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The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming

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Abstract

We analyze the rate of local convergence of the augmented Lagrangian method in nonlinear semidefinite optimization. The presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and variational analysis on the projection operator in the symmetric matrix space. Without requiring strict complementarity, we prove that, under the constraint nondegeneracy condition and the strong second order sufficient condition, the rate of convergence is linear and the ratio constant is proportional to 1/c, where c is the penalty parameter that exceeds a threshold \(\overline{c} > 0\) .

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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, Singapore, Singapore

    Defeng Sun

  2. Department of Decision Sciences, National University of Singapore, Singapore, Singapore

    Jie Sun

  3. Department of Applied Mathematics, Dalian University of Technology, Dalian, 116024, China

    Liwei Zhang

Authors
  1. Defeng Sun
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  2. Jie Sun
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  3. Liwei Zhang
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Correspondence to Defeng Sun.

Additional information

The research of Defeng Sun is partly supported by the Academic Research Fund from the National University of Singapore. The research of Jie Sun and Liwei Zhang is partly supported by Singapore–MIT Alliance and by Grants RP314000-042/057-112 of the National University of Singapore. The research of Liwei Zhang is also supported by the National Natural Science Foundation of China under project grant no. 10471015 and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, China.

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Sun, D., Sun, J. & Zhang, L. The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Math. Program. 114, 349–391 (2008). https://doi.org/10.1007/s10107-007-0105-9

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  • DOI: https://doi.org/10.1007/s10107-007-0105-9

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