1998)

  • Chudak, F.A., Shmoys, D.B.: Improved approximation algorithms for a capacitated facility location problem. In: Proc. 10th Annu. ACM-SIAM Sympos. Discrete Algorithms, pp. S875–S876 (1999)

  • Delmaire, H., Diaz, J.A., Fernandez, E., Ortega, M.: Reactive grasp and tabu search based heuristics for the single source capacitated plant location problem. INFOR 37, 194–225 (1999)

    Google Scholar 

  • Drexl, A., Grunewald, J.: Nonpreemptive multi-mode resource-constrained project scheduling. IIE Trans. 25(5), 74–81 (1993)

    Article  Google Scholar 

  • Elmaghraby, S.E.: Activity Networks: Project Planning and Control by network Models. Wiley, New York (1977)

    MATH  Google Scholar 

  • Ganesh, A.J., Kermarrec, A.-M., Massoulié, L.: Peer-to-peer membership management for gossip-based protocols. IEEE Trans. Comput. 52(2) (February 2003)

  • Glover, F.: Future paths for integer programming and links to artificial intelligence. Comput. Oper. Res. 13, 533–549 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  • Glover, F., Laguna, M.: Tabu Search. Kluwer Academic, Boston (1997)

    MATH  Google Scholar 

  • Hartmann, S.: Project scheduling with multiple modes: a genetic algorithm. Ann. Oper. Res. 102, 111–135 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Holmberg, K., Ronnqvist, M., Yuan, D.: An exact algorithm for the capacitated facility location problems with single sourcing. Eur. J. Oper. Res. 113, 544–559 (1999)

    Article  MATH  Google Scholar 

  • Holt, J., Ronnqvist, M., Tragantalerngsak, S.: A repeated matching heuristic for the single source capacitated facility location problem. Eur. J. Oper. Res. 116, 51–68 (1999)

    Article  MATH  Google Scholar 

  • Jozefowska, J., Mika, M., Royzycki, R., Waligora, G., Weglarz, J.: Simulated annealing for multi-mode resource-constrained project scheduling. Ann. Oper. Res. 102, 137–155 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Klincewicz, J., Luss, H.: A Lagrangean relaxation heuristic for capacitated facility location with single-source constraints. J. Oper. Res. Soc. 37, 495–500 (1986)

    Article  MATH  Google Scholar 

  • Kolisch, R., Drexl, A.: Local search for nonpreemptive multi-mode resource constrained project scheduling. Technical Report 360, Institut fr Betriebswirtschaftslehre, Christian-Albrechts-Universitt zu Kiel, Kiel (1994)

  • Kolisch, R., Sprecher, A., Drexl, A.: Characterization and generation of a general class of resource constrained project scheduling problems. Manag. Sci. 41, 1693–1703 (1995)

    Article  MATH  Google Scholar 

  • Maniezzo, V., Mingozzi, A.: A heuristic procedure for the multi-mode project scheduling problem based on benders decomposition. In: Weglarz, J. (ed.) Project Scheduling: Recent Models, Algorithms and Applications, pp. 179–196. Kluwer Academic, Dordrecht (1999)

    Google Scholar 

  • Maniezzo, V., Boschetti, M.A., Jelasity, M.: An ant approach to membership overlay design. In: Ant Colony, Optimization and Swarm Intelligence: Proc. ANTS 2004. Lecture Notes in Computer Science, vol. 3172, p. 3748. Springer, Berlin (2004)

    Google Scholar 

  • Maniezzo, V., Boschetti, M.A., Jelasity, M.: A fully distributed Lagrangean metaheuristic for a P2P overlay network design problem. In: Proceedings of the 6th Metaheuristics International Conference (MIC 2005), Vienna, Austria (2005)

  • Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implementations. Wiley, New York (1990)

    MATH  Google Scholar 

  • Mingozzi, A., Maniezzo, V., Ricciardelli, S., Bianco, L.: An exact algorithm for the resource constrained project scheduling problem based on a new mathematical formulation. Manag. Sci. 44, 715–729 (1998)

    Article  Google Scholar 

  • Neebe, A., Rao, M.: An algorithm for the fixed-charge assigning users to sources problem. J. Oper. Res. Soc. 34, 1107–1113 (1983)

    Article  MATH  Google Scholar 

  • Peersim: A peer-to-peer simulator, http://peersim.sourceforge.net/ (2006)

  • Pirkul, H.: Efficient algorithm for the capacitated concentrator location problem. Comput. Oper. Res. 14, 197–208 (1987)

    Article  MATH  Google Scholar 

  • Planetlab: An open platform for developing, deploying, and accessing planetary-scale services, http://www.planet-lab.org/ (2006)

  • Polyak, B.T.: Minimization of unsmooth functionals. USSR Comput. Math. Phys. 9, 14–29 (1969)

    Article  Google Scholar 

  • Saroiu, S., Krishna Gummadi, P., Gribble, S.D.: A measurement study of peer-to-peer file sharing systems. In: Proceedings of Multimedia Computing and Networking 2002 (MMCN’02), San Jose, CA (2002)

  • Sherali, H.D., Choi, G.: Recovery of primal solutions when using subgradient optimization methods to solve Lagrangian duals of liner programs. Oper. Res. Lett. 19, 105–113 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Solomon, M.: Algorithms for the vehicle routing and scheduling problem with time window constraints. Oper. Res. 35, 254–365 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  • Sprecher, A., Drexl, A.: Solving multi-mode resource-constrained project scheduling by a simple, general and powerful sequencing algorithm. Eur. J. Oper. Res. 107, 431–450 (1998)

    Article  MATH  Google Scholar 

  • Sridharan, R.: A Lagrangian heuristic for the capacitated plant location problem with single source constraints. Eur. J. Oper. Res. 66, 305–312 (1991)

    Article  Google Scholar 

  • Talbot, B.: Resource-constrained project scheduling with time-resource tradeoffs: the nonpreemptive case. Manag. Sci. 28, 1197–1210 (1982)

    Article  MATH  Google Scholar 

  • Ulusoy, G., Ozdamar, L.: A constraint-based perspective in resource constrained project scheduling. Int. J. Prod. Res. 32, 693–705 (1994)

    Article  MATH  Google Scholar 

  • van Roy, T.J.: A cross decomposition algorithm for capacitated facility location. Oper. Res. 34(1), 145–163 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  • Voss, S.: Meta-heuristics: The state of the art. In: Nareyek, A. (ed.) Local Search for Planning and Scheduling. LNAI, vol. 2148, pp. 1–23. Springer, Berlin (2001)

    Chapter  Google Scholar 

  • Zhang, H., Tam, C.M., Li, H.: Multimode project scheduling based on particle swarm optimization. Comput. Aided Civ. Infrastruct. Eng. 21, 93–103 (2006)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Bologna, Bologna, Italy

    Marco Boschetti

  2. Department of Computer Science, University of Bologna, Bologna, Italy

    Vittorio Maniezzo

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  1. Marco Boschetti
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  2. Vittorio Maniezzo
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Correspondence to Marco Boschetti.

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Boschetti, M., Maniezzo, V. Benders decomposition, Lagrangean relaxation and metaheuristic design. J Heuristics 15, 283–312 (2009). https://doi.org/10.1007/s10732-007-9064-9

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  • DOI: https://doi.org/10.1007/s10732-007-9064-9

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