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Finding the global optimum of a class of quartic minimization problem

Author

Listed:
  • Pengfei Huang

    (Nankai University)

  • Qingzhi Yang

    (Nankai University)

  • Yuning Yang

    (Guangxi University)

Abstract

We consider a special nonconvex quartic minimization problem over a single spherical constraint, which includes the discretized energy functional minimization problem of non-rotating Bose-Einstein condensates (BECs) as one of the important applications. Such a problem is studied by exploiting its characterization as a nonlinear eigenvalue problem with eigenvector nonlinearity (NEPv). Firstly, we show that the NEPv has a unique nonnegative eigenvector, corresponding to the smallest nonlinear eigenvalue of NEPv, which is exactly the global minimizer to the optimization problem. Secondly, with these properties, we obtain that any algorithm converging to the nonnegative stationary point of this optimization problem finds its global optimum, such as the regularized Newton method. In particular, we obtain the convergence to the global optimum of the inexact alternating direction method of multipliers for this problem. Numerical experiments for applications in non-rotating BECs validate our theories.

Suggested Citation

  • Pengfei Huang & Qingzhi Yang & Yuning Yang, 2022. "Finding the global optimum of a class of quartic minimization problem," Computational Optimization and Applications, Springer, vol. 81(3), pages 923-954, April.
    • Handle: RePEc:spr:coopap:v:81:y:2022:i:3:d:10.1007_s10589-021-00345-9
    DOI: 10.1007/s10589-021-00345-9
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    References listed on IDEAS

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    1. Yuning Yang & Qingzhi Yang, 2012. "On solving biquadratic optimization via semidefinite relaxation," Computational Optimization and Applications, Springer, vol. 53(3), pages 845-867, December.
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