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Exact augmented Lagrangian functions for nonlinear semidefinite programming

Author

Listed:
  • Ellen H. Fukuda

    (Kyoto University)

  • Bruno F. Lourenço

    (University of Tokyo)

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.

Suggested Citation

  • Ellen H. Fukuda & Bruno F. Lourenço, 2018. "Exact augmented Lagrangian functions for nonlinear semidefinite programming," Computational Optimization and Applications, Springer, vol. 71(2), pages 457-482, November.
    • Handle: RePEc:spr:coopap:v:71:y:2018:i:2:d:10.1007_s10589-018-0017-z
    DOI: 10.1007/s10589-018-0017-z
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    References listed on IDEAS

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    1. Roberto Andreani & Ellen H. Fukuda & Paulo J. S. Silva, 2013. "A Gauss–Newton Approach for Solving Constrained Optimization Problems Using Differentiable Exact Penalties," Journal of Optimization Theory and Applications, Springer, vol. 156(2), pages 417-449, February.
    2. Florian Jarre, 2012. "Elementary Optimality Conditions for Nonlinear SDPs," 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 455-470, Springer.
    3. Thiago André & Paulo Silva, 2010. "Exact penalties for variational inequalities with applications to nonlinear complementarity problems," Computational Optimization and Applications, Springer, vol. 47(3), pages 401-429, November.
    4. Hiroshi Konno & Naoya Kawadai & Dai Wu, 2003. "Estimation of failure probability using semi-definite logit model," Computational Management Science, Springer, vol. 1(1), pages 59-73, December.
    5. Alexander Shapiro & Jie Sun, 2004. "Some Properties of the Augmented Lagrangian in Cone Constrained Optimization," Mathematics of Operations Research, INFORMS, vol. 29(3), pages 479-491, August.
    Full references (including those not matched with items on IDEAS)

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