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New SOCP relaxation and branching rule for bipartite bilinear programs

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Abstract

A bipartite bilinear program (BBP) is a quadratically constrained quadratic optimization problem where the variables can be partitioned into two sets such that fixing the variables in any one of the sets results in a linear program. We propose a new second order cone representable (SOCP) relaxation for BBP, which we show is stronger than the standard SDP relaxation intersected with the boolean quadratic polytope. We then propose a new branching rule inspired by the construction of the SOCP relaxation. We describe a new application of BBP called as the finite element model updating problem, which is a fundamental problem in structural engineering. Our computational experiments on this problem class show that the new branching rule together with an polyhedral outer approximation of the SOCP relaxation outperforms a state-of-the-art commercial global solver in obtaining dual bounds.

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Acknowledgements

The authors would like to thank Xinjun Dong in Civil and Environmental Engineering at Georgia Tech, for his assistance with preparing the structural example data. Santanu S. Dey would like to acknowledge the discussion on a preliminary version of this paper at Dagstuhl workshop #18081, that helped improve the paper.

Funding

This work was supported by the NSF CMMI [Grant No. 1149400]; the NSF CMMI [Grant No. 1150700]; and the CNPq-Brazil [Grant No. 248941/2013-5].

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Authors and Affiliations

  1. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, USA

    Santanu S. Dey & Asteroide Santana

  2. School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, USA

    Yang Wang

Authors
  1. Santanu S. Dey
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  2. Asteroide Santana
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  3. Yang Wang
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Correspondence to Asteroide Santana.

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Dey, S.S., Santana, A. & Wang, Y. New SOCP relaxation and branching rule for bipartite bilinear programs. Optim Eng 20, 307–336 (2019). https://doi.org/10.1007/s11081-018-9402-9

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  • DOI: https://doi.org/10.1007/s11081-018-9402-9

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