Skip to main content
SpringerLink
Log in

The sparsest solutions to Z-tensor complementarity problems

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

Finding the sparsest solutions to a tensor complementarity problem is generally NP-hard due to the nonconvexity and noncontinuity of the involved \(\ell _0\) norm. In this paper, a special type of tensor complementarity problems with Z-tensors has been considered. Under some mild conditions, we show that to pursuit the sparsest solutions is equivalent to solving polynomial programming with a linear objective function. The involved conditions guarantee the desired exact relaxation and also allow to achieve a global optimal solution to the relaxed nonconvex polynomial programming problem. Particularly, in comparison to existing exact relaxation conditions, such as RIP-type ones, our proposed conditions are easy to verify.

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

Notes

  1. More precisely, here the tensor should be called the hypermatrix. In physics and mechanics, tensors are geometric objects equipped with a transformation law that details how the components of the tensor respond to a change of basis, while hypermatrices are simply multi-dimensional arrays. However, for most papers in tensor decomposition (e.g., the survey paper [25] in SIAM Review, and the numerical optimization methods in [24] in Math. Programming), and in tensor complementarity problems (e.g., [34]), the word “tensors” is used for multi-dimensional arrays. We will then roughly use the word “tensors” throughout this paper.

References

  1. Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces. SIAM Rev. 18(4), 620–709 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blumensath, T.: Compressed sensing with nonlinear observations and related nonlinear optimization problems. IEEE Trans. Inf. Theory 59(6), 3466–3474 (2013)

    Article  MathSciNet  Google Scholar 

  3. Candes, E.J.: The restricted isometry property and its implications for compressed sensing. Comptes Rendus Math. 346(9), 589–592 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Candes, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Candes, E.J., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Candes, E.J., Wakin, M.B., Boyd, S.P.: Enhancing sparsity by reweighted \(\ell _1\) minimization. J. Fourier Anal. Appl. 14(5–6), 877–905 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Che,M., Qi,L., Wei,Y.: Positive definite tensors to nonlinear complementarity problems. J Optim Theory Appl. 168, 475–487 (2016)

  8. Chen, X., Ge, D., Wang, Z., Ye, Y.: Complexity of unconstrained \(l_2-l_p\) minimization. Math. Program. 143(1–2), 371–383 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, X., Ng, M.K., Zhang, C.: Non-Lipschitz-regularization and box constrained model for image restoration. IEEE Trans. Image Process. 21(12), 4709–4721 (2012)

    Article  MathSciNet  Google Scholar 

  10. Chen, X., Xiang, S.: Sparse solutions of linear complementarity problems. Preprint (2014)

  11. Comon, P., Golub, G., Lim, L.H., Mourrain, B.: Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. Appl. 30(3), 1254–1279 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. De Lathauwer, L., De Moor, B., Vandewalle, J.: On the best rank-1 and rank-\((R_1, R_2,\cdots , R_N)\) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 21(4), 1324–1342 (electronic) (2000)

  13. Ding, W., Qi, L., Wei, Y.: \(M\)-tensors and nonsingular \(M\)-tensors. Linear Algebra Appl. 439(10), 3264–3278 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ding, W., Wei, Y.: Solving multi-linear systems with \(\cal M\)-tensors. Preprint (2015)

  15. Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Donoho, D.L., Tanner, J.: Precise undersampling theorems. Proc. IEEE 98(6), 913–924 (2010)

    Article  Google Scholar 

  17. Facchinei, F., Pang, J.S.: Finite-dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, vol. I. Springer-Verlag, New York (2003)

    MATH  Google Scholar 

  18. Foucart, S., Lai, M.J.: Sparsest solutions of underdetermined linear systems via \(l_q\)-minimization for \(0<q\le 1\). Appl. Comput. Harmon. Anal. 26(3), 395–407 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gowda, M.S., Luo, Z., Qi, L., Xiu, N.: \(Z\)-tensors and complementarity problems. arXiv preprint arXiv:1510.07933 (2015)

  20. Harmany, Z.T., Marcia, R.F., Willett, R.M.: Spiral out of convexity: sparsity-regularized algorithms for photon-limited imaging. In: IS&T/SPIE Electronic Imaging, pp. 75330R–75330R. International Society for Optics and Photonics (2010)

