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Projection algorithms for nonconvex minimization with application to sparse principal component analysis

Author

Listed:
  • William W. Hager

    (University of Florida)

  • Dzung T. Phan

    (IBM T. J. Watson Research Center)

  • Jiajie Zhu

    (University of Florida
    Boston College)

Abstract

We consider concave minimization problems over nonconvex sets. Optimization problems with this structure arise in sparse principal component analysis. We analyze both a gradient projection algorithm and an approximate Newton algorithm where the Hessian approximation is a multiple of the identity. Convergence results are established. In numerical experiments arising in sparse principal component analysis, it is seen that the performance of the gradient projection algorithm is very similar to that of the truncated power method and the generalized power method. In some cases, the approximate Newton algorithm with a Barzilai–Borwein Hessian approximation and a nonmonotone line search can be substantially faster than the other algorithms, and can converge to a better solution.

Suggested Citation

  • William W. Hager & Dzung T. Phan & Jiajie Zhu, 2016. "Projection algorithms for nonconvex minimization with application to sparse principal component analysis," Journal of Global Optimization, Springer, vol. 65(4), pages 657-676, August.
    • Handle: RePEc:spr:jglopt:v:65:y:2016:i:4:d:10.1007_s10898-016-0402-z
    DOI: 10.1007/s10898-016-0402-z
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    References listed on IDEAS

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    1. Akiko Takeda & Mahesan Niranjan & Jun-ya Gotoh & Yoshinobu Kawahara, 2013. "Simultaneous pursuit of out-of-sample performance and sparsity in index tracking portfolios," Computational Management Science, Springer, vol. 10(1), pages 21-49, February.
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    Cited by:

    1. Dzung T. Phan & Matt Menickelly, 2021. "On the Solution of ℓ 0 -Constrained Sparse Inverse Covariance Estimation Problems," INFORMS Journal on Computing, INFORMS, vol. 33(2), pages 531-550, May.

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