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Exact augmented Lagrangian functions for nonlinear semidefinite programming

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Abstract

In this paper, we study augmented Lagrangian functions for nonlinear semidefinite programming (NSDP) problems with exactness properties. The term exact is used in the sense that the penalty parameter can be taken appropriately, so a single minimization of the augmented Lagrangian recovers a solution of the original problem. This leads to reformulations of NSDP problems into unconstrained nonlinear programming ones. Here, we first establish a unified framework for constructing these exact functions, generalizing Di Pillo and Lucidi’s work from 1996, that was aimed at solving nonlinear programming problems. Then, through our framework, we propose a practical augmented Lagrangian function for NSDP, proving that it is continuously differentiable and exact under the so-called nondegeneracy condition. We also present some preliminary numerical experiments.

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Notes

  1. Namely, that \(\inf _{\Vert x\Vert = 1} \Vert Ax - Bx\Vert = \sigma _{\min }(A-B) \ge \sigma _{\min }(A) - \sigma _{\max }(B)\).

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Acknowledgements

We would like to thank the anonymous referees for their suggestions which improved the original version of the paper. We are also thankful to Akiko Kobayashi for valuable discussions about exact augmented Lagrangian functions.

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Authors and Affiliations

  1. Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan

    Ellen H. Fukuda

  2. Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo, 113-8656, Japan

    Bruno F. Lourenço

Authors
  1. Ellen H. Fukuda
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  2. Bruno F. Lourenço
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Correspondence to Ellen H. Fukuda.

Additional information

This work was supported by the Grant-in-Aid for Young Scientists (B) (26730012) and for Scientific Research (B) (15H02968) from Japan Society for the Promotion of Science.

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Fukuda, E.H., Lourenço, B.F. Exact augmented Lagrangian functions for nonlinear semidefinite programming. Comput Optim Appl 71, 457–482 (2018). https://doi.org/10.1007/s10589-018-0017-z

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