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Finding the global optimum of a class of quartic minimization problem

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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.

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Acknowledgements

The authors would like to thank the anonymous referees for their suggestions, which help us to improve the paper. And we are grateful to Professor Xinming Wu, Dr. Jinshan Zeng and Dr. Xudong Li for their inspiration and help.

Funding

The first author was supported by the Tianjin Graduate Research and Innovation Project 2019YJSB040. The second author was supported by the National Natural Science Foundation of China Grant 11671217 and 12071234. The third author was supported by the National Natural Science Foundation of China Grant 11801100 and the Fok Ying Tong Education Foundation Grant 171094.

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Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

    Pengfei Huang & Qingzhi Yang

  2. College of Mathematics and Information Science, Guangxi University, Nanning, 530004, Guangxi, People’s Republic of China

    Yuning Yang

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  1. Pengfei Huang
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Correspondence to Qingzhi Yang.

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Huang, P., Yang, Q. & Yang, Y. Finding the global optimum of a class of quartic minimization problem. Comput Optim Appl 81, 923–954 (2022). https://doi.org/10.1007/s10589-021-00345-9

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