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Optimization challenges in the structured low rank approximation problem

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  • Jonathan Gillard
  • Anatoly Zhigljavsky

Abstract

In this paper we illustrate some optimization challenges in the structured low rank approximation (SLRA) problem. SLRA can be described as the problem of finding a low rank approximation of an observed matrix which has the same structure as this matrix (such as Hankel). We demonstrate that the optimization problem arising is typically very difficult: in particular, the objective function is multiextremal even for simple cases. The main theme of the paper is to suggest that the difficulties described in approximating a solution of the SLRA problem open huge possibilities for the application of stochastic methods of global optimization. Copyright Springer Science+Business Media, LLC. 2013

Suggested Citation

  • Jonathan Gillard & Anatoly Zhigljavsky, 2013. "Optimization challenges in the structured low rank approximation problem," Journal of Global Optimization, Springer, vol. 57(3), pages 733-751, November.
    • Handle: RePEc:spr:jglopt:v:57:y:2013:i:3:p:733-751
    DOI: 10.1007/s10898-012-9962-8
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    References listed on IDEAS

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    1. 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.
    2. Anatoly Zhigljavsky & Antanas Žilinskas, 2008. "Stochastic Global Optimization," Springer Optimization and Its Applications, Springer, number 978-0-387-74740-8, June.
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    Cited by:

    1. Gillard, Jonathan & Usevich, Konstantin, 2018. "Structured low-rank matrix completion for forecasting in time series analysis," International Journal of Forecasting, Elsevier, vol. 34(4), pages 582-597.
    2. Sergeyev, Yaroslav D. & Kvasov, Dmitri E. & Mukhametzhanov, Marat S., 2017. "Operational zones for comparing metaheuristic and deterministic one-dimensional global optimization algorithms," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 141(C), pages 96-109.
    3. Yaroslav D. Sergeyev & Marat S. Mukhametzhanov & Dmitri E. Kvasov & Daniela Lera, 2016. "Derivative-Free Local Tuning and Local Improvement Techniques Embedded in the Univariate Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 186-208, October.
    4. Zhigljavsky, Anatoly & Golyandina, Nina & Gryaznov, Svyatoslav, 2016. "Deconvolution of a discrete uniform distribution," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 37-44.

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