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Two Relaxation Methods for Rank Minimization Problems

Author

Listed:
  • April Sagan

    (Rensselaer Polytechnic Institute)

  • Xin Shen

    (Yelp Inc.)

  • John E. Mitchell

    (Rensselaer Polytechnic Institute)

Abstract

The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be lifted to give an equivalent semidefinite program with complementarity constraints. The formulation requires two positive semidefinite matrices to be complementary. This is a continuous and nonconvex reformulation of the rank minimization problem. We develop two relaxations and show that constraint qualification holds at any stationary point of either relaxation of the rank minimization problem, and we explore the structure of the local minimizers.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:186:y:2020:i:3:d:10.1007_s10957-020-01731-9
    DOI: 10.1007/s10957-020-01731-9
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    References listed on IDEAS

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    1. Yi Zhang & Jia Wu & Liwei Zhang, 2015. "First order necessary optimality conditions for mathematical programs with second-order cone complementarity constraints," Journal of Global Optimization, Springer, vol. 63(2), pages 253-279, October.
    2. 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.
    3. Yulan Liu & Shujun Bi & Shaohua Pan, 2018. "Equivalent Lipschitz surrogates for zero-norm and rank optimization problems," Journal of Global Optimization, Springer, vol. 72(4), pages 679-704, December.
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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.

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