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Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming

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Abstract

This paper aims at showing that the class of augmented Lagrangian functions for nonlinear semidefinite programming problems can be derived, as a particular case, from a nonlinear separation scheme in the image space associated with the given problem. By means of the image space analysis, a global saddle point condition for the augmented Lagrangian function is investigated. It is shown that the existence of a saddle point is equivalent to a regular nonlinear separation of two suitable subsets of the image space. Without requiring the strict complementarity, it is proved that, under second order sufficiency conditions, the augmented Lagrangian function admits a local saddle point. The existence of global saddle points is then obtained under additional assumptions that do not require the compactness of the feasible set. Motivated by the result on global saddle points, we propose two modified primal-dual methods based on the augmented Lagrangian using different strategies and prove their convergence to a global solution and the optimal value of the original problem without requiring the boundedness condition of the multiplier sequence.

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Acknowledgments

The authors would like to thank the two anonymous referees for the detailed comments and valuable suggestions which have improved the final presentation of the paper.

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Correspondence to H. Z. Luo.

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This work was supported by the National Natural Science Foundation of China under grant 11071219, the Postdoctoral Key Research Foundation of China under grant 201003242, and the Zhejiang Provincial Natural Science Foundation of China under grants LY13A010012 and LY13A010017.

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Wu, H.X., Luo, H.Z. & Yang, J.F. Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming. J Glob Optim 59, 695–727 (2014). https://doi.org/10.1007/s10898-013-0093-7

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