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Tractable Relaxations for the Cubic One-Spherical Optimization Problem

  • Christoph Buchheim
  • Marcia FampaEmail author
  • Orlando Sarmiento
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)

Abstract

We consider the cubic one-spherical optimization problem, consisting in minimizing a homogeneous cubic function over the unit sphere. We propose different lower bounds that can be computed efficiently, using decompositions of the objective function and well-known results for the corresponding quadratic problem variant.

Keywords

Cubic one-spherical optimization problem Best rank-1 tensor approximation Trust region subproblem Convex relaxation 

Notes

Acknowledgments

C. Buchheim has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 764759. M. Fampa was supported in part by CNPq-Brazil grants 303898/2016-0 and 434683/2018-3. O. Sarmiento contributed much of his work while visiting the Technische Universität Dortmund, Dortmund, Germany, supported by a Research Fellowship from CAPES-Brazil - Finance Code 001.

References

  1. 1.
    Bader, B.W., Kolda, T.G., et al.: MATLAB Tensor Toolbox Version 3.0-dev, Oct 2017. https://www.tensortoolbox.org
  2. 2.
    Basser, P.J., Mattiello, J., LeBihan, D.: MR diffusion tensor spectroscopy and imaging. Biophys. J. 66, 259–267 (1994)Google Scholar
  3. 3.
    Basser, P.J., Mattiello, J., LeBihan, D.: Estimation of the effective seldiffusion tensor from the NMR spin echo. J. Magn. Reson. B 103, 247–254 (1994)Google Scholar
  4. 4.
    Basser, P.J., Jones, D.K.: Diffusion-tensor MRI: theory, experimental design and data analysis-a technical review. NMR Biomed. 15, 456–467 (2002)Google Scholar
  5. 5.
    Liu, C.L., Bammer, R., Acar, B., Moseley, M.E.: Characterizing non-gaussian diffusion by using generalized diffusion tensors. Magn. Reson. Med. 51, 924–937 (2004)Google Scholar
  6. 6.
    Lucidi, S., Palagi, L.: Solution of the trust region problem via a smooth unconstrained reformulation. Top. Semidefinite Inter. Point Methods Fields Inst. Commun. 18, 237–250 (1998)Google Scholar
  7. 7.
    Nesterov, Y.E.: Random walk in a simplex and quadratic optimization over convex polytopes. CORE Discussion Paper 2003/71 CORE-UCL (2003)Google Scholar
  8. 8.
    Nie, J., Wang, L.: Semidefinte relaxations for best rank-1 tensor approximations. SIAM J. Matrix Anal. Appl. 35, 1155–1179 (2014)Google Scholar
  9. 9.
    Stern, R.J., Wolkowicz, H.: Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations. SIAM J. Optim. 5, 286–313 (1995)Google Scholar
  10. 10.
    Zhang, X., Qi, L., Ye, Y.: The cubic spherical optimization problems. Math. Comput. 81(279), 1513–1525 (2012)Google Scholar
  11. 11.
    Zhang, X., Ling, C., Qi, L., Wu, E.X.: The measure of diffusion skewness and kurtosis in magnetic resonance imaging. Pac. J. Optim. 6, 391–404 (2010)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Technische Universität DortmundDortmundGermany
  2. 2.Universidade Federal do Rio de JaneiroRio de JaneiroBrazil

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