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Exact augmented Lagrangian duality for mixed integer linear programming

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Abstract

We investigate the augmented Lagrangian dual (ALD) for mixed integer linear programming (MIP) problems. ALD modifies the classical Lagrangian dual by appending a nonlinear penalty function on the violation of the dualized constraints in order to reduce the duality gap. We first provide a primal characterization for ALD for MIPs and prove that ALD is able to asymptotically achieve zero duality gap when the weight on the penalty function is allowed to go to infinity. This provides an alternative characterization and proof of a recent result in Boland and Eberhard (Math Program 150(2):491–509, 2015, Proposition 3). We further show that, under some mild conditions, ALD using any norm as the augmenting function is able to close the duality gap of an MIP with a finite penalty coefficient. This generalizes the result in Boland and Eberhard (2015, Corollary 1) from pure integer programming problems with bounded feasible region to general MIPs. We also present an example where ALD with a quadratic augmenting function is not able to close the duality gap for any finite penalty coefficient.

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Acknowledgments

We would like to thank Dr. Natashia Boland for the helpful discussions and feedback on our draft. This research has been supported in part by the National Science Foundation Grant 1331426 and the Office of Naval Research Grant N00014-15-1-2078.

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Authors and Affiliations

  1. Robinson College of Business, Georgia State University, Atlanta, GA, 30303, USA

    Mohammad Javad Feizollahi

  2. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Shabbir Ahmed & Andy Sun

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  1. Mohammad Javad Feizollahi
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  2. Shabbir Ahmed
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  3. Andy Sun
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Correspondence to Mohammad Javad Feizollahi.

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Feizollahi, M.J., Ahmed, S. & Sun, A. Exact augmented Lagrangian duality for mixed integer linear programming. Math. Program. 161, 365–387 (2017). https://doi.org/10.1007/s10107-016-1012-8

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