Abstract
The aim of this paper is to present some results for the augmented Lagrangian function in the context of constrained global optimization by means of the image space analysis. It is first shown that a saddle point condition for the augmented Lagrangian function is equivalent to the existence of a regular nonlinear separation in the image space. Local and global sufficient optimality conditions for the exact augmented Lagrangian function are then investigated by means of second-order analysis in the image space. Local optimality result for this function is established under second-order sufficiency conditions in the image space. Global optimality result is further obtained under additional assumptions. Finally, it is proved that the exact augmented Lagrangian method converges to a global solution–Lagrange multiplier pair of the original problem under mild conditions.
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Acknowledgements
The authors are very grateful to the two anonymous referees for the constructive comments and suggestions, which have improved the presentation of the paper. This work was supported by the National Natural Science Foundation of China under grant 11071219, the Zhejiang Provincial Natural Science Foundation of China under grants LY13A010012 and LY13A010017, and the Postdoctoral Key Research Foundation of China under grant 201003242.
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Luo, H., Wu, H. & Liu, J. Some Results on Augmented Lagrangians in Constrained Global Optimization via Image Space Analysis. J Optim Theory Appl 159, 360–385 (2013). https://doi.org/10.1007/s10957-013-0358-9
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DOI: https://doi.org/10.1007/s10957-013-0358-9