  21. Isac, G.: Complementarity problems. Springer-Verlag, Berlin (1992)

  22. Juditsky, A., Nemirovski, A.: On verifiable sufficient conditions for sparse signal recovery via \(\ell _1\) minimization. Math. Program. 127(1), 57–88 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Khajehnejad, M.A., Dimakis, A.G., Xu, W., Hassibi, B.: Sparse recovery of nonnegative signals with minimal expansion. IEEE Trans. Signal Process. 59(1), 196–208 (2011)

    Article  MathSciNet  Google Scholar 

  24. Kolda, T.G.: Numerical optimization for symmetric tensor decomposition. Math. Program. 151(1), 225–248 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lauer, F., Ohlsson, H.: Finding sparse solutions of systems of polynomial equations via group-sparsity optimization. J. Glob. Optim. 62(2), 319–349 (2015)

  27. Li, X., Voroninski, V.: Sparse signal recovery from quadratic measurements via convex programming. SIAM J. Math. Anal. 45(5), 3019–3033 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, Y.F., Dai, Y.H., Luo, Z.Q.: Joint power and admission control via linear programming deflation. IEEE Trans. Signal Process. 61(6), 1327–1338 (2013)

    Article  MathSciNet  Google Scholar 

  29. Luo, Z., Qin, L., Kong, L., Xiu, N.: The nonnegative zero-norm minimization under generalized \(Z\)-matrix measurement. J. Optim. Theory Appl. 160(3), 854–864 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40(6), 1302–1324 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rheinboldt, W.C.: On \(M\)-functions and their application to nonlinear Gauss–Seidel iterations and to network flows. J. Math. Anal. Appl. 32, 274–307 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shang, M., Zhang, C., Xiu, N.: Minimal zero norm solutions of linear complementarity problems. J. Optim. Theory Appl. 163(3), 795–814 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sheikh, M.A., Sarvotham, S., Milenkovic, O., Baraniuk, R.G.: Dna array decoding from nonlinear measurements by belief propagation. In: IEEE/SP 14th Workshop on statistical signal processing, 2007. SSP’07, pp. 215–219. IEEE (2007)

  34. Song, Y., Qi, L.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165(3), 854–873 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Song, Y., Qi, L.: Properties of tensor complementarity problem and some classes of structured tensors. arXiv preprint arXiv:1412.0113 (2014)

  36. Tamir, A.: Minimality and complementarity properties associated with \(Z\)-functions and \(M\)-functions. Math. Program. 7, 17–31 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  37. Xu, Z., Chang, X., Xu, F., Zhang, H.: regularization: a thresholding representation theory and a fast solver. IEEE Trans. Neural Netw. Learn. Syst. 23(7), 1013–1027 (2012)

    Article  Google Scholar 

  38. Yang, Y., Yang, Q.: Further results for Perron–Frobenius theorem for nonnegative tensors. SIAM J. Matrix Anal. Appl. 31(5), 2517–2530 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhang, L., Qi, L., Zhou, G.: \(M\)-tensors and some applications. SIAM J. Matrix Anal. Appl. 35(2), 437–452 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhang, Y.: Theory of compressive sensing via \(\ell _1\)-minimization: a non-RIP analysis and extensions. J. Oper. Res. Soc. China 1(1), 79–105 (2013)

    Article  Google Scholar 

  41. Zhao, Y.B.: RSP-based analysis for sparsest and least \(\ell _1\)-norm solutions to underdetermined linear systems. IEEE Trans. Signal Process. 61(22), 5777–5788 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank two anonymous referees and the associated editor for their valuable comments, and Mr. Weiyang Ding and Prof. Wei for the proof of Proposition 3 and their matlab codes of the Gauss-Seidel iteration method.

Author information

Authors and Affiliations

  1. State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, 100044, China

    Ziyan Luo

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

    Liqun Qi

  3. Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing, China

    Naihua Xiu

Authors
  1. Ziyan Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liqun Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Naihua Xiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ziyan Luo.

Additional information

This research was supported by the National Natural Science Foundation of China (11301022, 11431002), the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University (RCS2014ZT20, RCS2014ZZ01), and the Hong Kong Research Grant Council (Grant No. PolyU 502111, 501212, 501913 and 15302114).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luo, Z., Qi, L. & Xiu, N. The sparsest solutions to Z-tensor complementarity problems. Optim Lett 11, 471–482 (2017). https://doi.org/10.1007/s11590-016-1013-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-016-1013-9

Keywords

Search

Navigation