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A matrix nonconvex relaxation approach to unconstrained binary polynomial programs

Author

Listed:
  • Yitian Qian

    (South China University of Technology)

  • Shaohua Pan

    (South China University of Technology)

  • Shujun Bi

    (South China University of Technology)

Abstract

This paper is concerned with a class of unconstrained binary polynomial programs (UBPPs), which covers the classical binary quadratic program and has a host of applications in many science and engineering fields. We start with the global exact penalty of its DC constrained SDP reformulation, and propose a continuous relaxation approach by seeking a finite number of approximate stationary points for the factorized form of the global exact penalty with increasing penalty parameters. A globally convergent majorization-minimization method with extrapolation is developed to capture such stationary points. Under a mild condition, we show that the rank-one projection of the output for the relaxation approach is an approximate feasible solution of the UBPP and quantify the lower bound of its minus objective value from the optimal value. Numerical comparisons with the SDP relaxation method armed with a special random rounding technique and the DC relaxation approaches armed with the solvers for linear and quadratic SDPs confirm the efficiency of the proposed relaxation approach, which can solve the instance of 20,000 variables in 15 min and yield a lower bound for the optimal value and the known best value with a relative error at most 1.824 and 2.870%, respectively.

Suggested Citation

  • Yitian Qian & Shaohua Pan & Shujun Bi, 2023. "A matrix nonconvex relaxation approach to unconstrained binary polynomial programs," Computational Optimization and Applications, Springer, vol. 84(3), pages 875-919, April.
    • Handle: RePEc:spr:coopap:v:84:y:2023:i:3:d:10.1007_s10589-022-00443-2
    DOI: 10.1007/s10589-022-00443-2
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    References listed on IDEAS

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    1. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
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