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A penalty method for rank minimization problems in symmetric matrices

Author

Listed:
  • Xin Shen

    (Rensselaer Polytechnic Institute)

  • John E. Mitchell

    (Rensselaer Polytechnic Institute)

Abstract

The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be cast equivalently as a semidefinite program with complementarity constraints (SDCMPCC). The formulation requires two positive semidefinite matrices to be complementary. This is a continuous and nonconvex reformulation of the rank minimization problem. We investigate calmness of locally optimal solutions to the SDCMPCC formulation and hence show that any locally optimal solution is a KKT point. We develop a penalty formulation of the problem. We present calmness results for locally optimal solutions to the penalty formulation. We also develop a proximal alternating linearized minimization (PALM) scheme for the penalty formulation, and investigate the incorporation of a momentum term into the algorithm. Computational results are presented.

Suggested Citation

  • Xin Shen & John E. Mitchell, 2018. "A penalty method for rank minimization problems in symmetric matrices," Computational Optimization and Applications, Springer, vol. 71(2), pages 353-380, November.
    • Handle: RePEc:spr:coopap:v:71:y:2018:i:2:d:10.1007_s10589-018-0010-6
    DOI: 10.1007/s10589-018-0010-6
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    References listed on IDEAS

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    1. J. Zhai & X. X. Huang, 2014. "Calmness and Exact Penalization in Vector Optimization under Nonlinear Perturbations," Journal of Optimization Theory and Applications, Springer, vol. 162(3), pages 856-872, September.
    2. Nathan Krislock & Henry Wolkowicz, 2012. "Euclidean Distance Matrices and Applications," International Series in Operations Research & Management Science, in: Miguel F. Anjos & Jean B. Lasserre (ed.), Handbook on Semidefinite, Conic and Polynomial Optimization, chapter 0, pages 879-914, Springer.
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    Cited by:

    1. April Sagan & John E. Mitchell, 2021. "Low-rank factorization for rank minimization with nonconvex regularizers," Computational Optimization and Applications, Springer, vol. 79(2), pages 273-300, June.
    2. April Sagan & Xin Shen & John E. Mitchell, 2020. "Two Relaxation Methods for Rank Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 806-825, September.

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