Skip to main content
SpringerLink
Log in

Tighter bound estimation for efficient biquadratic optimization over unit spheres

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

Bi-quadratic programming over unit spheres is a fundamental problem in quantum mechanics introduced by pioneer work of Einstein, Schrödinger, and others. It has been shown to be NP-hard; so it must be solve by efficient heuristic algorithms such as the block improvement method (BIM). This paper focuses on the maximization of bi-quadratic forms with nonnegative coefficient tensors, which leads to a rank-one approximation problem that is equivalent to computing the M-spectral radius and its corresponding eigenvectors. Specifically, we propose a tight upper bound of the M-spectral radius for nonnegative fourth-order partially symmetric (PS) tensors. This bound, serving as an improved shift parameter, significantly enhances the convergence speed of BIM while maintaining computational complexity aligned with the initial shift parameter of BIM. Moreover, we elucidate that the computation cost of such upper bound can be further simplified for certain sets and delve into the nature of these sets. Building on the insights gained from the proposed bounds, we derive the exact solutions of the M-spectral radius and its corresponding M-eigenvectors for certain classes of fourth-order PS-tensors and discuss the nature of this specific category. Lastly, as a practical application, we introduce a testable sufficient condition for the strong ellipticity in the field of solid mechanics. Numerical experiments demonstrate the utility of the proposed results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Algorithm 1
Fig. 1
Fig. 2
Fig. 3
Algorithm 2
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Ling, C., Nie, J., Qi, L., Ye, Y.: bi-quadratic optimization over unit spheres and semidefinite programming relaxations. SIAM J. Optim. 20, 1286–1310 (2010)

    Article  Google Scholar 

  2. He, S., Li, Z., Zhang, S.: Approximation algorithms for homogeneous polynomial optimization with quadratic constraints. Math. Program. 125, 353–383 (2010)

    Article  MathSciNet  Google Scholar 

  3. Wang, Y., Qi, L., Zhang, X.: A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor. Numer. Linear Algebra Appl. 16, 589–601 (2009)

    Article  MathSciNet  Google Scholar 

  4. Zhang, X., Ling, C., Qi, L.: Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints. J. Global Optim. 49, 293–311 (2011)

    Article  MathSciNet  Google Scholar 

  5. Hu, S.L., Huang, Z.H.: Alternating direction method for bi-quadratic programming. J. Global Optim. 51, 429–446 (2011)

    Article  MathSciNet  Google Scholar 

  6. Yang, Y., Yang, Q.: On solving bi-quadratic optimization via semidefinite relaxation. Comput. Optim. Appl. 53, 845–867 (2012)

    Article  MathSciNet  Google Scholar 

  7. Yang, Y., Yang, Q., Qi, L.: Approximation bounds for trilinear and bi-quadratic optimization problems over nonconvex constraints. J. Optim. Theory Appl. 163, 841–858 (2014)

    Article  MathSciNet  Google Scholar 

  8. Wang, Y., Caccetta, L., Zhou, G.: Convergence analysis of a block improvement method for polynomial optimization over unit spheres. Numer. Linear Algebra Appl. 22, 1059–1076 (2015)

    Article  MathSciNet  Google Scholar 

  9. Qi, L., Hu, S., Zhang, X., Xu, Y.: Bi-quadratic tensors, bi-quadratic decompositions, and norms of bi-quadratic tensors. Front. Math. China 16, 1–15 (2021)

    Article  MathSciNet  Google Scholar 

  10. Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935)

    Article  Google Scholar 

  11. Schrödinger, E.: Die gegenwärtige situation in der quantenmechanik, Naturwissenschaften. 23, 807–812, 823–828, 844–849 (1935)

  12. Doherty, A.C., Parrilo, P.A., Spedalieri, F.M.: Distinguishing separable and entangled states. Phys. Rev. Lett. (2002). https://doi.org/10.1103/PhysRevLett.88.187904

    Article  Google Scholar 

  13. Dahl, G., Leinaas, J.M., Myrheim, J., Ovrum, E.: A tensor product matrix approximation problem in quantum physics. Linear Algebra Appl. 420, 711–725 (2007)

    Article  MathSciNet  Google Scholar 

  14. Leordeanu, M., Hebert, M.: A spectral technique for correspondence problems using pairwise constraints. In: Proceedings of 10th IEEE International Conference of Computer Vision, vol. 2, pp. 1482–1489 (2005)

