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Optimality Conditions for Problems over Symmetric Cones and a Simple Augmented Lagrangian Method

Author

Listed:
  • Bruno F. Lourenço

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

  • Ellen H. Fukuda

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

  • Masao Fukushima

    (Department of Systems and Mathematical Science, Faculty of Science and Engineering, Nanzan University, Nagoya, Aichi 466-8673, Japan)

Abstract

In this work, we are interested in nonlinear symmetric cone problems (NSCPs), which contain as special cases nonlinear semidefinite programming, nonlinear second-order cone programming, and the classical nonlinear programming problems. We explore the possibility of reformulating NSCPs as common nonlinear programs (NLPs), with the aid of squared slack variables. Through this connection, we show how to obtain second-order optimality conditions for NSCPs in an easy manner, thus bypassing a number of difficulties associated to the usual variational analytical approach. We then discuss several aspects of this connection. In particular, we show a “sharp” criterion for membership in a symmetric cone that also encodes rank information. Also, we discuss the possibility of importing convergence results from nonlinear programming to NSCPs, which we illustrate by discussing a simple augmented Lagrangian method for nonlinear symmetric cones. We show that, employing the slack variable approach, we can use the results for NLPs to prove convergence results, thus extending a special case (i.e., the case with strict complementarity) of an earlier result by Sun et al. [Sun D, Sun J, Zhang L (2008) The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Math. Programming 114(2):349–391] for nonlinear semidefinite programs.

Suggested Citation

  • Bruno F. Lourenço & Ellen H. Fukuda & Masao Fukushima, 2018. "Optimality Conditions for Problems over Symmetric Cones and a Simple Augmented Lagrangian Method," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1233-1251, November.
    • Handle: RePEc:inm:ormoor:v:43:y:2018:i:4:p:1233-1251
    DOI: 10.1287/moor.2017.0901
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    References listed on IDEAS

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    1. Ellen H. Fukuda & Masao Fukushima, 2016. "The Use of Squared Slack Variables in Nonlinear Second-Order Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 394-418, August.
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    Cited by:

    1. Roberto Andreani & Ellen H. Fukuda & Gabriel Haeser & Daiana O. Santos & Leonardo D. Secchin, 2024. "Optimality Conditions for Nonlinear Second-Order Cone Programming and Symmetric Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 1-33, January.
    2. R. Andreani & E. H. Fukuda & G. Haeser & D. O. Santos & L. D. Secchin, 2021. "On the use of Jordan Algebras for improving global convergence of an Augmented Lagrangian method in nonlinear semidefinite programming," Computational Optimization and Applications, Springer, vol. 79(3), pages 633-648, July.

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