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Lower-Order Penalization Approach to Nonlinear Semidefinite Programming

Author

Listed:
  • X. X. Huang

    (Chongqing Normal University, Chongqing, China and School of Management, Fudan University)

  • X. Q. Yang

    (Hong Kong Polytechnic University)

  • K. L. Teo

    (Curtin University of Technology)

Abstract

In this paper, we reformulate a nonlinear semidefinite programming problem into an optimization problem with a matrix equality constraint. We apply a lower-order penalization approach to the reformulated problem. Necessary and sufficient conditions that guarantee the global (local) exactness of the lower-order penalty functions are derived. Convergence results of the optimal values and optimal solutions of the penalty problems to those of the original semidefinite program are established. Since the penalty functions may not be smooth or even locally Lipschitz, we invoke the Ekeland variational principle to derive necessary optimality conditions for the penalty problems. Under certain conditions, we show that any limit point of a sequence of stationary points of the penalty problems is a KKT stationary point of the original semidefinite program.

Suggested Citation

  • X. X. Huang & X. Q. Yang & K. L. Teo, 2007. "Lower-Order Penalization Approach to Nonlinear Semidefinite Programming," Journal of Optimization Theory and Applications, Springer, vol. 132(1), pages 1-20, January.
    • Handle: RePEc:spr:joptap:v:132:y:2007:i:1:d:10.1007_s10957-006-9055-2
    DOI: 10.1007/s10957-006-9055-2
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    References listed on IDEAS

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    1. X. X. Huang & X. Q. Yang, 2003. "A Unified Augmented Lagrangian Approach to Duality and Exact Penalization," Mathematics of Operations Research, INFORMS, vol. 28(3), pages 533-552, August.
    2. NESTEROV, Yu. & WOLKOWICZ, Henry & YE, Yinyu, 2000. "Semidefinite programming relaxations of nonconvex quadratic optimization," LIDAM Reprints CORE 1471, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. J. Frédéric Bonnans & Roberto Cominetti & Alexander Shapiro, 1998. "Sensitivity Analysis of Optimization Problems Under Second Order Regular Constraints," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 806-831, November.
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    Cited by:

    1. X. X. Huang, 2012. "Calmness and Exact Penalization in Constrained Scalar Set-Valued Optimization," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 108-119, July.
    2. Huixian Wu & Hezhi Luo & Xiaodong Ding & Guanting Chen, 2013. "Global convergence of modified augmented Lagrangian methods for nonlinear semidefinite programming," Computational Optimization and Applications, Springer, vol. 56(3), pages 531-558, December.

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