  15. Cour, T., Srinivasan, P., Shi, J.: Balanced graph matching. In: Proceedings of Advances in Neural Information Processing Systems, pp. 313–320 (2006)

  16. Egozi, A., Keller, Y., Guterman, H.: A probabilistic approach to spectral graph matching. IEEE Trans. Pattern Anal. 35, 18–27 (2012)

    Article  Google Scholar 

  17. Chiriţă, S., Danescu, A., Ciarletta, M.: On the strong ellipticity of the anisotropic linearly elastic materials. J. Elast. 87, 1–27 (2007)

    Article  MathSciNet  Google Scholar 

  18. Han, D., Dai, H., Qi, L.: Conditions for strong ellipticity of anisotropic elastic materials. J. Elast. 97, 1–13 (2009)

    Article  MathSciNet  Google Scholar 

  19. Li, S., Li, C., Li, Y.: M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor. J. Comput. Appl. Math. 356, 391–401 (2019)

    Article  MathSciNet  Google Scholar 

  20. Wang, X., Che, M., Wei, Y.: Best rank-one approximation of fourth-order partially symmetric tensors by neural network. Numer. Math. Theory Methods Appl. 11, 673–700 (2018)

    Article  MathSciNet  Google Scholar 

  21. Miao, Y., Wei, Y., Chen, Z.: Fourth-order tensor Riccati equations with the Einstein product. Linear Multilinear Algebra (2020). https://doi.org/10.1080/03081087.2020.1777248

    Article  Google Scholar 

  22. Qi, L., Dai, H., Han, D.: Conditions for strong ellipticity and M-eigenvalues. Front. Math. China 4, 349–364 (2009)

    Article  MathSciNet  Google Scholar 

  23. Waki, H., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite program relaxations for polynomial optimization problems with constructed sparsity. SIAM J. Optim. 17, 218–242 (2006)

    Article  MathSciNet  Google Scholar 

  24. Luo, Z.Q., Zhang, S.: A semidefinite relaxation scheme for multivariate quartic polynomial optimization with quadratic constraints. SIAM J. Optim. 20, 1716–1736 (2010)

    Article  MathSciNet  Google Scholar 

  25. Ng, M., Qi, L., Zhou, G.: Finding the largest eigenvalue of a nonnegative tensor. SIAM J. Matrix Anal. Appl. 31, 1090–1099 (2010)

    Article  MathSciNet  Google Scholar 

  26. Gloub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins Universtiy Press, Baltimore (1996)

    Google Scholar 

  27. Kolda, T.G., Mayo, J.R.: Shifted power method for computing tensor eigenpairs. SIAM J. Matrix Anal. Appl. 32, 1095–1124 (2012)

    Article  MathSciNet  Google Scholar 

  28. Che, H., Chen, H., Wang, Y.: On the M-eigenvalue estimation of fourth-order partially symmetric tensors. J. Ind. Manag. Optim. 16, 309–324 (2020)

    Article  MathSciNet  Google Scholar 

  29. He, J., Li, C., Wei, Y.: M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity. Appl. Math. Lett. (2019). https://doi.org/10.1016/j.aml.2019.106137

    Article  Google Scholar 

  30. Li, S., Chen, Z., Li, C., Zhao, J.: Eigenvalue bounds of third-order tensors via the minimax eigenvalue of symmetric matrices. Comput. Appl. Math. (2020). https://doi.org/10.1007/s40314-020-01245-0

    Article  MathSciNet  Google Scholar 

  31. Li, S., Chen, Z., Liu, Q., Lu, L.: Bounds of M-eigenvalues and strong ellipticity conditions for elasticity tensors. Linear Multilinear Algebra (2021). https://doi.org/10.1080/03081087.2021.1885600

    Article  Google Scholar 

  32. Ling, C., He, H., Qi, L.: Improved approximation results on standard quartic polynomial optimization. Optim. Lett. 11, 1767–1782 (2017)

    Article  MathSciNet  Google Scholar 

  33. Chen, H., He, H., Wang, Y., Zhou, G.: An efficient alternating minimization method for fourth degree polynomial optimization. J. Global Optim. 82, 83–103 (2022)

