Skip to main content
SpringerLink
Log in

On solving biquadratic optimization via semidefinite relaxation

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

In this paper, we study a class of biquadratic optimization problems. We first relax the original problem to its semidefinite programming (SDP) problem and discuss the approximation ratio between them. Under some conditions, we show that the relaxed problem is tight. Then we consider how to approximately solve the problems in polynomial time. Under several different constraints, we present variational approaches for solving them and give provable estimation for the approximation solutions. Some numerical results are reported at the end of this paper.

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

Similar content being viewed by others

References

  1. Alon, N., Naor, A.: Approximating the cut-norm via Grothendieck’s inequality. SIAM J. Comput. 35, 787–803 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bro, R., Jong, S.: A fast non-negativity constrained least squares algorithm. J. Chemom. 11, 393–401 (1997)

    Article  Google Scholar 

  3. Bro, R., Sidiropoulos, N.: Least squares algorithms under unimodality and non-negativity constraints. J. Chemom. 12, 223–247 (1998)

    Article  Google Scholar 

  4. Cardoso, J.F.: High-order contrasts for independent component analysis. Neural Comput. 11, 157–192 (1999)

    Article  Google Scholar 

  5. Cichocki, A., Zdunek, R., Phan, A.H., Amari, S.: Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation. Wiley, New York (2009)

    Google Scholar 

  6. Comon, P.: Independent component analysis, a new concept? Signal Process. 36, 287–314 (1994)

    Article  MATH  Google Scholar 

  7. 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  MATH  Google Scholar 

  8. De Lathauwer, L., Moor, B.D., Vandewalle, J.: On the best rank-1 and rank-(R 1,R 2,…,R N ) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 21, 1324–1342 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  10. Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grant, M., Boyd, S., CVX: Matlab Software for Disciplined Convex Programming, version 1.2 (2010). http://cvxr.com/cvx

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

    Article  MathSciNet  MATH  Google Scholar 

  13. He, S., Li, Z., Zhang, S.: General constrained polynomial optimization: An approximation approach. Technical Report SEEM2009-06, Department of Systems Engineering & Engineering Management, The Chinese University of Hong Kong (2009)

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

    Article  MathSciNet  MATH  Google Scholar 

  15. He, S., Li, Z., Zhang, S.: Approximation algorithms for discrete polynomial optimization. Working paper (2010)

  16. Khot, S., Naor, A.: Linear equations modulo 2 and the L1 diameter of convex bodies. SIAM J. Comput. 38, 1448–1463 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kim, S., Kojima, M.: Exact solutions of some nonconvex quadratic optimization problems via SDP and SOCP relaxations. Comput. Optim. Appl. 26, 143–154 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Kolday, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. (2010)

  20. Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lasserre, J.B.: Polynomials nonnegative on a grid and discrete representations. Trans. Am. Math. Soc. 354, 631–649 (2001)

    Article  MathSciNet  Google Scholar 

  22. Lim, L.-H.: Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, vol. 1, pp. 129–132 (2005)

    Google Scholar 

  23. Ling, C., Nie, J., Qi, L., Ye, Y.: Biquadratic optimization over unit spheres and semidefinite programming relaxations. SIAM J. Optim. 20, 1286–1310 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ling, C., Zhang, X., Qi, L.: Semidefinite relaxation approximation for multivariate bi-quadratic optimization with quadratic constraints. Numer. Linear Algebra Appl. (2011). doi:10.1002/nla.781

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Nesterov, Y.: Semidefinite relaxation and nonconvex quadratic optimization. Optim. Methods Softw. 12, 1–20 (1997)

    MathSciNet  Google Scholar 

  27. Paatero, P.: A weighted non-negative least squares algorithm for three-way parafac factor analysis. Chemom. Intell. Lab. Syst. 38, 223–242 (1997)

    Article  Google Scholar 

  28. Parrilo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD Dissertation, California Institute of Technology, CA (2000)

  29. Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program., Ser. B 96, 293–320 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Qi, L., Wang, F., Wang, Y.: Z-eigenvalue methods for a global polynomial optimization problem. Math. Program., Ser. A 118, 301–316 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rosakis, P.: Ellipticity and deformations with discontinuous deformation gradients in finite elastostatics. Arch. Ration. Mech. Anal. 109, 1–37 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  34. So, A.M.: Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems. Math. Program. (2011). doi:10.1007/s10107-011-0464-0

    Google Scholar 

  35. Wang, Y., Aron, M.: A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media. J. Elast. 44, 89–96 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  36. 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  MATH  Google Scholar 

  37. Zhang, S.: Quadratic maximization and semidefinite relaxation. Math. Program., Ser. A 87, 453–465 (2000)

    Article  MATH  Google Scholar 

  38. Zhang, X., Ling, C., Qi, L.: Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints. J. Glob. Optim. 49, 293–311 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their suggestions, which help us to improve the paper.

Author information

Authors and Affiliations

  1. School of Mathematics Science and LPMC, Nankai University, Tianjin, 300071, P.R. China

    Yuning Yang & Qingzhi Yang

Authors
  1. Yuning Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qingzhi Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qingzhi Yang.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant No. 10871105), Academic Scholarship for Doctoral Candidates, Ministry of Education of China (Grant No. (190)H0511009) and Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, Y., Yang, Q. On solving biquadratic optimization via semidefinite relaxation. Comput Optim Appl 53, 845–867 (2012). https://doi.org/10.1007/s10589-012-9462-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-012-9462-2

Keywords

Search

Navigation