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Polynomial Optimization: Tightening RLT-Based Branch-and-Bound Schemes with Conic Constraints

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Abstract

This paper explores the potential of (nonlinear) conic constraints to tighten the relaxations of spatial branch-and-bound algorithms. More precisely, we contribute to the literature on the use of conic optimization for the efficient solution, to global optimality, of nonconvex polynomial optimization problems. Taking as baseline an RLT-based algorithm, we present different families of well-known conic-driven constraints: linear SDP-cuts, second-order cone constraints, and SDP constraints. We integrate these constraints in the baseline algorithm and present a thorough computational study to assess their performance, both with respect to each other and with respect to the standard RLT relaxations for polynomial optimization problems. Our main finding is that the different variants of nonlinear constraints (second-order cone and semidefinite) are the best performing ones in around \(50\%\) of the instances in widely used test sets. Additionally, we discuss how one can benefit from the use of machine learning to decide on the most suitable constraints to add to a given instance. The computational results show that the machine learning approach significantly outperforms each of the individual approaches.

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Data Availability

Most of the details on the test instances used in this research are included in this manuscript. Any additional information is available from the authors upon reasonable request.

Notes

  1. Refer to Section 5.4 in [29] for additional details.

  2. For further discussion on this and other approaches, the reader is referred to [50].

  3. Instances can be downloaded from https://raposa.usc.es/files/DS-TS.zip (DS), https://raposa.usc.es/files/MINLPLib-TS.zip (MINLPLib), and https://raposa.usc.es/files/QPLIB-TS.zip (QPLIB).

  4. The specific RAPOSa options to control the conic relaxations are: sdp, sdpcuts, sdpsolver, socp, and socpsolver. Refer to https://raposa.usc.es/requirements/ for further details.

  5. Refer to [29] for additional details.

  6. In our analysis we consider \(\varepsilon =0.001\).

  7. The reason for using \(\log (\text {time}+1)\) in the y-axis instead of \(\log (\text {time})\) is that solve times are often close to 0 and, with \(\log (\text {time}+1)\), these times are mapped again to (nonnegative) values close to 0.

  8. Similarly, the percentages of binding constraints at the root node do not provide much insight on what might be special for these subclasses of problems.

  9. VIG and CMIG stand for two graphs that can be associated to any given polynomial optimization problem: variables intersection graph and constraints-monomials intersection graph, and whose precise definitions is given in [26].

  10. Refer to [26] and [29] for further details. It is worth noting that in the former reference there is another branching rule that plays an important role: the dual rule. It is based on the dual values of the constraints in which each variable appears, but computing such values in the context of conic constraints is not straightforward and, therefore, we have chosen to exclude this rule from the analysis.

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Acknowledgements

The authors thank the referees, the AE, and the Editor for their thorough and thoughtful comments that have contributed to improvements to the original submission. This work is part of the R+D+I project grants MTM2017-87197-C3 and PID2021-124030NB-C32, funded by MCIN/AEI/10.13039/501100011033/ and by “ERDF A way of making Europe”/EU. This research was also funded by Grupos de Referencia Competitiva ED431C-2021/24 from the Consellería de Cultura, Educación e Universidades, Xunta de Galicia. Brais González-Rodríguez acknowledges support from the Spanish Ministry of Education through FPU grant 17/02643. Raúl Alvite-Pazó and Samuel Alvite-Pazó acknowledge support from CITMAga through project ITMATI-R-7-JGD. Bissan Ghaddar’s research is supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant 2017-04185 and by the Thompson Chair of Leadership and Innovation.

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Authors and Affiliations

  1. Department of Statistics and Operations Research, University of Vigo Ourense, Ourense, Galicia, Spain

    Brais González-Rodríguez

  2. CITMAga (Galician Center for Mathematical Research and Technology), Santiago de Compostela, Galicia, Spain

    Raúl Alvite-Pazó, Samuel Alvite-Pazó & Julio González-Díaz

  3. Ivey Business School, Western University London, London, Ontario, Canada

    Bissan Ghaddar

  4. Department of Statistics, Mathematical Analysis and Optimization and MODESTYA Research Group, University of Santiago de Compostela, Santiago de Compostela, Galicia, Spain

    Julio González-Díaz

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González-Rodríguez, B., Alvite-Pazó, R., Alvite-Pazó, S. et al. Polynomial Optimization: Tightening RLT-Based Branch-and-Bound Schemes with Conic Constraints. J Optim Theory Appl 204, 12 (2025). https://doi.org/10.1007/s10957-024-02558-4

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