    Article  MathSciNet  Google Scholar 

  34. Ding, W., Liu, J., Qi, L., Yan, H.: Elasticity M-tensors and the strong ellipticity condition. Appl. Math. Comput. (2020). https://doi.org/10.1016/j.amc.2019.124982

    Article  MathSciNet  Google Scholar 

  35. Bhatia, R.: Matrix Analysis, vol. 169. Springer (2013)

    Google Scholar 

  36. Dong, X., Thanou, D., Frossard, P., Vandergheynst, P.: Learning Laplacian matrix in smooth graph signal representations. IEEE Trans. Signal Process. 64, 6160–6173 (2016)

    Article  MathSciNet  Google Scholar 

  37. Ye, K., Lim, L.H.: Every matrix is a product of Toeplitz matrices. Found. Comput. Math. 16, 577–598 (2016)

    Article  MathSciNet  Google Scholar 

  38. Kriegeskorte, N., Mur, M., Bandettini, P.A.: Representational similarity analysis-connecting the branches of systems neuroscience. Front. Syst. Neurosci. (2008). https://doi.org/10.3389/neuro.06.004.2008

    Article  Google Scholar 

  39. Feng, H., Qiu, X., Miao, H.L.: Hypothesis Testing for Two Sample Comparison of Network Data (2021). https://doi.org/10.48550/arXiv.2106.13931

  40. Zubov, L., Rudev, A.: A criterion for the strong ellipticity of the equilibrium equations of an isotropic non-linearly elastic material. J. Appl. Math. Mech. 75, 432–446 (2011)

    Article  MathSciNet  Google Scholar 

  41. Li, S., Li, Y.: Checkable criteria for the M-positive definiteness of fourth-order partially symmetric tensors. Bull. Iran. Math. Soc. 46, 1455–1463 (2020)

    Article  MathSciNet  Google Scholar 

  42. Huang, Z., Qi, L.: Positive definiteness of paired symmetric tensors and elasticity tensors. J. Comput. Appl. Math. 338, 22–43 (2018)

    Article  MathSciNet  Google Scholar 

  43. Qi, L., Chen, H., Chen, Y.: Tensor Eigenvalues and Their Applications. Springer, Singapore (2018)

    Book  Google Scholar 

  44. Li, S., Li, Y.: Programmable sufficient conditions for the strong ellipticity of partially symmetric tensors. Appl. Math. Comput. (2021). https://doi.org/10.1016/j.amc.2021.126134

    Article  MathSciNet  Google Scholar 

  45. Gurtin, M.: The linear theory of elasticity. In: Linear Theories of Elasticity and Thermoelasticity, pp. 1–295. Springer, Berlin, Heidelberg (1973)

    Google Scholar 

  46. Knowles, J.K., Sternberg, E.: On the ellipticity of the equations of nonlinear elastostatics for a special material. J. Elast. 5, 341–361 (1975)

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work was supported in part by grants from the National Science Foundation of China (61571005, 12061025, 12161020), the fundamental research program of Guangdong, China (2020B1515310023, 2023A1515011281), and the Science and Technology Foundation of Guizhou Province ([2020]1Z002). Author Delu Zeng was supported by the National Science Foundation of China (61571005) and the fundamental research program of Guangdong, China (2020B1515310023, 2023A1515011281). Author Linzhang Lu was supported by the National Science Foundation of China (12161020). Author Zhen Chen was supported by the National Science Foundation of China (12061025) and the Science and Technology Foundation of Guizhou Province ([2020]1Z002).

Author information

Authors and Affiliations

  1. School of Mathematics, South China University of Technology, Guangzhou, China

    Shigui Li

  2. School of Mathematical Sciences, Guizhou Normal University, Guiyang, China

    Linzhang Lu & Zhen Chen

  3. School of Mathematical Sciences, Xiamen University, Xiamen, China

    Linzhang Lu

  4. Department of Biostatistics and Computational Biology, University of Rochester, New York, USA

    Xing Qiu

  5. School of Electronic and Information Engineering, South China University of Technology, Guangzhou, China

    Delu Zeng

Authors
  1. Shigui Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Linzhang Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xing Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhen Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Delu Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Delu Zeng.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, S., Lu, L., Qiu, X. et al. Tighter bound estimation for efficient biquadratic optimization over unit spheres. J Glob Optim (2024). https://doi.org/10.1007/s10898-024-01401-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10898-024-01401-4

Keywords

Search

Navigation