On the strength of Lagrangian duality in multiobjective integer programming
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10107_2024_Article_2121.pdf
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Author(s)
Brun, Matthew
Perini, Tyler
Sinha, Saumya
Schaefer, Andrew J.
Date Issued
August 20, 2024
Journal
Mathematical Programming
Publisher
Springer Berlin Heidelberg
Citation
Brun, M., Perini, T., Sinha, S. et al. On the strength of Lagrangian duality in multiobjective integer programming. Math. Program. (2024).
Version
Final published version
Abstract
This paper investigates the potential of Lagrangian relaxations to generate quality bounds on non-dominated images of multiobjective integer programs (MOIPs). Under some conditions on the relaxed constraints, we show that a set of Lagrangian relaxations can provide bounds that coincide with every bound generated by the convex hull relaxation. We also provide a guarantee of the relative quality of the Lagrangian bound at unsupported solutions. These results imply that, if the relaxed feasible region is bounded, some Lagrangian bounds will be strictly better than some convex hull bounds. We demonstrate that there exist Lagrangian multipliers which are sparse, satisfy a complementary slackness property, and generate tight relaxations at supported solutions. However, if all constraints are dualized, a relaxation can never be tight at an unsupported solution. These results characterize the strength of the Lagrangian dual at efficient solutions of an MOIP.
MIT Department
Massachusetts Institute of Technology. Operations Research Center
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1007/s10107-024-02121-z
