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Matrix product constraints by projection methods

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  • Veit Elser

    (Cornell University)

Abstract

The decomposition of a matrix, as a product of factors with particular properties, is a much used tool in numerical analysis. Here we develop methods for decomposing a matrix C into a product XY, where the factors X and Y are required to minimize their distance from an arbitrary pair $$X_0$$ X 0 and $$Y_0$$ Y 0 . This type of decomposition, a projection to a matrix product constraint, in combination with projections that impose structural properties on X and Y, forms the basis of a general method of decomposing a matrix into factors with specified properties. Results are presented for the application of these methods to a number of hard problems in exact factorization.

Suggested Citation

  • Veit Elser, 2017. "Matrix product constraints by projection methods," Journal of Global Optimization, Springer, vol. 68(2), pages 329-355, June.
    • Handle: RePEc:spr:jglopt:v:68:y:2017:i:2:d:10.1007_s10898-016-0466-9
    DOI: 10.1007/s10898-016-0466-9
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    References listed on IDEAS

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    1. Gillis, Nicolas & Glineur, François & Tuyttens, Daniel & Vandaele, Arnaud, 2015. "Heuristics for exact nonnegative matrix factorization," LIDAM Discussion Papers CORE 2015006, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. GILLIS, Nicolas & GLINEUR, François, 2010. "On the geometric interpretation of the nonnegative rank," LIDAM Discussion Papers CORE 2010051, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Francisco J. Aragón Artacho & Jonathan M. Borwein & Matthew K. Tam, 2016. "Global behavior of the Douglas–Rachford method for a nonconvex feasibility problem," Journal of Global Optimization, Springer, vol. 65(2), pages 309-327, June.
    4. William P. Orrick, 2005. "The maximal {-1,1}-determinant of order 15," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 62(2), pages 195-219, November.
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    Cited by:

    1. Badenbroek, Riley & de Klerk, Etienne, 2020. "An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix," Other publications TiSEM 876ff1ab-036c-4635-9688-1, Tilburg University, School of Economics and Management.
    2. Riley Badenbroek & Etienne de Klerk, 2022. "An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 1115-1125, March.
    3. Heinz H. Bauschke & Manish Krishan Lal & Xianfu Wang, 2023. "Projections onto hyperbolas or bilinear constraint sets in Hilbert spaces," Journal of Global Optimization, Springer, vol. 86(1), pages 25-36, May.
    4. Badenbroek, Riley & de Klerk, Etienne, 2022. "An analytic center cutting plane method to determine complete positivity of a matrix," Other publications TiSEM 088da653-b943-4ed0-9720-6, Tilburg University, School of Economics and Management.

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