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A Feasibility-Ensured Lagrangian Heuristic for General Decomposable Problems

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  • Kouhei Harada

    (NTTDATA Mathematical Systems Inc.)

Abstract

The Lagrangian relaxation method is a popular and useful tool to solve large-scale optimization problems with decomposable structures. A drawback of this approach is that a primal solution obtained by solving the Lagrangian relaxation problem is often primal infeasible even if an optimal dual solution is given. To recover primal feasibility, we rely on specialized heuristic-based approaches, each of which is tailored for a particular optimization problem. To overcome the difficulty, a general heuristic method has been proposed recently. The method ensures primal feasibility, but it is not tailored for some particular optimization problems. Furthermore, the method can estimate the gap of the objective between an optimal solution and the feasible solution obtained using the heuristic. Although the method provides important results, it relies on the assumption that both objective and constraint functions are concave. We generalize the results: the method ensures primal feasibility for a rather general class of problems. We have demonstrated that the fundamentally important requirement here is the extremity of the solution set of the Lagrangian relaxation problem. Our numerical experiments support the efficiency and validity of our generalized results.

Suggested Citation

  • Kouhei Harada, 2021. "A Feasibility-Ensured Lagrangian Heuristic for General Decomposable Problems," SN Operations Research Forum, Springer, vol. 2(4), pages 1-26, December.
    • Handle: RePEc:spr:snopef:v:2:y:2021:i:4:d:10.1007_s43069-021-00094-9
    DOI: 10.1007/s43069-021-00094-9
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    References listed on IDEAS

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    1. Li, Jing-Quan & Mirchandani, Pitu B. & Borenstein, Denis, 2009. "A Lagrangian heuristic for the real-time vehicle rescheduling problem," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 45(3), pages 419-433, May.
    2. Diego Fiorotto & Silvio Araujo, 2014. "Reformulation and a Lagrangian heuristic for lot sizing problem on parallel machines," Annals of Operations Research, Springer, vol. 217(1), pages 213-231, June.
    3. Marshall L. Fisher, 1981. "The Lagrangian Relaxation Method for Solving Integer Programming Problems," Management Science, INFORMS, vol. 27(1), pages 1-18, January.
    4. Patriksson, Michael & Strömberg, Christoffer, 2015. "Algorithms for the continuous nonlinear resource allocation problem—New implementations and numerical studies," European Journal of Operational Research, Elsevier, vol. 243(3), pages 703-722.
    5. Cynthia Barnhart & Ellis L. Johnson & George L. Nemhauser & Martin W. P. Savelsbergh & Pamela H. Vance, 1998. "Branch-and-Price: Column Generation for Solving Huge Integer Programs," Operations Research, INFORMS, vol. 46(3), pages 316-329, June.
    6. Duan Li & Xiaoling Sun, 2006. "Nonlinear Integer Programming," International Series in Operations Research and Management Science, Springer, number 978-0-387-32995-6, September.
    7. Valentina Cacchiani & Alberto Caprara & Matteo Fischetti, 2012. "A Lagrangian Heuristic for Robustness, with an Application to Train Timetabling," Transportation Science, INFORMS, vol. 46(1), pages 124-133, February.
    8. Michael Held & Richard M. Karp, 1970. "The Traveling-Salesman Problem and Minimum Spanning Trees," Operations Research, INFORMS, vol. 18(6), pages 1138-1162, December.
    9. Beasley, J. E., 1993. "Lagrangean heuristics for location problems," European Journal of Operational Research, Elsevier, vol. 65(3), pages 383-399, March.
    10. J. P. Aubin & I. Ekeland, 1976. "Estimates of the Duality Gap in Nonconvex Optimization," Mathematics of Operations Research, INFORMS, vol. 1(3), pages 225-245, August.
    Full references (including those not matched with items on IDEAS)

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