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A variational approach of the rank function

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  • Jean-Baptiste Hiriart-Urruty
  • Hai Le

Abstract

In the same spirit as the one of the paper (Hiriart-Urruty and Malick in J. Optim. Theory Appl. 153(3):551–577, 2012 ) on positive semidefinite matrices, we survey several useful properties of the rank function (of a matrix) and add some new ones. Since the so-called rank minimization problems are the subject of intense studies, we adopt the viewpoint of variational analysis, that is the one considering all the properties useful for optimizing, approximating or regularizing the rank function. Copyright Sociedad de Estadística e Investigación Operativa 2013

Suggested Citation

  • Jean-Baptiste Hiriart-Urruty & Hai Le, 2013. "A variational approach of the rank function," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 207-240, July.
    • Handle: RePEc:spr:topjnl:v:21:y:2013:i:2:p:207-240
    DOI: 10.1007/s11750-013-0283-y
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    References listed on IDEAS

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    1. Bomze, Immanuel M., 2012. "Copositive optimization – Recent developments and applications," European Journal of Operational Research, Elsevier, vol. 216(3), pages 509-520.
    2. Carl Eckart & Gale Young, 1936. "The approximation of one matrix by another of lower rank," Psychometrika, Springer;The Psychometric Society, vol. 1(3), pages 211-218, September.
    3. Jean-Baptiste Hiriart-Urruty & Jérôme Malick, 2012. "A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World," Journal of Optimization Theory and Applications, Springer, vol. 153(3), pages 551-577, June.
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    Cited by:

    1. Roger Behling & Douglas S. Gonçalves & Sandra A. Santos, 2019. "Local Convergence Analysis of the Levenberg–Marquardt Framework for Nonzero-Residue Nonlinear Least-Squares Problems Under an Error Bound Condition," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 1099-1122, December.
    2. Liqun Qi & Ziyan Luo & Qing-Wen Wang & Xinzhen Zhang, 2022. "Quaternion Matrix Optimization: Motivation and Analysis," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 621-648, June.
    3. Kim Christensen & Mikkel Slot Nielsen & Mark Podolskij, 2021. "High-dimensional estimation of quadratic variation based on penalized realized variance," Papers 2103.03237, arXiv.org.
    4. Tim Hoheisel & Elliot Paquette, 2023. "Uniqueness in Nuclear Norm Minimization: Flatness of the Nuclear Norm Sphere and Simultaneous Polarization," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 252-276, April.

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